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Copyright©2017byLeonardSusskindandArtFriedman

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Contents

CoverTitlePageCopyrightDedicationPrefaceIntroduction

Lecture1:TheLorentzTransformationLecture2:Velocitiesand4-VectorsLecture3:RelativisticLawsofMotionLecture4:ClassicalFieldTheoryLecture5:ParticlesandFieldsInterlude:CrazyUnitsLecture6:TheLorentzForceLawLecture7:FundamentalPrinciplesandGaugeInvarianceLecture8:Maxwell’sEquationsLecture9:PhysicalConsequencesofMaxwell’sEquationsLecture10:MaxwellFromLagrangeLecture11:FieldsandClassicalMechanics

AbouttheAuthorAlsobytheAuthorsAppendixA:MagneticMonopoles:LennyFoolsArtAppendixB:Reviewof3-VectorOperatorsIndex

Tomyfatherandmyhero,amanofcourage,BenjaminSusskind

—LS

Tomywife,Maggie,andherparents,DavidandBarbaraSloan

—AF

Thisbookis thethirdvolumeof theTheoreticalMinimumseries.Thefirstvolume,TheTheoreticalMinimum:WhatYouNeedtoKnowtoStartDoingPhysics,coveredclassicalmechanics,which is thecoreofanyphysicseducation.Wewill refer to it from time totimesimplyasVolumeI.Thesecondbook(VolumeII)explainsquantummechanicsandits relationship to classical mechanics. This third volume covers special relativity andclassicalfieldtheory.

The books in this series run parallel to Leonard Susskind’s videos, available on theWebthroughStanfordUniversity(seewww.theoreticalminimum.comforalisting).Whilecovering the samegeneral topicsas thevideos, thebookscontainadditionaldetails andtopicsthatdon’tappearinthevideos.

Preface

ThisbookisoneofseveralthatcloselyfollowmyInternetcourseseries,TheTheoreticalMinimum.Mycoauthor,ArtFriedman,wasastudentinthesecourses.Thebookbenefitedfrom the fact thatArtwas learning the subject andwas therefore sensitive to the issuesthatmightbeconfusingtothebeginner.Duringthecourseofwriting,wehadalotoffun,andwe’ve tried to convey some of that spirit with a bit of humor. If you don’t get it,ignoreit.

The two previous books in this series cover classicalmechanics and basic quantummechanics. So far, we have not studied light, and that’s because light is a relativisticphenomenon—aphenomenonthathastodowiththespecialtheoryofrelativity,orSRaswe’ll sometimes call it. That’s our goal for this book: SR and classical field theory.Classicalfieldtheorymeanselectromagnetictheory—waves,forcesonchargedparticles,andsoon—inthecontextofSR.Specialrelativityiswherewe’llbegin.

LeonardSusskind

Myparents,thechildrenofimmigrants,werebilingual.TheytaughtuskidssomeYiddishwordsandphrasesbutmainlyreservedthat languagefor themselves,oftentosaythingstheydidnotwantustounderstand.Manyoftheirsecretconversationswereaccompaniedbyloudpealsoflaughter.

Yiddishisanexpressivelanguage;it’swellsuitedtogreatliteratureaswellastodailylifeanddown-to-earthhumor.Itbothersmethatmyowncomprehensionissolimited.I’dlovetoreadallthegreatworksintheoriginal,butfranklyI’dbehappyenoughjusttogetthejokes.

Alotofushavesimilarfeelingsaboutmathematicalphysics.Wewant tounderstandthegreatideasandproblemsandengageourowncreativity.Weknowthere’spoetrytoberead and written, and we’re eager to participate in some fashion. All we lack is that“secret”language.Inthisseries,ourgoalistoteachyouthelanguageofphysicsandshowyousomeofthegreatideasintheirnativehabitat.

Ifyou joinus,you’llbeable towrapyourheadaroundagoodportionof twentieth-centuryphysics.You’llcertainlybeequippedtounderstandmuchofEinstein’searlywork.Ataminimum,you’ll“getthejokes”andtheseriousideasthatunderliethem.Togetyoustarted,we’vethrowninafewjokesofourown,includingsomerealgroaners.

I’mdelighted to acknowledge everyonewhohelped and supported us along theway. Itmaybeaclichétosay“wecouldn’thavedoneitwithoutyou,”butitalsohappenstobe

true.

Working with the professionals at Brockman, Inc., and Basic Books is always apleasureaswellasalearningexperience.JohnBrockman,MaxBrockman,andMichaelHealeyplayedacritical role in transformingour idea intoarealproject.Fromthere,TJKelleher,HélèneBarthélemy,CarrieNapolitano,andMelissaVeronesiwalkedusthroughtheeditorialandproductionprocesswithgreatskillandunderstanding.LauraStickneyofPenguinBookscoordinatedthepublicationoftheUKeditionsosmoothly,wehardlysawithappening.CopyeditorAmyJ.Schneidermadesubstantialimprovementstoourinitialmanuscript,asdidproofreadersLorGehretandBenTedoff.

AnumberofLeonard’sformerstudentsgenerouslyofferedtoreviewthemanuscript.Thiswasnosmalltask.Theirinsightsandsuggestionswereinvaluable,andthebookisfarbetterasaresult.OursincerethanksgotoJeremyBranscome,ByronDom,JeffJustice,ClintonLewis,JohanShamrilSosa,andDawnMarciaWilson.

Asalways,thewarmthandsupportI’vereceivedfromfamilyandfriendshasseenmethrough this project. My wife, Maggie, spent hours creating and re-creating the twoHermann’sHideawaydrawings,gettingthemdoneontimewhiledealingwiththeillnessandpassingofhermother.

Thisprojecthasaffordedmetheluxuryofpursuingtwoofmylifepassionsatthesametime:graduatelevelphysicsandfourth-gradehumor.Inthisrespect,LeonardandIareaperfectteam,andcollaboratingwithhimisanunmitigatedpleasure.

ArtFriedman

Introduction

DearreadersandstudentsofTheTheoreticalMinimum,

Hellothere,andwelcomebacktoLenny&Art’sExcellentAdventure.Welastlefttheintrepid pair recovering from a wild rollicking roller coaster ride through the quantumworldofentanglementanduncertainty.Theywerereadyforsomethingsedate,somethingreliableanddeterministic,somethingclassical.But theridecontinues inVolumeIII,andit’s no less wild. Contracting rods, time dilation, twin paradoxes, relative simultaneity,stretchlimousinesthatdoanddon’tfitintoVolkswagen-sizegarages.LennyandArtarehardlyfinishedwiththeirmadcapadventure.AndattheendoftherideLennytricksArtwithafakemonopole.

Well,maybe that is abitoverwrought,but to thebeginner the relativisticworld is astrangeandwondrous funhouse, full ofdangerouspuzzles and slipperyparadoxes.Butwe’ll be there to hold your handwhen the going gets tough. Some basic grounding incalculusandlinearalgebrashouldbegoodenoughtogetyouthrough.

Ourgoalasalwaysistoexplainthingsinacompletelyseriousway,withoutdumbingthemdownatall,butalsowithoutexplainingmorethanisnecessarytogotothenextstep.Depending on your preference, that could be either quantum field theory or generalrelativity.

It’sbeenawhilesinceArtandIpublishedVolumeIIonquantummechanics.We’vebeen tremendouslygratifiedby the thousandsofe-mailsexpressingappreciation forourefforts,thusfar,todistillthemostimportanttheoreticalprinciplesofphysicsintoTTM.

Thefirstvolumeonclassicalmechanicswasmostlyaboutthegeneralframeworkforclassical physics that was set up in the nineteenth century by Lagrange, Hamilton,Poisson,andothergreats.Thatframeworkhaslasted,andprovidestheunderpinningforallmodernphysics,evenasitgrewintoquantummechanics.

Quantummechanics percolated into physics starting from the year 1900,whenMaxPlanckdiscoveredthelimitsofclassicalphysics,until1926whenPaulDiracsynthesizedtheideasofPlanck,Einstein,Bohr,deBroglie,Schrödinger,Heisenberg,andBornintoaconsistentmathematical theory. That great synthesis (which, by theway,was based onHamilton’s and Poisson’s framework for classical mechanics) was the subject of TTMVolumeII.

InVolumeIIIwetakeahistoricalstepbacktothenineteenthcenturytotheoriginsofmodernfieldtheory.I’mnotahistorian,butIthinkIamaccurateintracingtheideaofa

field to Michael Faraday. Faraday’s mathematics was rudimentary, but his powers ofvisualizationwereextraordinaryandledhimtotheconceptsofelectromagneticfield,linesofforce,andelectromagneticinduction.InhisintuitivewayheunderstoodmostofwhatMaxwell later combined into his unified equations of electromagnetism. Faraday waslackingoneelement,namelythatachangingelectricfieldleadstoeffectssimilartothoseofanelectriccurrent.

ItwasMaxwellwholaterdiscoveredthisso-calleddisplacementcurrent,sometimeinthe early 1860s, and thenwent on to construct the first true field theory: the theory ofelectromagnetismandelectromagneticradiation.ButMaxwell’stheorywasnotwithoutitsowntroublingconfusions.

TheproblemwithMaxwell’s theorywas that itdidnot seem tobeconsistentwithabasic principle, attributed to Galileo and clearly spelled out by Newton: All motion isrelative.No(inertial)frameofreferenceismoreentitledtobethoughtofasatrest thananyotherframe.However thisprinciplewasatoddswithelectromagnetic theory,whichpredicted that lightmoveswith adistinctvelocityc = 3×108meters per second.Howcoulditbepossibleforlighttohavethesamevelocityineveryframeofreference?Howcoulditbethat light travelswiththesamevelocityintherestframeofthetrainstation,andalsointheframeofthespeedingtrain?

Maxwellandothersknewabouttheclash,andresolveditthesimplestwaytheyknewhow:bytheexpedientoftossingoutGalileo’sprincipleofrelativemotion.Theypicturedtheworld as being filledwith a peculiar substance—the ether—which, like an ordinarymaterial, would have a rest frame in which it was not moving. That’s the only frame,accordingtotheetherists,inwhichMaxwell’sequationswerecorrect.Inanyotherframe,movingwithrespecttotheether,theequationshadtobeadjusted.

Thiswas the statusuntil1887whenAlbertMichelsonandEdwardMorleydid theirfamousexperiment,attemptingtomeasurethesmallchangesinthemotionoflightduetothemotionofEarththroughtheether.Nodoubtmostreadersknowwhathappened;theyfailed to find any. People tried to explain away Michelson and Morley’s result. Thesimplest ideawas called ether drag, the idea being that the ether is dragged alongwithEarthsothattheMichelson-Morleyexperimentwasreallyatrestwithrespecttotheether.Butnomatterhowyoutriedtorescueit,theethertheorywasuglyandungainly.

According tohisown testimony,Einsteindidnotknowabout theMichelson-Morleyexperiment when in 1895 (at age sixteen), he began to think about the clash betweenelectromagnetism and the relativity ofmotion. He simply felt intuitively that the clashsomehow was not real. He based his thinking on two postulates that together seemedirreconcilable:

1.Thelawsofnaturearethesameinallframesofreference.Thustherecanbenopreferredether-frame.

2.Itisalawofnaturethatlightmoveswithvelocityc.

As uncomfortable as it probably seemed, the two principles together implied that lightmustmovewiththesamevelocityinallframes.

Ittookalmosttenyears,butby1905Einsteinhadreconciledtheprinciplesintowhathecalledthespecialtheoryofrelativity.Itisinterestingthatthetitleofthe1905paperdidnotcontainthewordrelativityatall;itwas“OntheElectrodynamicsofMovingBodies.”Gonefromphysicswastheevermorecomplicatedether;initsplacewasanewtheoryofspaceand time.However, to thisdayyouwill still finda residueof theether theory intextbooks, where you will find the symbol 0, the so-called dielectric constant of thevacuum,asifthevacuumwereasubstancewithmaterialproperties.Studentsnewtothesubjectoftenencounteragreatdealofconfusionoriginatingfromconventionsandjargonthattracebacktotheethertheory.IfI’vedonenothingelseintheselectures,Itriedtogetridoftheseconfusions.

AsintheotherbooksinTTMI’vekeptthematerialtotheminimumneededtomovetothe next step—depending on your preference, either quantum field theory or generalrelativity.

You’ve heard this before: Classical mechanics is intuitive; things move in predictableways.Anexperiencedballplayercantakeaquicklookataflyballandfromitslocationanditsvelocityknowwheretoruninordertobetherejust intimetocatchtheball.Ofcourse, a sudden unexpected gust of wind might fool him, but that’s only because hedidn’t take into account all the variables. There is an obvious reason why classicalmechanicsisintuitive:Humans,andanimalsbeforethem,havebeenusingitmanytimeseverydayforsurvival.

Inourquantummechanicsbook,weexplainedingreatdetailwhylearningthatsubjectrequiresustoforgetourphysicalintuitionandreplaceitwithsomethingentirelydifferent.Wehadtolearnnewmathematicalabstractionsandanewwayofconnectingthemtothephysicalworld.Butwhataboutspecialrelativity?WhilequantummechanicsexplorestheworldoftheVERYSMALL,special relativity takesus into therealmof theVERYFAST,andyes,italsoforcesustobendourintuition.Buthere’sthegoodnews:Themathematicsofspecial relativity is far less abstract, andwe don’t need brain surgery to connect thoseabstractionstothephysicalworld.SRdoesstretchourintuition,butthestretchisfarmoregentle.Infact,SRisgenerallyregardedasabranchofclassicalphysics.

Special relativity requires us to rethink our notions of space, time, and especiallysimultaneity.Physicistsdidnotmaketheserevisionsfrivolously.Aswithanyconceptualleap, SRwas resisted bymany.You could say that some physicists had to be draggedkickingandscreamingtoanacceptanceofSR,andothersneveraccepteditatall.1Whydidmostofthemultimatelyrelent?Asidefromthemanyexperimentsthatconfirmedthepredictions made by SR, there was strong theoretical support. The classical theory ofelectromagnetism,perfectedbyMaxwellandothersduringthenineteenthcentury,quietlyproclaimedthat“thespeedoflightisthespeedoflight.”Inotherwords,thespeedoflightisthesameineveryinertial(nonaccelerating)referenceframe.Whilethisconclusionwasdisturbing, it could not just be ignored—the theory of electromagnetism is far toosuccessful to be brushed aside. In this book, we’ll explore SR’s deep connections toelectromagnetictheory,aswellasitsmanyinterestingpredictionsandparadoxes.

1NotablyAlbertMichelson,thefirstAmericantowinaNobelPrizeinphysics,andhiscollaboratorEdwardMorley.TheirprecisemeasurementsprovidedstrongconfirmationofSR.

Lecture1

TheLorentzTransformation

WeopenVolumeIIIwithArtandLennyrunningfortheirlives.

Art:Geez,Lenny,thankheavenswegotoutofHilbert’splacealive!Ithoughtwe’dnevergetdisentangled.Can’twefindamoreclassicalplacetohangout?

Lenny:Goodidea,Art.I’vehaditwithallthatuncertainty.Let’sheadovertoHermann’sHideawayandseewhat’sdefinitelyhappening.

Art:Where?WhoisthisguyHermann?

Lenny: Minkowski? Oh, you’ll love him. I guarantee there won’t be any bras inMinkowski’sspace.Noketseither.

LennyandArtsoonwindupatHermann’sHideaway,atavernthatcaterstoafast-pacedcrowd.

Art:WhydidHermannbuildhisHideawaywayouthereinthemiddleof—what?Acowpasture?Aricepaddy?

Lenny:Wejustcallitafield.Youcangrowjustaboutanythingyoulike;cows,rice,sourpickles,younameit.Hermann’sanoldfriend,andIrentthelandouttohimataverylowprice.

Art:Soyou’reagentlemanfarmer!Whoknew?Bytheway,howcomeeveryonehereissoskinny?Isthefoodthatbad?

Lenny: The food is great. They’re skinny because they’re moving so fast. Hermannprovidesfreejetpackstoplaywith.Quick!Lookout!Duck!DUCK!

Art:Goose!Let’stryoneout!Wecouldbothstandtogetabitthinner.

Morethananythingelse,thespecialtheoryofrelativityisatheoryaboutreferenceframes.If we say something about the physical world, does our statement remain true in adifferentreferenceframe?Isanobservationmadebyapersonstandingstillonthegroundequally valid for a person flying in a jet? Are there quantities or statements that areinvariant—that do not depend at all on the observer’s reference frame?The answers toquestions of this sort turn out to be interesting and surprising. In fact, they sparked arevolutioninphysicsintheearlyyearsofthetwentiethcentury.

1.1ReferenceFramesYoualreadyknowsomethingaboutreferenceframes.ItalkedabouttheminVolumeIonclassicalmechanics.Cartesian coordinates, for example, are familiar tomost people.ACartesianframehasasetofspatialcoordinatesx,y,andz,andanorigin.Ifyouwanttothinkconcretelyaboutwhatacoordinatesystemmeans,thinkofspaceasbeingfilledupwithalatticeofmeterstickssothateverypointinspacecanbespecifiedasbeingacertainnumberofmeterstotheleft,acertainnumberofmetersup,acertainnumberofmetersinorout,fromtheorigin.That’sacoordinatesystemforspace.Itallowsustospecifywhereaneventhappens.

In order to specify when something happens we also need a time coordinate. Areferenceframeisacoordinatesystemforbothspaceandtime.Itconsistsofanx-,y-,z-,andtaxis.Wecanextendournotionofconcretenessbyimaginingthatthere’saclockatevery point in space.We also imagine that we have made sure that all the clocks aresynchronized,meaningthattheyallreadt=0atthesameinstant,andthattheclocksallrunatthesamerate.Thusareferenceframe(orRFforsimplicity)isarealorimaginedlatticeofmeterstickstogetherwithasynchronizedsetofclocksateverypoint.

Therearemanywaystospecifypointsinspaceandtime,ofcourse,whichmeanswecanhavedifferentRFs.Wecantranslatetheoriginx=y=z=t=0tosomeotherpoint,so thata location inspaceand time ismeasuredrelative to theneworigin.Wecanalsorotatethecoordinatesfromoneorientationtoanother.Finally,weconsiderframesthataremovingrelativetosomeparticularframe.Wecanspeakofyourframeandmyframe,andherewecometoakeypoint:Besidesthecoordinateaxesandtheorigin,areferenceframemaybeassociatedwithanobserverwhocanuseallthoseclocksandmeterstickstomakemeasurements.

Let’sassumeyou’resittingstillat thecenterof thefrontrowinthelecturehall.Thelecturehall is filledwithmetersticks and clocks at rest in your frame.Every event thattakesplacein theroomisassignedapositionandatimebyyoursticksandclocks.I’malsointhelecturehall,butinsteadofstandingstillImovearound.Imightmarchpastyoumovingtotheleftortotheright,andasIdoIcarrymylatticeofclocksandmeterstickswithme.AteveryinstantI’matthecenterofmyownspacecoordinates,andyouareatthe centerofyours.Obviouslymycoordinates aredifferent fromyours.You specify aneventbyanx,y,z,andt,andIspecifythesameeventbyadifferentsetofcoordinatesinordertoaccountforthefactthatImaybemovingpastyou.InparticularifIammovingalongthexaxisrelative toyou,wewon’tagreeaboutourxcoordinates. I’llalwayssay

thattheendofmynoseisatx=5,meaningthatitisfiveinchesinfrontofthecenterofmyhead.However,youwillsaymynoseisnotatx=5;you’llsaymynoseismoving,andthatitspositionchangeswithtime.

Imightalsoscratchmynoseatt=2,bywhichImeanthattheclockattheendofmynose indicated 2 seconds into the lecture when my nose was scratched. Youmight betempted to think thatyourclockwouldalso read t=2at thepointwheremynosewasscratched.But that’s exactlywhere relativisticphysicsdeparts fromNewtonianphysics.The assumption that all clocks in all frames of reference can be synchronized seemsintuitivelyobvious,butitconflictswithEinstein’sassumptionofrelativemotionandtheuniversalityofthespeedoflight.

We’llsoonelaborateonhow,andtowhatextent,clocksatdifferentplacesindifferentreference frames can be synchronized, but for nowwe’ll just assume that at any giveninstantoftimeallofyourclocksagreewitheachother,andtheyagreewithmyclocks.InotherwordswetemporarilyfollowNewtonandassumethatthetimecoordinateisexactlythe same for you as it is for me, and there’s no ambiguity resulting from our relativemotion.

1.2InertialReferenceFramesThelawsofphysicswouldbeveryhardtodescribewithoutcoordinatestolabeltheeventsthat take place.Aswe’ve seen, there aremany sets of coordinates and thereforemanydescriptionsofthesameevents.Whatrelativitymeant,toGalileoandNewtonaswellasEinstein, is that the laws governing those events are the same in all inertial referenceframes.1Aninertialframeisoneinwhichaparticle,withnoexternalforcesactingonit,movesinastraightlinewithuniformvelocity.Itisobviousthatnotallframesareinertial.Suppose your frame is inertial so that a particle, thrown through the room,moveswithuniformvelocitywhenmeasuredbyyoursticksandclocks.IfIhappentobepacingbackandforth,theparticlewilllooktomelikeitaccelerateseverytimeIturnaround.ButifIwalkwith steadymotion along a straight line, I toowill see the particle with uniformvelocity.Whatwemaysay ingeneral is thatany twoframes thatareboth inertialmustmovewithuniformrelativemotionalongastraightline.

It’safeatureofNewtonianmechanicsthatthelawsofphysics,F=matogetherwithNewton’slawofgravitationalattraction,arethesameineveryIRF.Iliketodescribeitthisway:SupposethatIamanaccomplishedjuggler.Ihavelearnedsomerulesforsuccessfuljuggling,suchasthefollowing:IfIthrowaballverticallyupwarditwillfallbacktothesamepointwhereitstarted.Infact,Ilearnedmyruleswhilestandingontheplatformofarailwaystationwaitingforthetrain.

WhenthetrainstopsatthestationIjumponandimmediatelystarttojuggle.Butasthetrainpullsoutofthestation,theoldlawsdon’twork.Foratimetheballsseemtomoveinoddways, fallingwhere I don’t expect them.However, once the train gets goingwithuniformvelocity,thelawsstartworkingagain.IfI’minamovingIRFandeverythingissealedsothatIcan’tseeoutside,IcannottellthatI’mmoving.IfItrytofindoutbydoing

somejuggling,I’llfindoutthatmystandardlawsofjugglingwork.ImightassumethatI’mat rest,but that’snotcorrect; all I can really say is that I’m inan inertial referenceframe.

Theprinciple of relativity states that the lawsof physicsare the same in every IRF.ThatprinciplewasnotinventedbyEinstein;itexistedbeforehimandisusuallyattributedtoGalileo.Newtoncertainlywouldhaverecognizedit.WhatnewingredientdidEinsteinadd?Headdedonelawofphysics:thelawthatthespeedoflightisthespeedoflight,c.Inunitsofmetersper second, the speedof light is approximately3×108. Inmilesperseconditisabout186,000,andinlight-yearsperyearitisexactly1.Butoncetheunitsarechosen,Einstein’snewlawstatesthatthevelocityoflightisthesameforeveryobserver.

Whenyoucombine these two ideas—that the lawsofphysics are the same in everyIRF,andthatit’salawofphysicsthatlightmovesatafixedvelocity—youcometotheconclusionthatlightmustmovewiththesamevelocityinevery IRF.Thatconclusion istruly puzzling. It led some physicists to reject SR altogether. In the next section,we’llfollowEinstein’slogicandfindouttheramificationsofthisnewlaw.

1.2.1Newtonian(Pre-SR)FramesIn this section I will explain how Newton would have described the relation betweenreferenceframes,andtheconclusionshewouldhavemadeaboutthemotionoflightrays.Newton’sbasicpostulatewouldhavebeenthatthereexistsauniversaltime,thesameinallreferenceframes.

Let’s begin by ignoring they and z directions and focus entirely on the x direction.We’ll pretend that theworld is one-dimensional and that all observers are free tomovealong thexaxisbutare frozen in theother twodirectionsofspace.Fig.1.1 follows thestandardconventioninwhichthexaxispointstotheright,andthetaxispointsup.Theseaxesdescribe themetersticksandclocks inyour frame—the frameat rest in the lecturehall.(Iwillarbitrarilyrefertoyourframeastherestframeandmyframeas themovingframe.)We’llassumethatinyourframelightmoveswithitsstandardspeedc.Adiagramofthiskindiscalledaspacetimediagram.Youcanthinkofitasamapoftheworld,butamapthatshowsallpossibleplacesandallpossibletimes.Ifalightrayissentoutfromtheorigin,movingtowardtheright,itwillmovewithatrajectorygivenbytheequation

Figure1.1:NewtonianFrames.

x=ct.

Similarlyalightraymovingtotheleftwouldberepresentedby

x=−ct.

Anegativevelocityjustmeansmovingtotheleft.InthevariousfiguresthatfollowIwilldraw them as if my frame ismoving to the right (v positive). As an exercise you canredrawthemfornegativev.

InFig.1.1,thelightrayisshownasadashedline.Iftheunitsfortheaxesaremetersandseconds,thelightraywillappearalmosthorizontal;itwillmove3×108meterstotherightwhilemoving vertically by only 1 second! But the numerical value of c dependsentirelyontheunitswechoose.Therefore,itisconvenienttousesomeotherunitsforthespeedoflight—unitsinwhichweseemoreclearlythattheslopeofthelight-raytrajectoryisfinite.

Nowlet’saddmyframe,movingalongthexaxisrelativetoyourframe,withauniformvelocityv.2 The velocity could be either positive (inwhich case Iwould bemoving toyour right), negative (inwhichcase Iwouldbemoving toyour left), or zero (inwhichcaseweareatrestrelativetoeachother,andmytrajectorywouldbeaverticallineinthefigure).

I’ll call the coordinates inmy frame x′ and t′ instead of x and t. The fact that I ammoving relative to you with a constant velocity implies that my trajectory throughspacetimeisastraightline.Youmaydescribemymotionbytheequation

x=vt

or

x−vt=0,

wherevismyvelocityrelativetoyou,asshowninFig.1.1.HowdoIdescribemyownmotion?That’s easy; I am always at the origin ofmy own coordinate system. In otherwords, Idescribemyselfby theequationx′=0.The interestingquestion is,howdowetranslatefromoneframetotheother;inotherwords,what’stherelationshipbetweenyourcoordinatesandmine?AccordingtoNewton,thatrelationwouldbe

The first of these equations isNewton’s assumption of universal time, the same for allobservers.Thesecondjustshowsthatmycoordinatex′isdisplacedfromyourcoordinatebyourrelativevelocity times the time,measuredfromtheorigin.Fromthiswesee thattheequations

x−vt=0

and

x′=0

have the same meaning. Eqs. 1.1 and 1.2 comprise the Newtonian transformation ofcoordinatesbetweentwoinertialreferenceframes.Ifyouknowwhenandwhereaneventhappens in your coordinates, you can tell me in my coordinates when and where ithappens.Canweinverttherelationship?That’seasyandIwillleaveittoyou.Theresultis

Nowlet’slookatthelightrayinFig.1.1.Accordingtoassumption,itmovesalongthepathx=ctinyourframe.HowdoIdescribeitsmotioninmyframe?IsimplysubstitutethevaluesofxandtfromEqs.1.3and1.4intotheequationx=ct,toget

x′+vt′=ct′,

whichwerewriteintheform

x′=(c−v)t′.

Notsurprisingly,thisshowsthelightraymovingwithvelocity(c−v)inmyframe.ThatspellstroubleforEinstein’snewlaw—thelawthatalllightraysmovewiththesamespeedc in every IRF. If Einstein is correct, then something is seriously wrong. Einstein andNewtoncannotbothberight:Thespeedoflightcannotbeuniversalifthereisauniversaltimethatallobserversagreeon.

Beforemovingon,let’sjustseewhatwouldhappentoalightraymovingtotheleft.Inyourframesuchalightraywouldhavetheequation

x=−ct.

It’seasytoseethatinmyframetheNewtonianruleswouldgive

x′=−(c+v)t.

In other words, if I’mmoving to your right, a light ray moving in the same directiontravelsalittleslower(withspeedc−v),andalightraymovingintheoppositedirectiontravelsa little faster (withspeedc+v) relative tome.That’swhatNewtonandGalileowould have said.That’swhateveryonewould have said until the endof the nineteenthcenturywhenpeoplestartedmeasuringthespeedoflightwithgreatprecisionandfoundoutthatit’salwaysthesame,nomatterhowtheinertialobserversmove.

Theonlyway to reconcile thisconflict is to recognize that something iswrongwithNewton’stransformationlawbetweencoordinatesindifferentframes.3WeneedtofigureouthowtorepairEqs.1.1and1.2sothatthespeedoflightisthesameforbothofus.

1.2.2SRFramesBefore deriving our new transformation equations, let’s revisit one of Newton’s keyassumptions.Theassumptionthatismostatrisk,andinfacttheonethat’swrong,isthatsimultaneity means the same thing in every frame—that if we begin with our clockssynchronized, and then I start tomove,my clockswill remain synchronizedwith your

clocks.We’reabouttoseethattheequation

t′=t

isnot the correct relationship betweenmoving clocks and stationary clocks.Thewholeideaofsimultaneityisframe-dependent.

SynchronizingOurClocksHere’swhatIwantyoutoimagine.We’reinalecturehall.You,astudent,aresittinginthefrontrow,whichisfilledwitheagerattentivestudents,andeachstudentinthefrontrowhasaclock.Theclocksareallidenticalandcompletelyreliable.Youinspecttheseclockscarefully,andmakesuretheyallreadthesametimeandtickatthesamerate.Ihaveanequivalentcollectionofclocksinmyframethatarespreadoutrelativetomeinthesamewayasyourclocks.Eachofyourclockshasacounterpart inmysetup,andviceversa.I’vemadesurethatmyclocksaresynchronizedwitheachotherandalsowithyourclocks.ThenI,togetherwithallmyclocks,startmovingrelativetoyouandyourclocks.Aseachofmyclockspasseseachofyours,wecheckeachother’sclockstoseeiftheystillreadthesametimeandifnot,howfaroutofwhackeachclockiscomparedtoitscounterpart.Theanswermaydependoneachclock’spositionalongtheline.

Ofcourse,wecouldasksimilarquestionsaboutourmetersticks,suchas,“AsIpassyou,doesmymeterstickmeasure1inyourcoordinates?”ThisiswhereEinsteinmadehisgreatleap.Herealizedthatwehavetobemuchmorecarefulabouthowwedefinelengths,times,andsimultaneity.Weneed to thinkexperimentally abouthow to synchronize twoclocks.Buttheoneanchorthatheheldontoisthepostulatethatthespeedoflightisthesame in every IRF.For that, he had to give upNewton’s postulate of a universal time.Insteadhefoundthat“simultaneityisrelative.”Wewillfollowhislogic.

Whatexactlydowemeanwhenwesaythattwoclocks—let’scallthemAandB—aresynchronized?Ifthetwoclocksareatthesamelocation,movingwiththesamevelocity,itshouldbeeasytocomparethemandseeiftheyreadthesamevalueoftime.ButevenifAandBarestandingstill, say inyour frame,butarenotat thesameposition,checking iftheyaresynchronizedrequiressomethought.TheproblemisthatlighttakestimetotravelbetweenAandB.

Einstein’sstrategywastoimagineathirdclock,C,locatedmidwaybetweenAandB.4Tobespecific,let’simagineallthreeclocksbeinglocatedinthefrontrowofthelecturehall.ClockAisheldbythestudentattheleftendoftherow,clockBisheldbythestudentattherightend,andclockCisatthecenteroftherow.GreatcarehasbeentakentomakesurethatthedistancefromAtoCisthesameasthedistancefromBtoC.

AtexactlythetimewhentheAclockreadsnoonitactivatesaflashoflighttowardC.SimilarlywhenBreadsnoonitalsosendsaflashoflighttoC.Ofcourse,bothflasheswilltakesometimetoreachC,butsincethevelocityoflightisthesameforbothflashes,andthedistancetheyhavetotravelisthesame,theybothtakethesametimetogettoC.Whatwemeanby sayingAandB are synchronized is that the two flasheswill arriveatCatexactly the same time. Of course, if they don’t arrive simultaneously, student C will

concludethatAandBwerenotsynchronized.ShemaythensendamessagetoeitherAorBwithinstructionsforhowmuchtochangetheirsettingstogetsynchronized.

SupposeclocksAandBaresynchronizedinyourframe.Whathappensinmymovingframe?Let’s say I’mmoving to the right, and Ihappen to reach themidpointC just asthesetwoflashesareemitted.Butthelightdoesn’tgettoCatnoon;itgetsthereslightlylater.By that time, I’ve alreadymoved a little to the right of center. Since I’m right ofcenter, the light ray coming from the leftwill reachme a little later than the light raycomingfromtheright.Therefore,Iwillconcludethatyourclocksarenot synchronized,becausethetwolightflashesreachmeattwodifferenttimes.

Evidentlywhat you and I call synchronous—occurring at the same time—is not thesame.Twoevents that takeplaceat the same time inyour frame takeplaceatdifferenttimesinmyframe.Oratleastthat’swhatEinstein’stwopostulatesforceustoaccept.

UnitsandDimensions:AQuickDetourBeforemovingahead,weshouldpausebrieflytoexplainthatwe’llbeusingtwosystemsofunits.Eachsystemiswellsuitedtoitspurpose,andit’sfairlyeasytoswitchfromonesystemtotheother.

Thefirstsystemusesfamiliarunitssuchasmeters,seconds,andsoon.We’llcallthemcommonorconventionalunits.Theseunitsareexcellentfordescribingtheordinaryworldinwhichmostvelocitiesarefarsmaller thanthespeedof light.Avelocityof1 in thoseunitsmeans1meterpersecond,ordersofmagnitudelessthanc.

Thesecondsystemisbasedonthespeedof light. In thissystem,unitsof lengthandtimearedefinedinawaythatgivesthespeedoflightadimensionlessvalueof1.Wecallthemrelativisticunits.Relativisticunitsmakeiteasiertocarryoutderivationsandnoticethe symmetries in our equations. We’ve already seen that conventional units areimpracticalforspacetimediagrams.Relativisticunitsworkbeautifullyforthispurpose.

In relativistic units, not only does c have a value of 1, but all velocities aredimensionless. For this to work out, we have to choose appropriately defined units oflength and time—after all, velocity is a length divided by a time. If our time units areseconds,wechooselight-secondsasourlengthunits.Howbigisalight-second?Weknowthat it’s186,000miles, but forourpurposes that’sunimportant.Here’swhatmatters:Alight-secondisaunitoflength,andbydefinitionlighttravels1light-secondpersecond!Ineffect,we’remeasuringbothtimeandlengthinunitsofseconds.That’showvelocity—a lengthdividedby a time—gets tobedimensionless.Whenweuse relativisticunits, avelocity variable such as v is a dimensionless fraction of the speed of light. That’sconsistentwithcitselfhavingavalueof1.

In a spacetime diagram such as Fig. 1.2, the x and t axes are both calibrated inseconds.5Thetrajectoryofalightraymakesequalangleswiththexaxisandwiththetaxis.Conversely, any trajectory thatmakes equal angleswith the two axes represents alightray.InyourstationaryRF,thatangleis45degrees.

Knowinghowtoswitcheasilybetweenthetwotypesofunitswillpayoff.Theguidingprincipleisthatmathematicalexpressionsneedtobedimensionallyconsistentinwhateversystemofunitswe’reusingatthetime.Themostcommonandusefultrickingoingfromrelativistictoconventionalunitsistoreplacevwithv/c.Thereareotherpatternsaswell,whichtypicallyinvolvemultiplyingordividingbysomeappropriatepowerofc,thespeedoflight.We’llshowexamplesaswego,andyou’llfindthattheseconversionsarefairlysimple.

SettingUpOurCoordinates—Again!Let’sgoback toour twocoordinate systems.This time,we’llbeverycarefulabout theexactmeaningofthewordsynchronousinthemovingRF.InthestationaryRF,twopointsare synchronous (or simultaneous) if they’re both on the same horizontal level in aspacetimediagram.Thetwopointsbothhavethesametcoordinate,andalineconnectingthemisparalleltothexaxis.ThatmuchNewtonwouldhaveagreedwith.

Butwhataboutthemovingframe?We’llfindoutinaminutethatinthemovingframe,thepoint

x=0,t=0

isnotsynchronouswiththeotherpointsonthexaxis,butwithanentirelydifferentsetofpoints.Infact,thewholesurfacethatthemovingframecalls“synchronous”issomeplaceelse. How can we map out this surface? We’ll use the synchronization procedure,described inaprevioussubsection(SynchronizingOurClocks)andfurther illustrated inFig.1.2.

Drawing a spacetime diagram is usually the best way to understand a problem inrelativity.Thepictureisalwaysthesame;xisthehorizontalaxis,andtisvertical.Thesecoordinates represent a RF that you can think of as stationary. In other words, theyrepresentyourframe.Alinethatrepresentsthetrajectoryofanobservermovingthroughspacetimeiscalledaworldline.

Withouraxesinplace,thenextthingstodrawarethelightrays.InFig.1.2,thesearerepresentedbythelineslabeledx=ctandx=−ct.Thedashedlinefrompointatopointbinthefigureisalsoalightray.

BacktotheMainRoadGettingback toFig. 1.2, let’s sketch in anobserver,Art,who’s sitting in a railroad carmoving to the rightwithconstant speedv.Hisworld line is labeledwith equations thatdescribehismotion.Onceagain,Art’sframewillbemovingsuchthatx′=x−vt,exactlylikethemovingobserverinFig.1.1.

Now let’s figure out how to draw Art’s x′ axis. We begin by adding two moreobservers,MaggieandLenny.MaggieissittingintherailcardirectlyinfrontofArt(toyour right),andLenny’s railcar isdirectly in frontofMaggie’s.Adjacentobserversareseparated from each other by one unit of length as measured in your frame (the rest

frame).EquationsforMaggie’sandLenny’sworldlinesareshowninthefigure.BecauseMaggieislocatedoneunittotherightofArt,hertrajectoryisjustx=vt+1.Likewise,Lenny’strajectoryisx=vt+2.Art,Maggie,andLennyareinthesamemovingframe.They’reatrestwithrespecttoeachother.

Ourfirstobserver,Art,hasaclock,andhisclockhappenstoread12noonjustashearrivesat theorigin.We’llassumetheclockin therest framealsoreads12noonat thisevent.Webothagree tocall12noonour“timezero,”andwe labelourcommonoriginwith(x=0,t=0)inyourcoordinates,and(x′=0,t′=0)inArt’scoordinates.Themovingobserverandthestationaryobserveragree,byassumption,onthemeaningof t=0.Foryou(thestationaryobserver),tequalszeroallalongthehorizontalaxis.Infact,that’sthedefinitionofthehorizontalaxis:it’sthelinewherealltimesforthestationaryobserverarezero.

Figure1.2:SRFramesUsingRelativisticUnits(c=1).TheequationsassociatedwithArtaretwodifferentwaystocharacterizehisworldline.Thedashedlinesaretheworldlinesoflightrays.Theconstants1and2inthe

equationsforMaggie’sandLenny’sworldlinesarenotpurenumbers.Theirrelativistic

unitsareseconds.

SupposeArt sendsout a light signal toMaggie from theorigin.At somepoint—wedon’tknowyetwhatthatpointis—LennywillalsosendoutalightsignaltowardMaggie.He’ll somehowarrange to do this in away that both signals reachMaggie at the sameinstant.IfArt’slightsignalstartsattheorigin,MaggiewillreceiveitatpointainFig.1.2.FromwhatpointmustLennysendhislightsignal,ifitistoreachatMaggieatthesametime?Wecanfindoutbyworkingbackward.TheworldlineofanylightsignalthatLennysends toMaggie must make a 45-degree angle to the x axis. So all we need to do isconstructalineatpointathatslopes45degreesdownwardtowardtheright,andextendituntilitcrossesLenny’spath.Thisispointbinthefigure.Aswecaneasilyseefromthefigure,pointbliesabovethexaxisandnotonit.

Whatwe’vejustshownisthattheoriginandpointbaresimultaneouseventsinArt’sframe! In otherwords, themovingobserver (Art)will say that t′= 0 at pointb.Why?Becausein themovingframeofreference,ArtandLenny,whoareequidistantfromthecentralobserverMaggie,sentherlightsignalsthatarrivedatthesameinstantoftime.SoMaggie will say, “You guys sent me light signals at exactly the same instant of time,because they both arrived here at the same instant of time, and I happen to know thatyou’reatequaldistancesfromme.”

Findingthex′AxisWe’ve established thatArt’s (andMaggie’s and Lenny’s) x axis is a line that joins thecommon origin (i.e., common to both frames) to point b. Our next task is to find outexactlywhere pointb is.Oncewe figure out pointb’s coordinates,we’ll know how tospecifythedirectionofArt’sx′axis.We’llgo throughthisexercise indetail. It’sa littlecumbersome, but quite easy. There are two steps involved, the first being to find thecoordinatesofpointa.

Pointasitsattheintersectionoftwolines;therightwardmovinglightrayx=ct,andthe line x = vt + 1, which is Maggie’s world line. To find the intersection, we justsubstituteoneequationintotheother.Becausewe’reusingrelativisticunitsforwhichthespeedoflightcisequalto1,wecanwritetheequation

x=ct

inanevensimplerway,

SubstitutingEq.1.5intoMaggie’sworldline,

x=vt+1,

gives

t=vt+1

or

t(1−v)=1.

Or,evenbetter,

Now thatweknow the time coordinate of pointawe can find itsx coordinate.This iseasily done by noticing that all along the light ray, x= t. In other words, we can justreplacetwithxinEq.1.6,andwrite

xa=1/(1−v).

Voilà!—we’vefoundpointa.

Withthecoordinatesofpointainhand,let’slookatlineab.Oncewehaveanequationforlineab,wecanfigureoutwhereitintersectsLenny’sworldline,x=vt+2.Ittakesa

fewsteps,butthey’refunandIdon’tknowofanyshortcuts.

Everylinethatslopesat45degrees,pointingdownwardtotheright,hasthepropertythatx+tisconstantalongthatline.Everylineslopingupwardtotherightat45degreeshasthepropertythatx−tisconstant.Let’stakethelineab.Itsequationis

x+t=someconstant.

Whatistheconstant?Aneasywaytofindoutistotakeonepointalongthelineandpluginspecificvaluesxandt.Inparticular,wehappentoknowthatatpointa,

xa+ta=2/(1−v).

Therefore,weknowthatthisistrueallalonglineab,andtheequationforthatlinemustbe

Nowwecanfindb’scoordinatesbysolvingthesimultaneousequationsforlineabandforLenny’sworldline.Lenny’sworldlineisx=vt+2,whichwerewriteasx−vt=2.Oursimultaneousequationsare

x+t=2/(1−v)

and

x−vt=2.

Thesolution,afterabitofeasyalgebra,is

First, themost importantpoint: tb isnot zero.Thereforepointb,which is simultaneouswiththeorigininthemovingframe,isnotsimultaneouswiththeoriginintherestframe.

Next,consider thestraight lineconnecting theoriginwithpointb.Bydefinition, theslopeofthatlineistb/xb,andusingEqs.1.8weseethattheslopeisv.Thislineisnothingbutthex′axis,anditisgiven,verysimply,bytheequation

KeepinmindaswegoalongthatthevelocityvcanbepositiveornegativedependingonwhetherIammovingtotherightortotheleftrelativetoyourframe.Fornegativevelocityyouwillhavetoredrawdiagramsorjustflipthemhorizontally.

Fig.1.3showsourspacetimediagramwiththex′andt′axesdrawnin.Thelinet=vx(orwhat is reallya three-dimensional surface in the spacetimemapwhen the twoothercoordinatesy,zareaccountedfor)hastheimportantpropertythatonit,alltheclocksinthe moving frame record the same value of t′. To give it a name, it is a surface ofsimultaneityinthemovingframe.Itplaysthesameroleasthesurfacet=0doesfortherestframe.

Sofarinthissection,we’veworkedinrelativisticunitswherethespeedoflightisc=1.HereisagoodopportunitytopracticeyourskillsindimensionalanalysisandfigureoutwhatEq.1.9wouldlooklikeinconventionalunitsofmetersandseconds.Inthoseunits,Eq.1.9isnotdimensionallyconsistent; theleftsidehasunitssecondsandtherightsidehasunitsmeterssquaredpersecond.Torestoreconsistency,wehavetomultiplytherightsidebyanappropriatepowerofc.Thecorrectfactoris1/c2:

Figure1.3:SRFrameswithx′andt′AxesShown.

TheinterestingthingaboutEq.1.10isthatitdescribesastraightlinewiththeincrediblytiny slope . For example, if v were 300 meters per second (roughly the speed of ajetliner)theslopewouldbev/c2=3×10−15.Inotherwords,thex′axisinFig.1.3wouldbealmostexactlyhorizontal.ThesurfacesofsimultaneityintherestandmovingframeswouldalmostexactlycoincidejustastheywouldinNewtonianphysics.

This is an example of the fact that Einstein’s description of spacetime reduces toNewton’s if the relativevelocityof the reference frames ismuch less than the speedoflight.This,ofcourse,isanimportant“sanity”check.

Nowwecanreturntorelativisticunitswithc=1.Let’ssimplifyourdiagramandkeeponlythefeatureswe’llneedgoingforward.ThedashedlineinFig.1.4representsalightray,whoseworldlinemakesa45-degreeanglewithboththetandxaxes.Art’sworldlineisshownasthet′axis.Hisx′axisisalsolabeled.BothofArt’saxesarealsolabeledwithappropriateequations.Noticethesymmetryofhistwoaxes:x=vtandt=vx.Thesetwolines are reflections of each other about the dashed light trajectory. They’re related byinterchangingtandx.Anotherwaytosayit is that theyeachmakethesameanglewiththeirnearestunprimedaxis—thexaxisinthecaseoft=vx,andthetaxisinthecaseofx=vt.

Figure1.4:SRFramesSimplified.

We’vediscoveredtwointerestingthings.First,ifthespeedoflightreallyisthesameineveryframe,andyouuselightraystosynchronizeclocks,thenthepairsofeventsthataresynchronousinoneframearenotthesamepairsthataresynchronousintheotherframe.Second, we’ve found what synchronicity actually means in the moving frame. Itcorrespondstosurfacesthatarenothorizontal,butaretiltedwithslopev.Wehavefiguredout thedirectionsof thex′and t′axes inArt’smovingframe.Lateron,we’ll figureouthowtomarkofftheintervalsalongtheseaxes.

SpacetimeLet’s pause for a moment to contemplate what we’ve found about space and time.Newton, of course, knew about both space and time but regarded them as entirelyseparate. ToNewton, three-dimensional spacewas space, and timewas universal time.Theywereentirelyseparate,thedifferencebeingabsolute.

ButmapslikeFigs.1.3and1.4indicatesomethingthatNewtoncouldnothaveknown,namely that in going from one inertial reference frame to another, the space and timecoordinatesgetmixedupwitheachother.Forexample, inFig.1.3 the intervalbetweentheoriginandpointbrepresentstwopointsatthesametimeinthemovingframe.Butintherestframe,pointbisnotonlyshiftedinspacefromtheorigin;italsoisshiftedintime.

Three years afterEinstein’smighty 1905paper that introduced the special theory ofrelativity,Minkowski completed the revolution. In an address to the 80thAssembly ofGermanNaturalScientistsandPhysicians,hesaid,

Spacebyitself,andtimebyitself,aredoomedtofadeawayintomereshadows,andonlyakindofunionofthetwowillpreserveanindependentreality.

Thatunion is four-dimensional,withcoordinates t,x,y,z.Depending on ourmood,wephysicists sometimescall thatunionof spaceand time spacetime.Sometimeswecall itMinkowskispace.Minkowskihadanothernameforit.Hecalledittheworld.

Minkowski called the points of spacetime events. An event is labeled by the fourcoordinatest,x,y,z.Bycallingapointofspacetimeanevent,Minkowskididnotmeanto

implythatsomethingactuallytookplaceatt,x,y,z.Onlythatsomethingcouldtakeplace.Hecalled the linesorcurvesdescribing trajectoriesofobjectsworldlines.Forexample,thelinex′=0inFig.1.3isArt’sworldline.

Thischangeofperspectivefromspaceandtimetospacetimewasradicalin1908,buttodayspacetimediagramsareasfamiliartophysicistsasthepalmsoftheirhands.

LorentzTransformationsAn event, in other words a point of spacetime, can be labeled by the values of itscoordinates in the rest frameor by its coordinates in themoving frame.Weare talkingabouttwodifferentdescriptionsofasingleevent.Theobviousquestionis,howdowegofromonedescriptiontotheother?Inotherwords,whatisthecoordinate transformationrelating the rest framecoordinates t,x,y,z to thecoordinates t′,x′,y′,z′ of themovingframe?

OneofEinstein’sassumptionswasthatspacetimeiseverywherethesame,inthesamesense that an infinite plane is everywhere the same. The sameness of spacetime is asymmetry that says no event is different from any other event, and one can choose theorigin anywhere without the equations of physics changing. It has a mathematicalimplicationforthenatureoftransformationsfromoneframetoanother.Forexample,theNewtonianequation

islinear;itcontainsonlyfirstpowersofthecoordinates.Equation1.11willnotsurviveinitssimpleform,butitdoesgetonethingright,namelythatx′=0wheneverx=vt.InfactthereisonlyonewaytomodifyEq.1.11andstillretainitslinearproperty,alongwiththefactthatx′=0isthesameasx=vt. It is tomultiplytherightsidebyafunctionofthevelocity:

Atthemomentthefunctionf(v)couldbeanyfunction,butEinsteinhadonemoretrickuphis sleeve: another symmetry—the symmetry between left and right. To put it anotherway, nothing in physics requires movement to the right to be represented by positivevelocityandmovement to the leftbynegativevelocity.That symmetry implies that f(v)mustnotdependonwhethervispositiveornegative.Thereisasimplewaytowriteanyfunctionthatisthesameforpositiveandnegativev.Thetrickistowriteitasafunctionofthesquareofthevelocityv2.6Thus,insteadofEq.1.12,Einsteinwrote

Tosummarize,writingf(v2)insteadoff(v)emphasizesthepointthatthereisnopreferreddirectioninspace.

Whataboutt′?We’llreasonthesamewayhereaswedidforx′.Weknowthatt′=0whenevert=vx.Inotherwords,wecanjustinverttherolesofxandt,andwrite

whereg(v2)issomeotherpossiblefunction.Equations1.13and1.14tellusthatx′iszero

wheneverx=vt,andthatt′iszerowhenevert=vx.Becauseofthesymmetryinthesetwoequations,thet′axisisjustareflectionofthex′axisaboutthelinex=t,andviceversa.

Whatweknowso far is thatour transformationequations should take the followingform:

Ournext task is to figureoutwhat the functions f(v2)andg(v2) actually are.Todo thatwe’llconsider thepathofa lightrayinthesetwoframes,andapplyEinstein’sprinciplethatthespeedoflightisthesameinbothofthem.Ifthespeedoflightcequals1inthestationaryframe,itmustalsoequal1inthemovingframe.Torephrasethis:Ifwestartoutwithalightraythatsatisfiesx=tinthestationaryframe,itmustalsosatisfyx′=t′inthemovingframe.Toputitanotherway,if

x=t,

thenitmustfollowthat

x′=t′.

Let’sgobacktoEqs.1.15.Settingx=tandrequiringx′=t′givesthesimplerequirementthat

f(v2)=g(v2).

Inotherwordstherequirementthatthespeedoflightisthesameinyourframeandmyframe leads to the simple condition that the two functions f(v2) andg(v2) are the same.ThuswemaysimplifyEqs.1.15

Tofindf(v2),Einsteinusedonemoreingredient.Ineffect,hesaid“Waitaminute,who’stosaywhichframeismoving?Who’stosayifmyframeismovingrelativetoyouwithvelocity v, or your frame is moving relative to me with velocity −v?” Whatever therelationship is between the two frames of reference, itmust be symmetrical. Followingthisapproach,wecould invertourentireargument; insteadofstartingwithxand t,andderivingx′andt′,wecoulddoexactlytheopposite.Theonlydifferencewouldbethatforme,you’removingwithvelocity−v,butforyou,I’mmovingwithvelocity+v.BasedonEqs.1.16,whichexpressx′andt′intermsofxandt,wecanimmediatelywritedowntheinversetransformations.Theequationsforxandtintermsofx′andt′are

WeshouldbeclearthatwewroteEqs.1.17byreasoningaboutthephysicalrelationshipbetweenthetworeferenceframes.They’renotjustacleverwaytosolveEqs.1.16fortheunprimedvariableswithoutdoing thework. In fact,nowthatwehave these twosetsof

equations,weneedtoverifythatthey’recompatiblewitheachother.

Let’sstartwithEqs.1.17andplugintheexpressionsforx′andt′fromEqs.1.16.Thismayseemcircular,butyou’llseethatitisn’t.Aftermakingoursubstitutions,we’llrequirethe results to be equivalent tox= x and t= t.How could they be anything else if theequationsarevalidtobeginwith?Fromthere,we’llfindouttheformoff(v2).Thealgebrais a little tedious but straightforward. Startingwith the first of Eqs. 1.17, the first fewsubstitutionsforxunfoldasfollows:

x=(x′+vt′)f(v2)

x={(x−vt)f(v2)+v(t−vx)f(v2)}f(v2)

x=(x−vt)f2(v2)+v(t−vx)f2(v2).

Expandingthelastlinegives

x=xf2(v2)−v2xf2(v2)−vtf2(v2)+vtf2(v2),

whichsimplifiesto

x=xf2(v2)(1−v2).

Cancelingxonbothsidesandsolvingforf(v2)gives

Nowwehaveeverythingweneedtotransformcoordinatesintherestframetocoordinatesinthemovingframeandviceversa.PluggingintoEqs.1.16gives

These are, of course, the famous Lorentz transformations between the rest andmovingframes.

Art:“Wow,that’sincrediblyclever,Lenny.Didyoufigureitalloutyourself?”

Lenny:“Iwish.No,I’msimplyfollowingEinstein’spaper.Ihaven’treaditforfiftyyears,butitleftanimpression.”

Art:“Okay,buthowcomethey’recalledLorentztransformationsiftheywerediscoveredbyEinstein?”

1.2.3HistoricalAsideToanswerArt’squestion,EinsteinwasnotthefirsttodiscovertheLorentztransformation.That honor belongs to the Dutch physicist Hendrik Lorentz. Lorentz, and others evenbefore him—notably George FitzGerald—had speculated that Maxwell’s theory ofelectromagnetism required moving objects to contract along the direction of motion, aphenomenonthatwenowcallLorentzcontraction.By1900Lorentzhadwrittendownthe

Lorentztransformationsmotivatedbythiscontractionofmovingbodies.ButtheviewsofEinstein’spredecessorsweredifferentandinasenseathrowbacktoolderideasratherthana new starting point. Lorentz and FitzGerald imagined that the interaction between thestationaryetherandthemovingatomsofallordinarymatterwouldcauseapressurethatwouldsqueezematteralongthedirectionofmotion.Tosomeapproximationthepressurewouldcontractallmatterbythesameamountsothattheeffectcouldberepresentedbyacoordinatetransformation.

JustbeforeEinstein’spaper,thegreatFrenchmathematicianHenriPoincarépublisheda paper in which he derived the Lorentz transformation from the requirement thatMaxwell’sequationstakethesameformineveryinertialframe.Butnoneoftheseworkshadtheclarity,simplicity,andgeneralityofEinstein’sreasoning.

1.2.4BacktotheEquationsIfweknowthecoordinatesofanevent in the rest frame,Eqs.1.19and1.20 tellus thecoordinatesof the sameevent in themoving frame.Canwego theotherway? Inotherwords,canwepredict thecoordinates in the rest frame ifweknowthemin themovingframe?Todosowemightsolvetheequationsforxandtintermsofx′andt′,butthereisaneasierway.

Allweneedistorealizethatthereisasymmetrybetweentherestandmovingframes.Who,afterall,istosaywhichframeismovingandwhichisatrest?Tointerchangetheirroles,wemight just interchange theprimedandunprimedcoordinates inEqs. 1.19 and1.20.That’salmostcorrectbutnotquite.

Considerthis:IfIammovingtotherightrelativetoyou,thenyouaremovingtotheleftrelativetome.Thatmeansyourvelocityrelativetomeis−v.ThereforewhenIwritetheLorentztransformationsforx,tintermsofx′,t′,Iwillneedtoreplacevwith−v.Theresultis

SwitchingtoConventionalUnitsWhatifthespeedoflightisnotchosentobe1?Theeasiestwaytoswitchfromrelativisticunitsbacktoconventionalunitsistomakesureourequationsaredimensionallyconsistentin those units. For example, the expression x − vt is dimensionally consistent as it isbecausebothxandvthaveunitsof length—saymeters.Ontheotherhand t−vx isnotdimensionallyconsistentinconventionalunits;thasunitsofsecondsandvxhasunitsofmeterssquaredoverseconds.Thereisauniquewaytofixtheunits.Insteadoft−vxwereplaceitwith

Nowbothtermshaveunitsoftime,butifwehappentouseunitswithc=1itreducestotheoriginalexpressiont−vx.

Similarlythefactorinthedenominators, ,isnotdimensionallyconsistent.Tofixtheunitswereplacevwithv/c.WiththesereplacementstheLorentztransformationscanbewritteninconventionalunits:

Noticethatwhenvisverysmallcomparedtothespeedoflight,v2/c2isevensmaller.Forexample,Ifv/cis1/10,thenv2/c2is1/100.Ifv/c=10−5,thenv2/c2istrulyasmallnumber,and the expression in the denominator is very close to 1.7 To a very goodapproximationwecanwrite

x′=x−vt.

That’sthegoodoldNewtonianversionofthings.Whathappenstothetimeequation(Eq.1.24)whenv/cisverysmall?Supposevis100meterspersecond.Weknowthatcisverybig,about3×108meterspersecond.Sov/c2isaverytinynumber.Ifthevelocityofthemovingframeissmall,thesecondterminthenumerator,vx/c2,isnegligibleandtoahighdegreeofapproximation thesecondequationof theLorentz transformationbecomes thesameastheNewtoniantransformation

t′=t.

Forframesmovingslowlyrelativetoeachother,theLorentztransformationboilsdowntotheNewtonianformula.That’sagoodthing;as longaswemoveslowlycomparedtoc,we get the old answer. But when the velocity becomes close to the speed of light thecorrectionsbecomelarge—hugewhenvapproachesc.

TheOtherTwoAxesEqs.1.23and1.24aretheLorentztransformationequationsincommon,orconventional,units.Ofcourse,thefullsetofequationsmustalsotellushowtotransformtheothertwocomponentsofspace,yandz.We’vebeenveryspecificaboutwhathappenstothexandtcoordinateswhen framesare in relativemotionalong thex axis.Whathappens to theycoordinate?

We’ll answer thiswith a simple thought experiment. Suppose your arm is the samelengthasmyarmwhenwe’rebothatrestinyourframe.ThenIstartmovingatconstantvelocityinthexdirection.Aswemovepasteachother,weeachholdoutanarmatarightangletothedirectionofourrelativemotion.Question:Aswemovepasteachother,wouldourarmsstillbeequalinlength,orwouldyoursbelongerthanmine?Bythesymmetryofthissituation,it’sclearthatourarmsaregoingtomatch,becausethere’snoreasonforone

tobelongerthantheother.Therefore,therestoftheLorentztransformationmustbey′=yandz′=z.Inotherwords,interestingthingshappenonlyinthex,tplanewhentherelativemotionisalongthexaxis.Thexandtcoordinatesgetmixedupwitheachother,butyandzarepassive.

For easy reference,here’s the completeLorentz transformation in conventionalunitsforareferenceframe(theprimedframe)movingwithvelocityvinthepositivexdirectionrelativetotheunprimedframe:

1.2.5NothingMovesFasterthanLightAquicklookatEqs.1.25and1.26indicatesthatsomethingstrangehappensiftherelativevelocity of two frames is larger than c. In that case 1 − v2/c2 becomes negative and

becomes imaginary.That’sobviousnonsense.Metersticksandclockscanonlydefinereal-valuedcoordinates.

Einstein’s resolutionof this paradoxwas an additional postulate: nomaterial systemcanmovewithavelocitygreaterthanlight.Moreaccurately,nomaterialsystemcanmovefasterthanlightrelativetoanyothermaterialsystem.Inparticularnotwoobserverscanmove,relativetoeachother,fasterthanlight.

Thusweneverhaveneedforthevelocityvtobegreaterthanc.Todaythisprincipleisa cornerstone ofmodern physics. It’s usually expressed in the form that no signal cantravel faster than light.But since signals are composed ofmaterial systems, even if nomoresubstantialthanaphoton,itboilsdowntothesamething.

1.3GeneralLorentzTransformationThesefourequationsremindusthatwehaveconsideredonlythesimplestkindofLorentztransformation: a transformation where each primed axis is parallel to its unprimedcounterpart, and where the relative motion between the two frames is only along theshareddirectionofthexandx′axes.

Uniformmotionissimple,butit’snotalwaysthatsimple.There’snothingtopreventthetwosetsofspaceaxesfrombeingorienteddifferently,witheachprimedaxisatsomenonzero angle to its unprimed counterpart.8 It’s also easy to visualize the two framesmoving with respect to each other not only in the x direction, but along the y and zdirections as well. This raises a question: By ignoring these factors, have we missedsomethingessentialaboutthephysicsofuniformmotion?Happily,theanswerisno.

Suppose you have two frames in relativemotion along some oblique direction, notalonganyofthecoordinateaxes.Itwouldbeeasytomaketheprimedaxeslineupwiththeunprimedaxesbyperformingasequenceofrotations.Afterdoingthoserotations,youwouldagainhaveuniformmotioninthexdirection.ThegeneralLorentztransformation—wheretwoframesarerelatedtoeachotherbyanarbitraryangleinspace,andaremovingrelativetoeachotherinsomearbitrarydirection—isequivalentto:

1.Arotationofspacetoaligntheprimedaxeswiththeunprimedaxes.

2.AsimpleLorentztransformationalongthenewxaxis.

3.Asecondrotationofspacetorestoretheoriginalorientationoftheunprimedaxesrelativetotheprimedaxes.

As long as you make sure that your theory is invariant with respect to the simpleLorentz transformation along, say, the x axis, and with respect to rotations, it will beinvariantwithrespecttoanyLorentztransformationatall.

Asamatterofterminology,transformationsinvolvingarelativevelocityofoneframemoving relative to another are calledboosts. For example, the Lorentz transformationslikeEqs.1.25and1.26arereferredtoasboostsalongthexaxis.

1.4LengthContractionandTimeDilationSpecial relativity, until you get used to it, is counterintuitive—perhaps not ascounterintuitiveasquantummechanics,butnevertheless fullofparadoxicalphenomena.My advice is that when confronted with one of these paradoxes, you should draw aspacetimediagram.Don’t ask your physicist friend, don’t emailme—drawa spacetimediagram.

Figure1.5:LengthContraction.

LengthContractionSupposeyou’reholdingameterstickandI’mwalkingpastyouinthepositivexdirection.Youknowthatyourstickis1meterlong,butI’mnotsosure.AsIwalkpast,Imeasure

yourmeterstickrelativetothelengthsofmymetersticks.SinceI’mmoving,Ihavetobevery careful; otherwise, I could bemeasuring the end points of yourmeterstick at twodifferent times. Remember, events that are simultaneous in your frame are notsimultaneousinmine.Iwanttomeasuretheendpointsofyourmeterstickatexactlythesametimeinmyframe.That’swhatImeanbythelengthofyourmeterstickinmyframe.

Fig.1.5showsaspacetimediagramofthissituation.Inyourframe,themeterstickisrepresented by a horizontal line segment along the x axis, which is a surface ofsimultaneityforyou.Themeterstickisatrest,andtheworldlinesofitsendpointsaretheverticallinesx=0andx=1inyourframe.

Inmymovingframe,thatsamemeterstickataninstantoftimeisrepresentedbylinesegment alongthex′axis.Thex′axisisasurfaceofsimultaneityformeandistiltedinthediagram.OneendofthemeterstickisatourcommonoriginOasIpassit.Theotherendofthestickattimet′=0islabeledPinthediagram.

Tomeasurethelocationofbothendsat time t′=0 inmyframe, Ineed toknowthecoordinatevaluesofx′atpointsOandP.ButIalreadyknowthatx′iszeroatpointO,soall Ineedtodo iscalculate thevalueofx′atpointP.We’lldo this theeasyway,usingrelativistic units (speed of light equals 1). In other words, we’ll use the LorentztransformationofEqs.1.19and1.20.

First,noticethatpointPsitsat theintersectionoftwolines,x=1andt′=0.Recall(basedonEq.1.20)thatt′=0meansthatt=vx.SubstitutingvxfortinEq.1.19gives

Pluggingx=1intotheprecedingequationgives

or

Sothereit is!Themovingobserverfindsthatataninstantoftime—whichmeansalongthesurfaceofsimultaneityt′=0—thetwoendsofthemeterstickareseparatedbydistance

;inthemovingframe,themeterstickisalittleshorterthanitwasatrest.

Itmayseemlikeacontradictionthatthesamemeterstickhasonelengthinyourframeand a different length in my frame. Notice, though, that the two observers are reallytalkingabouttwodifferentthings.Fortherestframe,wherethemeterstickitselfisatrest,we’re talking about the distance from pointO to pointQ, as measured by stationarymetersticks. In themovingframe,we’re talkingabout thedistancebetweenpointOandpointP,measuredbymovingmeasuringrods.PandQaredifferentpointsinspacetime,sothere’snocontradictioninsayingthat isshorterthan .

Trydoingtheoppositecalculationasanexercise:Startingwithamovingmeterstick,find its length in the rest frame.Don’t forget—begin by drawing a diagram. If you getstuckyoucancheatandcontinuereading.

Thinkofamovingmeterstickbeingobservedfromtherestframe.Fig.1.6showsthissituation.Ifthemeterstickis1unitlonginitsownrestframe,anditsleadingendpassesthroughpointQ,whatdoweknowaboutitsworldline?Isitx=1?No!Themeterstickis1meterlonginthemovingframe,whichmeanstheworldlineofitsleadingendisx′=1.Theobserveratrestnowseesthemeterstickbeingthelengthoflinesegment ,andthexcoordinateofpointQisnot1. It’ssomevaluecalculatedbytheLorentz transformation.Whenyoudothiscalculation,you’llfindthatthislengthisalsoshortenedbythefactor

.

The moving metersticks are short in the stationary frame, and the stationarymetersticks are short in the moving frame. There’s no contradiction. Once again, theobserversare just talkingaboutdifferent things.Thestationaryobserver is talkingaboutlengthsmeasuredataninstantofhistime.Themovingobserveristalkingaboutlengthsmeasuredatan instantof theother time.Therefore, theyhavedifferentnotionsofwhattheymeanbylengthbecausetheyhavedifferentnotionsofsimultaneity.

Figure1.6:LengthContractionExercise.

Exercise1.1:ShowthatthexcoordinateofpointQinFig.1.6is .

TimeDilationTime dilation works pretty much the same way. Suppose I have a moving clock—myclock.Assumemyclockismovingalongwithmeatuniformvelocity,asinFig.1.7.

Here’sthequestion:Attheinstantwhenmyclockreadst′=1inmyframe,whatisthetime in your frame? By the way, my standard wristwatch is an excellent timepiece, aRolex.9 I want to know the corresponding value of t measured by your Timex. Thehorizontalsurfaceinthediagram(thedashedline)isthesurfacethatyoucallsynchronous.Weneed two things inorder topindown thevalueof t inyour frame.First,myRolexmovesonthet′axiswhichisrepresentedbytheequationx′=0.Wealsoknowthatt′=1.Tofigureoutt,allweneedisoneoftheLorentztransformationequations(Eq.1.22),

Figure1.7:TimeDilation.

Plugginginx′=0andt′=1,wefind

Becausethedenominatorontherightsideissmallerthan1,titselfisbiggerthan1.Thetime interval measured along the t axis (your Timex) is bigger than the time intervalmeasuredbythemovingobserveralongthet′axis(myRolex)byafactorof .Inshort,t>t′.

Toputitanotherway,asviewedfromtherestframe,movingclocksrunslowerbyafactorof

TheTwinParadoxLenny:HeyArt!SayhellotoLorentzoverhere.Hehasaquestion.

Art:Lorentzhasaquestionforus?

Lorentz:PleasecallmeLor’ntz.It’stheoriginalLorentzcontraction.InalltheyearsI’vebeencomingtoHermann’sHideaway,I’veneverseeneitheroneofyouguyswithouttheother.Areyoubiologicaltwins?

Art:What?Look,ifwewerebiologicalweretwins,eitherI’dbeagenius—don’tchokeonyoursausage,Lor’ntz,it’snotthatfunny.AsIwassaying,eitherI’dbeageniusorLennywouldbesomewiseguyfromtheBronx.Waitaminute…10

Timedilationistheoriginoftheso-calledtwinparadox.InFig.1.8,Lennyremainsatrest,whileArttakesahigh-speedjourneyinthepositivexdirection.Atthepointlabeledt′=1inthediagram,Artattheageof1turnsaroundandheadsbackhome.

We’vealreadycalculatedtheamountofrestframetimethatelapsesbetweentheoriginandthepointlabeledtinthediagram.It’s .Inotherwords,wefindthatlesstimeelapsesalongthepathofthemovingclockthanalongthepathofthestationaryclock.Thesamethingistrueforthesecondlegofthejourney.WhenArtreturnshome,hefindsthathistwinLennyisolderthanhimself.

Figure1.8:TwinParadox.

We’vecalibrated theagesofArtandLennyby the time registeredon theirwatches.ButthesametimedilationsthatslowedArt’swatchfromtheviewpointoftherestframewouldaffectanyclock,includingthebiologicalagingclock.Thusinanextremecase,ArtcouldreturnhomestillaboywhileLennywouldhavealonggraybeard.

Twoaspectsofthetwinparadoxoftenleavepeopleconfused.First,itseemsnaturaltoexpecttheexperiencesofthetwotwinstobesymmetrical.IfLennyseesArtmovingawayfromhim,thenArtalsoseesLennytravelingaway,butintheoppositedirection.Therearenopreferreddirectionsinspace,sowhyshouldtheyageanydifferently?Butinfact,theirexperiencesarenotsymmetricalatall.Thetravelingtwinundergoesalargeaccelerationin order to change directions, while the stay-at-home twin does not. This difference iscrucial. Because of the abrupt reversal, Art’s frame is not a single inertial frame, butLenny’sis.Weinviteyoutodevelopthisideafurtherinthefollowingexercise.

Exercise1.2:InFig.1.8,thetravelingtwinnotonlyreversesdirectionsbutswitchestoadifferentreferenceframewhenthereversalhappens.

a)UsetheLorentztransformationtoshowthatbeforethereversalhappens,therelationshipbetweenthetwinsissymmetrical.Eachtwinseestheotherasagingmoreslowlythanhimself.

b)Usespacetimediagramstoshowhowthetraveler’sabruptswitchfromoneframetoanotherchangeshisdefinitionofsimultaneity.Inthetraveler’snewframe,histwinissuddenlymucholderthanhewasinthetraveler’soriginalframe.

Anotherpoint of confusion arises fromsimplegeometry.Referringback toFig. 1.7,recallthatwecalculatedthe“timedistance”frompointOtothepointlabeledt′=1tobesmallerthanthedistancefromOtothepointlabeled alongthetaxis.Basedon these twovalues, thevertical legof this right triangle is longer than its hypotenuse.

Manypeoplefindthispuzzlingbecausethenumericalcomparisonseemstocontradictthevisualmessageinthediagram.Infact,thispuzzleleadsustooneofthecentralideasinrelativity,theconceptofaninvariant.We’lldiscussthisideaextensivelyinSection1.5.

TheStretchLimoandtheBugAnother paradox is sometimes called thePole in theBarn paradox.But inPoland theyprefertocallittheParadoxoftheLimoandtheBug.

Art’scarisaVWBug.It’sjustunder14feetlong.HisgaragewasbuilttojustfittheBug.

Lenny has a reconditioned stretch limo. It’s 28 feet long. Art is going on vacation andrentinghishousetoLenny,butbeforehegoesthetwofriendsget togethertomakesurethatLenny’scarwill fit inArt’sgarage.Lenny isskeptical,butArthasaplan.Art tellsLenny to back up and get a good distance from the garage. Then step on the gas andaccelerate likeblazes. IfLennycanget the limoup to161,080milesper secondbeforegettingtothebackendofthegarage,itwilljustfitin.Theytryit.

Art watches from the sidewalk as Lenny backs up the limo and steps on the gas. Thespeedometerjumpsto172,000mps,plentyofspeedtospare.ButthenLennylooksoutatthegarage.“Holycow!Thegarageiscomingatmereallyfast,andit’slessthanhalfitsoriginalsize!I’llneverfit!”

“Sureyouwill,Lenny.According tomycalculation, in therest frameof thegarageyouarejustabitlongerthanthirteenfeet.Nothingtoworryabout.”

“Geez,Art,Ihopeyou’reright.”

Fig. 1.9 is a spacetime diagram including Lenny’s stretch limo shown in the darkshadedregion,andthegarageshownaslightlyshaded.Thefrontendof thelimoentersthegarageataandleaves(assumingthatArtleft thebackdoorofthegarageopen)justabovec.Thebackendofthelimoentersatbandleavesatd.Nowlookattheline Itispartofasurfaceofsimultaneity in therest frameof thegarage,andasyoucansee, theentirelimoiscontainedinthegarageatthattime.That’sArt’sclaim:Inhisframethelimocouldbemadetofitthegarage.ButnowlookatLenny’ssurfacesofsimultaneity.Theline is such a surface, and asyou can also see, the limooverflows thegarage.AsLenny

worried,thelimodoesnotfit.

Figure1.9:StretchLimo-GarageSpacetime

Diagram.

Thefiguremakesclearwhattheproblemis.Tosaythatthelimoisinthegaragemeansthatthefrontandbackaresimultaneouslyinthegarage.There’sthatwordsimultaneousagain. Simultaneous according to whom, Art? Or Lenny? To say that the car is in thegarage simply means different things in different frames. There is no contradiction insayingthatatsomeinstantinArt’sframethelimowasindeedinthegarage—andthatatnoinstantinLenny’sframewasthelimowhollyinthegarage.

Almostallparadoxesofspecialrelativitybecomeobviouswhenstatedcarefully.Watchoutforthepossiblyimplicituseofthewordsimultaneous.That’susuallythegiveaway—simultaneousaccordingtowhom?

1.5Minkowski’sWorldOneofthemostpowerfultoolsinthephysicist’stoolbagistheconceptofaninvariant.Aninvariant is a quantity that doesn’t change when looked at from different perspectives.Herewemeansomeaspectofspacetimethathasthesamevalueineveryreferenceframe.

Togettheidea,we’lltakeanexamplefromEuclideangeometry.Let’sconsideratwo-dimensional plane with two sets of Cartesian coordinates, x, y and a second set x′, y′.Assumethattheoriginsofthetwocoordinatesystemsareatthesamepoint,butthatthex′,y′axes(theprimedaxes)arerotatedcounterclockwisebyafixedanglewithrespecttotheunprimedaxes.Thereisnotimeaxisinthisexample,andtherearenomovingobservers;justtheordinaryEuclideanplaneofhighschoolgeometry.Fig.1.10givesyouthepicture.

ConsideranarbitrarypointPinthisspace.ThetwocoordinatesystemsdonotassignthesamecoordinatevaluestoP.Obviously,thexandtheyofthispointarenotthesamenumbersasthex′andthey′,eventhoughbothsetsofcoordinatesrefertothesamepointPinspace.Wewouldsaythatthecoordinatesarenotinvariant.

Figure1.10:EuclideanPlane.

However, there’s a property that is the same,whether you calculate it in primed orunprimed coordinates:P’s distance from the origin. That distance is the same in everycoordinatesystemregardlessofhowit’soriented.Thesameis truefor thesquareof thedistance.Tocalculate thisdistance in theunprimedcoordinatesweuse thePythagoreantheorem,d2 = x2 + y2, to get the square of the distance. If we use primed coordinatesinstead,thesamedistancewouldbegivenbyx′2+y′2.Thereforeitfollowsthat

x2+y2=x′2+y′2.

Inotherwords,foranarbitrarypointPthequantityx2+y2isinvariant.Invariantmeansthat it doesn’t depend onwhich coordinate systemyou use towork it out.You get thesameanswernomatterwhat.

OnefactaboutrighttrianglesinEuclideangeometryisthatthehypotenuseisgenerallylargerthaneitherside(unlessonesideiszero,inwhichcasethehypotenuseisequaltotheother side). This tells us that the distance d is at least as large as x or y. By the sameargumentitisatleastaslargeasx′ory’.

Circlingbacktorelativity,ourdiscussionofthetwinparadoxinvolvedsomethingthatlookedalotlikearighttriangle.GobacktoFig.1.8andconsiderthetriangleformedfromthe lines connecting the three black dots—the horizontal dashed line, the first half ofLenny’sverticalworld line,andthehypotenuseformedbythefirst legofArt’s journey.Wecanthinkofthedashed-linedistancebetweenthetwolaterdotstodefineaspacetimedistancebetweenthem.11ThetimealongLenny’ssideofthetrianglecanalsobethoughtofasaspacetimedistance. Its lengthwouldbe .And finally the time tickedoffduringthefirsthalfofArt’stripisthespacetimelengthofthehypotenuse.Butamoment’sinspectionshowssomethingunusual—theverticallegislongerthanthehypotenuse(that’swhyLennyhadtimetogrowabeardwhileArtremainedaboy).ThisimmediatelytellsusthatMinkowskispaceisnotgovernedbythesamelawsasEuclideanspace.

Neverthelesswemayask:IsthereananalogousinvariantquantityinMinkowskispace,associated with the Lorentz transformation—a quantity that stays the same in everyinertialreferenceframe?Weknowthatthesquareofthedistancefromtheorigintoafixedpoint P is invariant under simple rotations of Euclidean coordinates. Could a similarquantity, possibly t2 + x2, be invariant under Lorentz transformations? Let’s try it out.

ConsideranarbitrarypointP ina spacetimediagram.Thispoint is characterizedbya tvalueandanxvalue,andinsomemovingreferenceframeit’salsocharacterizedbya t′andanx′.WealreadyknowthatthesetwosetsofcoordinatesarerelatedbytheLorentztransformation.Let’sseeifourguess,

iscorrect.UsingtheLorentztransformation(Eqs.1.19and1.20)tosubstitutefort′andx′,wehave

whichsimplifiesto

Doestherightsideequalt2+x2?Noway!Youcanseeimmediatelythatthetxterminthefirst expression adds to the tx term in the second expression. They do not cancel, andthere’snotxtermontheleftsidetobalancethingsout.Theycan’tbethesame.

Butifyoulookcarefully,you’llnoticethatifwetakethedifferenceofthetwotermson the right side rather than their sum, the tx terms would cancel. Let’s define a newquantity

τ2=t2−x2.

Theresultofsubtractingx′2fromt′2gives

Afterabitofrearrangementitisexactlywhatwewant.

Bingo!We’ve discovered an invariant, τ2, whose value is the same under any Lorentztransformationalong thex axis.The square root of this quantity, τ, is called thepropertime.Thereasonforthisnamewillbecomeclearshortly.

Uptonowwe’veimaginedtheworldtobea“railroad”inwhichallmotionisalongthex axis. Lorentz transformations are all boosts along the x axis. By now youmay haveforgottenabout theother twodirectionsperpendicular to the tracks:directionsdescribedbythecoordinatesyandz.Let’sbringthembacknow.InSection1.2.4,IexplainedthatthefullLorentztransformation(withc=1)forrelativemotionalongthexaxis—aboostalongx—hasfourequations:

y′=y

z′=z

Whataboutboostsalongotheraxes?AsIexplainedinSection1.3,theseotherboostscanbe represented as combinations of boosts alongx and rotations that rotate the x axis toanother direction. As a consequence, a quantity will be invariant under all Lorentztransformationsifitisinvariantwithrespecttoboostsalongxandwithrespecttorotationsofspace.Whataboutthequantityτ2=t2−x2?We’veseenthatitisinvariantwithrespecttox-boosts,butitchangesifspaceisrotated.Thisisobviousbecauseitinvolvesxbutnoty and z. Fortunately it is easy to generalize τ to a full-fledged invariant. Consider thegeneralizedversionofEq.1.30,

Let’sfirstarguethatτisinvariantwithrespecttoboostsalongthexaxis.We’vealreadyseen that the term t2 − x2 is invariant. To that we add the fact that the perpendicularcoordinates y and z don’t change under a boost along x. If neither t2 − x2 nor y2 + z2changewhentransformingfromoneframetoanother,thenobviouslyt2−x2−y2−z2willalsobeinvariant.Thattakescareofboostsinthexdirection.

Nowlet’sseewhyitdoesnotchangeifthespaceaxesarerotated.Againtheargumentcomesintwoparts.Thefirstisthatarotationofspatialcoordinatesmixesupx,y,andzbuthasnoeffectontime.Thereforetisinvariantwithrespecttorotationsofspace.Nextconsider thequantityx2+y2+z2.A three-dimensionalversionofPythagoras’s theoremtellsusthatx2+y2+z2isthesquareofthedistanceofthepointx,y,ztotheorigin.Againthisissomethingthatdoesnotchangeunderarotationofspace.Combiningtheinvarianceoftimeandtheinvarianceofthedistancetotheorigin(underrotationsinspace),wecometo the conclusion that the proper time τ defined by Eq. 1.31 is an invariant that allobserverswill agree on. This holds not only for observersmoving in any direction butobserverswhosecoordinateaxesareorientedinanyway.

Figure1.11:MinkowskiLightCone.

1.5.1MinkowskiandtheLightConeThe invarianceof theproper time τ is apowerful fact. Idon’tknow if itwasknown toEinstein,butintheprocessofwritingthissectionIlookedthroughmyancientwornandfaded copyof theDover edition containing the1905paper (thepriceon the coverwas$1.50). I found no mention of Eq. 1.31 or of the idea of spacetime distance. It wasMinkowskiwho first understood that the invariance of proper time,with its unintuitiveminus signs, would form the basis for an entirely new four-dimensional geometry ofspacetime—Minkowskispace.IthinkitisfairtosaythatMinkowskideservesthecreditforcompleting in1908 thespecial relativity revolution thatEinstein set inmotion threeyearsearlier.ItistoMinkowskithatweowetheconceptoftimeasthefourthdimensionofafour-dimensionalspacetime.ItstillgivesmeshiverswhenIreadthesetwopapers.

Let’s followMinkowski and consider the path of light rays that start at the origin.Imaginea flashof light—aflashbulbevent—beingsetoffat theoriginandpropagatingoutward.Afteratimetitwillhavetraveledadistancect.Wemaydescribetheflashbytheequation

TheleftsideofEq.1.32isthedistancefromtheorigin,andtherightsideisthedistancetraveledbythelightsignalintimet.Equatingthesegivesthelocusofallpointsreachedbytheflash.Theequationcanbevisualized,albeitonlywiththreedimensionsinsteadoffour, as defining a cone in spacetime. Although he didn’t quite draw the cone, he diddescribe it in detail. Here we drawMinkowski’s Light Cone (Fig. 1.11). The upward-pointingbranchiscalledthefuturelightcone.Thedownward-pointingbranchisthepastlightcone.

Nowlet’sreturntotherailroadworldofmotionstrictlyalongthexaxis.

1.5.2ThePhysicalMeaningofProperTimeTheinvariantquantityτ2isnotjustamathematicalabstraction.Ithasaphysical—evenanexperimental—meaning.Tounderstand it, considerLenny,asusual,movingalong thexaxisandArtatrestintherestframe.TheypasseachotherattheoriginO.Wealsomarkasecond pointD along Lenny’sworld line,which represents Lennymoving along the t′axis.12 All of this is shown in Fig. 1.12. The starting point along theworld line is thecommonoriginO.BydefinitionLennyislocatedatx′=0andhemovesalongthet′axis.

Figure1.12:ProperTime.Pleasereadthefootnotethatexplainsthetwomeaningsoft′inthis

diagram.

Thecoordinates(x,t) refer toArt’sframe,andtheprimedcoordinates(x′, t′) refer toLenny’s frame.The invariant τ2 is defined as t′2 −x′2 in Lenny’s frame. By definition,Lennyalwaysremainsatx′=0inhisownrestframe,andbecausex′=0atpointD,t′2−x′2isthesameast′2.Therefore,theequation

τ2=t′2−x′2

becomes

τ2=t′2,

and

τ=t′.

Butwhatist′?ItistheamountoftimethathaspassedinLenny’sframesincehelefttheorigin.Thuswefindthattheinvariantτhasaphysicalmeaning:

The invariant proper time along a world line represents the ticking of a clock movingalongthatworldline.InthiscaseitrepresentsthenumberofticksofLenny’sRolexashemovesfromtheorigintopointD.

Tocompleteourdiscussionofpropertime,wewriteitinconventionalcoordinates:

1.5.3SpacetimeIntervalThetermpropertimehasaspecificphysicalandquantitativemeaning.Ontheotherhand,I’vealsousedthetermspacetimedistanceasagenericversionof thesame idea.Goingforward,we’llstartusingamorepreciseterm,spacetimeinterval,(Δs)2,definedas

(Δs)2=−Δt2+(Δx2+Δy2+Δz2).

Todescribethespacetimeintervalbetweenanevent(t,x,y,z)andtheorigin,wewrite

s2=−t2+(x2+y2+z2).

In otherwords, s2 is just the negative of τ2, and is therefore an invariant.13 So far, thedistinctionbetweenτ2ands2hasnotbeenimportant,butitwillsooncomeintoplay.

1.5.4Timelike,Spacelike,andLightlikeSeparationsAmong the many geometric ideas that Minkowski introduced into relativity were theconcepts of timelike, spacelike, and lightlike separations between events. Thisclassificationmaybebasedontheinvariant

τ2=t2−(x2+y2+z2),

oronitsalterego

s2=−t2+(x2+y2+z2),

thespacetimeintervalthatseparatesanevent(t,x,y,z)fromtheorigin.We’lluses2.Theintervals2maybenegative,positive,orzero,andthat’swhatdetermineswhetheraneventistimelike,spacelike,orlightlikeseparatedfromtheorigin.

To gain some intuition about these categories, think of a light signal originating atAlphaCentauriattimezero.IttakesaboutfouryearsforthatsignaltoreachusonEarth.Inthisexample,thelightflashatAlphaCentauriisattheorigin,andwe’reconsideringitsfuturelightcone(thetophalfofFig.1.11).

TimelikeSeparationFirst,considerapointthatliesinsidethecone.Thatwillbethecaseifthemagnitudeofitstimecoordinate|t|isgreaterthanthespatialdistancetotheevent—inotherwords,if

−t2+(x2+y2+z2)<0.

Sucheventsarecalledtimelikerelativetotheorigin.Allpointsonthetaxisaretimelikerelative to the origin (I’ll just call them timelike). The property of being timelike isinvariant:Ifaneventistimelikeinanyframe,itistimelikeinallframes.

If aneventonEarthhappensmore than fouryears after the flashwas sent, then it’stimelike relative to the flash.Thoseeventswillbe too late tobestruckby thesignal. It

willhavepassedalready.

SpacelikeSeparationSpacelikeeventsaretheonesoutsidethecone.14Inotherwords,they’retheeventssuchthat

−t2+(x2+y2+z2)>0.

For these events the space separation from theorigin is larger than the time separation.Again,thespacelikepropertyofaneventisinvariant.

Spacelikeeventsaretoofarawayforthelightsignaltoreach.AnyeventonEarththattakesplaceearlierthanfouryearsafterthelightsignalbeganitsjourneycannotpossiblybeaffectedbytheeventthatcreatedtheflash.

LightlikeSeparationFinally,therearetheeventsonthelightcone;thoseforwhich

−t2+(x2+y2+z2)=0.

Thosearethepointsthatalightsignal,startingattheorigin,wouldreach.Apersonwhoisatalightlikeeventrelativetotheoriginwouldseetheflashoflight.

1.6HistoricalPerspective1.6.1EinsteinPeopleoftenwonderwhetherEinstein’sdeclarationthat“cisalawofphysics”wasbasedon theoretical insight or prior experimental results—in particular theMichelson-Morleyexperiment.Ofcourse,wecan’tbecertainoftheanswer.Noonereallyknowswhat’sinanotherperson’smind.EinsteinhimselfclaimedthathewasnotawareofMichelson’sandMorley’sresultwhenhewrotehis1905paper.Ithinkthere’severyreasontobelievehim.

EinsteintookMaxwell’sequationstobealawofphysics.Heknewthattheygiverisetowavelikesolutions.Atagesixteen,hepuzzledoverwhatwouldhappenifyoumovedalong with a light ray. The “obvious” answer is that you’d see a static electric andmagneticfieldwithawavelikestructurethatdoesn’tmove.Somehow,heknewthatwaswrong—that itwasnot a solution toMaxwell’sequations.Maxwell’sequationssay thatlightmovesat thespeedof light. I’m inclined tobelieve that,consistentwithEinstein’sown account, he didn’t know of theMichelson-Morley experiment when he wrote hispaper.

In modern language we would explain Einstein’s reasoning a little differently. Wewould say that Maxwell’s equations have a symmetry of some kind—some set ofcoordinate transformations under which the equations have the same form in everyreferenceframe.IfyoutakeMaxwell’sequations,whichcontainx’sandt’s,andplug in

theoldGalileanrules,

x′=x−vt

t′=t,

youwouldfindthattheseequationstakeadifferentformintheprimedcoordinates.Theydon’thavethesameformasintheunprimedcoordinates.

However, if you plug the Lorentz transformation into Maxwell’s equations, thetransformedMaxwellequationshaveexactlythesameformintheprimedcoordinatesasintheunprimedcoordinates.Inmodernlanguage,Einstein’sgreataccomplishmentwastorecognize that the symmetry structure of Maxwell’s equations is not the Galileotransformation but the Lorentz transformation. He encapsulated all of this in a singleprinciple.Inasense,hedidn’tneedtoactuallyknowMaxwell’sequations(thoughhedidknowthem,ofcourse).Allheneeded toknowis thatMaxwell’sequationsarea lawofphysics,andthat thelawofphysicsrequireslight tomovewithacertainvelocity.Fromtherehecouldjustworkwiththemotionoflightrays.

1.6.2LorentzLorentz did know about theMichelson-Morley experiment.He came upwith the sametransformationequationsbut interpreted themdifferently.Heenvisioned themaseffectsonmovingobjectscausedbytheirmotionthroughtheether.Becauseofvariouskindsofetherpressures,objectswouldbesqueezedandthereforeshortened.

Washewrong? I supposeyoucould say that in somewayhewasn’twrong.Buthecertainlydidn’thaveEinstein’svisionofasymmetrystructure—thesymmetryrequiredofspaceandtimeinorderthatitagreewiththeprincipleofrelativityandthemotionofthespeedoflight.NobodywouldhavesaidthatLorentzdidwhatEinsteindid.15Furthermore,Lorentz didn’t think it was exact. He regarded the transformation equations as a firstapproximation.Anobjectmovingthroughafluidofsomekindwouldbeshortened,andthefirstapproximationwouldbetheLorentzcontraction.LorentzfullyexpectedthattheMichelson-Morley experimentwas not exact.He thought therewould be corrections tohigherpowersofv/c,andthatexperimentaltechniqueswouldeventuallybecomepreciseenoughtodetectdifferencesinthevelocityoflight.ItwasEinsteinwhosaidthisisreallyalawofphysics,aprinciple.

1Sometimeswe’llusetheabbreviationIRFforinertialreferenceframe.

2Wecouldalsodescribethisas“mytrajectoryinyourframe.”

3Thisstatementmaysoundglib.Thefactisthatmanyoftheworld’smosttalentedphysiciststriedtomakethingsworkoutwithoutgivingupontheequationt′=t.Theyallfailed.

4ThisisactuallyaslightvariationofEinstein’sapproach.

5Youcanthinkofthetaxisbeingcalibratedinlight-secondsinsteadofsecondsifyouprefer,butitamountstothesamething.

6Onceagain,thisisactuallyaslightvariationofEinstein’sapproach.

7Forv/c=10−5onefinds

8We’re talking about a fixed difference in orientation, not a situationwhere either frame is rotatingwith a nonzeroangularvelocity.

9Ifyoudon’tbelieveme,asktheguywhosoldittomeonCanalStreetfortwenty-fivebucks.

10InwhatfollowswewillpretendthatArtandLennywerebothbornatthesamespacetimeevent(labeledO)inFig.1.8.

11We’reusingthetermspacetimedistanceinagenericsense.Lateron,we’llswitchtothemoreprecisetermspropertimeandspacetimeinterval.

12Weusethelabelt′intwoslightlydifferentwaysinthisdiscussion.Itsprimarymeaningis“Lenny’st′coordinate.”Butwealsouseittolabelthet′axis.

13Signconventionsinrelativityarenotasconsistentaswewouldlike;someauthorsdefines2tohavethesamesignasτ2.

14Onceagain,we’reusing theshorthand term“spacelikeevent” tomean“event that is spacelikeseparated fromtheorigin.”

15IncludingLorentzhimself,Ibelieve.

Lecture2

Velocitiesand4-Vectors

Art:Thatstuffisincrediblyfascinating!Ifeelcompletelytransformed.

Lenny:Lorentztransformed?

Art:Yeah,positivelyboosted.

It’s true,when thingsmove at relativistic velocities they become flat, at least from theviewpointoftherestframe.Infact,astheyapproachthespeedoflighttheyshrinkalongthedirectionofmotiontoinfinitelythinwafers,althoughtothemselvestheylookandfeelfine.Cantheyshrinkevenpastthevanishingpointbymovingfasterthanlight?Well,no,forthesimplereasonthatnophysicalobjectcanmovefasterthanlight.Butthatraisesaparadox:

ConsiderArt at rest in the railway station.The train containingLennywhizzes pasthimat90percentofthespeedoflight.Theirrelativevelocityis0.9c.InthesamerailcarwithLenny,Maggieisridingherbicyclealongtheaislewithavelocityof0.9crelativetoLenny.Isn’titobviousthatsheismovingfasterthanlightrelativetoArt?InNewtonianphysics we would add Maggie’s velocity to Lenny’s in order to compute her velocityrelative toArt.Wewould findhermovingpastArtwithvelocity1.8c,almost twice thespeedoflight.Clearlythereissomethingwronghere.

2.1AddingVelocitiesTo understand what’s wrong, we will have to make a careful analysis of how Lorentztransformations combine.Our setup nowconsists of three observers:Art at rest,LennymovingrelativetoArtwithvelocityv,andMaggiemovingrelativetoLennywithvelocityu.Wewillworkinrelativisticunitswithc=1andassumethatvanduarebothpositiveandlessthan1.OurgoalistodeterminehowfastMaggieismovingwithrespecttoArt.Fig.2.1showsthissetup.

There are three framesof reference and three sets of coordinates.Let (x, t) beArt’scoordinatesintherestframeoftherailstation.Let(x′,t′)becoordinatesinLenny’sframe—theframeatrestinthetrain.Andfinallylet(x″,t″)beMaggie’scoordinatesthatmovewith her bicycle. Each pair of frames is related by a Lorentz transformation with theappropriatevelocity.ForexampleLenny’sandArt’scoordinatesarerelatedby

Figure2.1:CombiningVelocities.

Wealsoknowhowtoinverttheserelations.Thisamountstosolvingforxandtintermsofx′andt′.I’llremindyouthattheresultis

2.1.1MaggieOurthirdobserverisMaggie.WhatweknowaboutMaggieisthatsheismovingrelativetoLennywithrelativevelocityu.WeexpressthisbyaLorentztransformationconnectingLenny’sandMaggie’scoordinates,thistimewithvelocityu:

Ourgoal is tofindthetransformationrelatingArt’sandMaggie’scoordinates,andfromthattransformationreadofftheirrelativevelocities.Inotherwords,wewanttoeliminateLenny.1Let’sstartwithEq.2.5,

Now,substituteforx′andt′ontherightside,usingEqs.2.1and2.2,

andcombinethedenominators,

Nowwecometothemainpoint:determiningMaggie’svelocityinArt’sframerelativetoArt. It’s not entirely obvious that Eq. 2.7 has the form of a Lorentz transformation (itdoes), but fortunately we can avoid the issue for the moment. Observe that Maggie’sworld line is givenby the equationx″=0.For that tobe true,weonlyneed to set thenumeratorofEq.2.7tozero.Solet’scombinethetermsinthenumerator,

(1+uv)x−(v+u)t=0,

whichresultsin

Nowitshouldbeobvious:Eq.2.8istheequationforaworldlinemovingwithvelocity(u+v)/(1+uv).Thus,callingMaggie’svelocityinArt’sframew,wefind

It’snowfairlyeasytocheckthatArt’sframeandMaggie’sframereallyarerelatedbyaLorentztransformation.Iwillleaveitasanexercisetoshowthat

and

Tosummarize: IfLennymoveswithvelocityv relative toArt, andMaggiemoveswithvelocityuwithrespecttoLenny,thenMaggiemoveswithvelocity

with respect to Art. We will analyze this shortly, but first let’s express Eq. 2.12 inconventional units by enforcing dimensional consistency. The numerator u + v isdimensionallycorrect.However,theexpression1+uvinthedenominatorisnotbecause1is dimensionless and both u and v are velocities.We can easily restore dimensions byreplacingu and v with u/c and v/c. This gives the relativistic law for the addition ofvelocities,

Let’s compare the resultwithNewtonian expectations.Newtonwould have said that tofindMaggie’svelocityrelativetoArtweshouldjustaddutov.Thatisindeedwhatwedoin the numerator in Eq. 2.13. But relativity requires a correction in the form of thedenominator,(1+uv/c2).

Let’slookatsomenumericalexamples.Firstwe’llconsiderthecasewhereuandvaresmallcompared to thespeedof light.Forsimplicity,we’lluseEq.2.9,wherevelocitiesaredimensionless.Justrememberthatuandvarevelocitiesmeasuredinunitsofthespeedof light.Supposeu=0.01,1percentof the speedof light, andv is also0.01.PluggingthesevaluesintoEq.2.9gives

or

TheNewtonian answerwould, of course, have been 0.02, but the relativistic answer isslightly less. In general, the smaller u and v, the closer will be the relativistic andNewtonianresult.

Butnowlet’sreturntotheoriginalparadox:IfLenny’strainmoveswithvelocityv=

0.9relativetoArt,andMaggie’sbicyclemoveswithu=0.9relativetoLenny,shouldwenotexpectthatMaggieismovingfasterthanlightrelativetoArt?Substitutingappropriatevaluesforvandu,weget

or

Thedenominatorisslightlybiggerthan1.8,andtheresultingvelocityisslightlylessthan1.Inotherwords,wehavenotsucceededinmakingMaggiegofaster thanthespeedoflightinArt’sframe.

Whilewe’reatit, let’ssatisfyourcuriosityaboutwhatwouldhappenifbothuandvequalthespeedoflight.Wesimplyfindthatwbecomes

or

Even if Lenny could somehowmove at the speed of light relative toArt, andMaggiecouldmove at the speedof light relative toLenny, still shewould still notmove fasterthanlightrelativetoArt.

2.2LightConesand4-VectorsAswesawinSection1.5,thepropertime

τ2=t2−(x2+y2+z2)

anditsalter-ego,thespacetimeintervalrelativetotheorigin,

s2=−t2+(x2+y2+z2),

are invariant quantities under general Lorentz transformations in four-dimensionalspacetime.Inotherwords,thesequantitiesareinvariantunderanycombinationofLorentzboostsandcoordinaterotations.2We’llsometimeswriteτinabbreviatedformas

Thisisprobablythemostcentralfactaboutrelativity.

2.2.1HowLightRaysMoveBackinLecture1,wediscussedspacetimeregionsandthetrajectoriesoflightrays.Fig.2.2 illustrates this idea in slightly greater detail. The different kinds of separationcorrespond to a negative, positive, or zero value of the invariant quantity s. We alsodiscoveredtheinterestingresultthatiftwopointsinspacetimehaveaseparationofzero,thisdoesnotmeantheyhavetobethesamepoint.Zeroseparationsimplymeansthatthetwopointsarerelatedbythepossibilityofalightraygoingfromoneofthemtotheother.

That’soneimportantconceptofhowalightraymoves—itmovesinsuchawaythatthepropertime(alternately,spacetimeinterval)alongitstrajectoryiszero.Thetrajectoryofalightraythatstartsattheoriginservesasakindofboundarybetweenregionsofspacetimethataretimelikeseparatedfromtheoriginandthosethatarespacelikeseparated.

Figure2.2:FutureLightCone.Relativetotheorigin:Pointaistimelikeseparated,pointbisspacelikeseparated,andpointPislightlikeseparated.Onlytwospacedimensionsare

shown.

2.2.2Introductionto4-VectorsWehavea reasonforbringing theyandz spatialdimensionsback into thepicture.Themathematicallanguageofrelativityreliesheavilyonsomethingcalleda4-vector,and4-vectorsincorporateallthreedimensionsofspace.We’llintroducethemhereanddevelopthemfurtherinLecture3.

Themostbasic exampleof avector in threedimensions is the intervalbetween twopointsinspace.3Giventwopoints,there’savectorthatconnectsthem.Ithasadirectionandamagnitude.Itdoesn’tmatterwherethevectorbegins.Ifwemoveitaround,it’sstillthesamevector.Youcanthinkofitasanexcursionbeginningattheoriginandendingatsomepoint inspace.Ourvectorhascoordinates, in thiscasex,y,andz, thatdefine thelocationofthefinalpoint.

NewNotationOfcourse,thenamesofourcoordinatesdon’tnecessarilyhavetobex,y,andz.We’refreeto rename them. For instance, we could call them Xi, with i being 1, 2, or 3. In thisnotation,wecouldwrite

Orwecouldwrite

instead,andweplantousethatnotationextensively.Sincewe’llbemeasuringspaceandtimerelative tosomeorigin,weneed toadda timecoordinate t.Asa result, thevectorbecomesafour-dimensionalobject,a4-vectorwithonetimecomponentandthreespacecomponents.Byconvention,thetimecomponentisfirstonthelist:

where(X0,X1,X2,X3)hasthesamemeaningas(t,x,y,z).Remember,thesesuperscriptsarenot exponents. The coordinateX3means “the third space coordinate, the one that’softenwrittenasz.”ItdoesnotmeanX×X×X.Thedistinctionbetweenexponentsandsuperscripts should be clear from the context. From now on, when we write four-dimensionalcoordinates,thetimecoordinatewillcomefirst.

Noticethatwe’reusingtwoslightvariantsofourindexnotation:

•Xμ:AGreekindexsuchasμmeansthattheindexrangesoverallfourvalues0,1,2,and3.

•Xi:ARomanindexsuchasimeansthattheindexonlyincludesthethreespatialcomponents1,2,and3.

Whataboutthepropertimeandspacetimeintervalfromtheorigin?Wecanwritethemas

τ2=(X0)2−(X1)2−(X2)2−(X3)2

and

s2=−(X0)2+(X1)2+(X2)2+(X3)2.

There’s no new content here, just notation.4 But notation is important. In this case, itprovidesawaytoorganizeour4-vectorsandkeepourformulassimple.Whenyouseeaμindex, the index runs over the four possibilities of space and time.When you see an iindex,theindexonlyrunsoverspace.JustasXicanbethoughtofasabasicversionofavector in space,Xμ, with four components, represents a 4-vector in spacetime. Just asvectorstransformwhenyourotatecoordinates,4-vectorsLorentz-transformwhenyougofrom one moving frame to another. Here is the Lorentz transformation in our newnotation:

We can generalize this to a rule for the transformation properties of any 4-vector. Bydefinitiona4-vectorisanysetofcomponentsAμthattransformaccordingto

under a boost along the x axis.We also assume that the spatial componentsA1,A2,A3transformasaconventional3-vectorunderrotationsofspaceandthatA0isunchanged.

Justlike3-vectors,4-vectorscanbemultipliedbyanordinarynumberbymultiplyingthe components by that number. We may also add 4-vectors by adding the individualcomponents.Theresultofsuchoperationsisa4-vector.

4-VelocityLet’slookatanother4-vector.Insteadoftalkingaboutcomponentsrelativetoanorigin,this timewe’ll consider a small interval along a spacetime trajectory. Eventually,we’llshrink this interval down to an infinitesimal displacement; for now think of it as beingsmallbut finite.Fig.2.3shows thepicturewehave inmind.The interval that separatespoints a and b along the trajectory is ΔXμ. This simplymeans the changes in the fourcoordinatesgoingfromoneendofavectortotheother.ItconsistsofΔt,Δx,Δy,andΔz.

Nowwe’rereadytointroducethenotionof4-velocity.Four-dimensionalvelocityisalittledifferentfromthenormalnotionofvelocity.Let’stakethecurveinFig.2.3tobethetrajectoryofaparticle.I’minterestedinanotionofvelocityataparticularinstantalongsegment .Ifwewereworkingwithanordinaryvelocity,wewouldtakeΔxanddivideitbyΔt. Thenwewould take the limit as Δt approaches zero. The ordinary velocity hasthreecomponents:thexcomponent,theycomponent,andthezcomponent.There isnofourthcomponent.

We’llconstructthefour-dimensionalvelocityinasimilarway.StartwiththeΔXμ.Butinstead of dividing them byΔt, the ordinary coordinate time,we’ll divide them by thepropertimeΔτ.ThereasonisthatΔτisinvariant.Dividinga4-vectorΔXμbyaninvariantpreserves the transformation properties of the 4-vector. In other words, ΔXμ/Δτ is a 4-vector,whileΔXμ/Δtisnot.

Figure2.3:SpacetimeTrajectory(particle).

Inorder todistinguishthe4-velocityfromtheordinary3-velocitywe’llwrite itasUinsteadofV.UhasfourcomponentsUμ,definedby

We’llhaveacloserlookat4-velocityinthenextlecture.Itplaysanimportantroleinthetheoryof themotionofparticles.Fora relativistic theoryofparticlemotion,we’llneednewnotionsofoldconceptssuchasvelocity,position,momentum,energy,kineticenergy,andsoon.WhenweconstructrelativisticgeneralizationsofNewton’sconcepts,we’lldoitintermsof4-vectors.

1Don’tpanic.It’sonlyLenny’svelocitywe’retryingtogetridof.

2Weonlyshowedthisexplicitlyforτ,butthesameargumentsapplytos.

3We’rejusttalkingaboutvectorsinspace,nottheabstractstatevectorsofquantummechanics.

4Youmaywonderwhywe’reusingsuperscripts insteadofsubscripts.Lateron(Sec.4.4.2)we’ll introducesubscriptnotationwhosemeaningisslightlydifferent.

Lecture3

RelativisticLawsofMotion

Lennywassittingonabarstoolholdinghisheadinhishandswhilehiscellphonewasopenedtoanemailmessage.

Art:Whatsamatter,Lenny?Toomuchbeermilkshake?

Lenny:Here,Art,takealookatthisemail.Igetacouplelikeiteveryday.

EmailMessage:1

DearProfessorSusskino[sic],

EinsteinmadeabadmistakeandIdiscovered it. Iwrote toyour friendHawkins[sic]buthedidn’tanswer.

LetmeexplainEinsteins’[sic]mistake.Forceequalsmasstimesacceleration.SoIpushsomethingwithaconstantforceforalongtimetheaccelerationisconstantsoifIdoitlongenoughthevelocitykeepsincreasing.IcalculatedthatifIpusha220 pound (that’s my weight. I should probably go on a diet) person with acontinuousforceof224.809poundsinahorizontaldirection,afterayearhewillbemovingfasterthanthespeedoflight.AllIusedwasNewtons’[sic]equationF=MA. So Einstein was wrong since he said that nothing canmove faster thanlight.IamhopingyouwillhelpmepublishthisasIamcertainthatthephycicist’s[sic]needtoknowit.IhavealotofmoneyandIcanpayyou.

Art:Geez,that’sawfullystupid.

Bytheway,what’swrongwithit?

The answer to Art’s question is that we’re not doing Newton’s theory; we’re doingEinstein’stheory.Physics,includingthelawsofmotion,force,andacceleration,allhadtoberebuiltfromthegroundup,inaccordwiththeprinciplesofspecialrelativity.

We’re now ready to tackle that project.We will be especially interested in particlemechanics—howparticlesmoveaccordingtospecialrelativity.Toaccomplishthis,we’llneed to corral awide rangeof concepts, includingmany fromclassicalmechanics.Ourplanistodiscusseachideaseparatelybeforeknittingthemalltogetherattheend.

Relativitybuildson the classicalnotionsof energy,momentum, canonicalmomenta,Hamiltonians,andLagrangians;theprincipleofleastactionplaysacentralrole.Thoughweoffer somebrief remindersof these ideasaswego,weassumeyou remember themfromthefirstbookofthisseries,TheTheoreticalMinimum:WhatYouNeedtoKnowtoStartDoingPhysics.Ifnot,thiswouldbeanexcellenttimetoreviewthatmaterial.

3.1MoreAboutIntervalsWediscussedtimelikeandspacelikeintervals inLectures1and2.Asweexplained, theinterval or separation between two points in spacetime is timelike when the invariantquantity

islessthanzero,thatis,whenthetimecomponentoftheintervalisgreaterthanthespacecomponent.2Ontheotherhand,whenthespacetimeinterval(Δs)2betweentwoeventsisgreater thanzero, theopposite is trueand the interval iscalledspacelike.This ideawaspreviouslyillustratedinFig.2.2.

Figure3.1:SpacelikeInterval.

3.1.1SpacelikeIntervalsWhen isgreater than(Δt)2, thespatial separationbetween the twoevents isgreaterthan their time separation and (Δs)2 is greater than zero. You can see this in Fig. 3.1,wheretheseparationbetweeneventsaandbisspacelike.Thelineconnectingthosetwopointsmakesananglesmallerthan45degreestothexaxis.

Spacelike intervalshavemore space in them than time.Theyalsohave thepropertythatyoucannot find a reference frame inwhich the twoevents are located at the samepositioninspace.Whatyoucandoinsteadisfindareferenceframe inwhich theybothhappenatexactlythesametimebutatdifferentplaces.Thisamountstofindingaframewhosex′axispassesthroughbothpoints.3Butthere’samuchbiggersurpriseinstore.Inthe t, x frame of Fig. 3.1, event a happens before event b. However, if we Lorentz-transformto the t′,x′ frameof thediagram,eventbhappensbeforeeventa.Their timeorder is actually reversed. This brings into sharp focus what wemean by relativity ofsimultaneity:There’snoinvariantsignificancetotheideathatoneeventhappenslaterorearlierthantheotheriftheyarespacelikeseparated.

Figure3.2:TimelikeTrajectory.

3.1.2TimelikeIntervalsParticleswithnonzeromassmovealongtimeliketrajectories.Toclarifywhatthismeans,Fig. 3.2 shows an example. If we follow the path from point a to point b, every littlesegment of that path is a timelike interval. Saying that a particle follows a timeliketrajectoryisanotherwayofsayingitsvelocitycanneverreachthespeedoflight.

Whenanintervalistimelike,youcanalwaysfindareferenceframeinwhichthetwoeventshappenatthesameplace—wheretheyhavethesamespacecoordinatesbutoccuratdifferent times. In fact,allyouneed todo ischoosea referenceframewhere the lineconnecting the two points is at rest—a framewhose t′ axis coincideswith the line thatconnectsthetwopoints.4

3.2ASlowerLookat4-VelocityBack inLecture2,we introducedsomedefinitionsandnotation for the4-velocity.Nowit’stimetodevelopthatideafurther.Thecomponentsof4-velocitydXμ/dτareanalogoustothecomponentsdXi/dtoftheusualcoordinatevelocity,exceptthat

•The4-velocityhas—bigsurprise—fourcomponentsinsteadofthree,and

•Insteadofreferringtoarateofchangewithrespecttocoordinatetime,4-velocity

referstorateofchangewithrespecttopropertime.

Asaproperlydefined4-vector,thecomponentsof4-velocitytransforminthesamewayastheprototypical4-vector

(t,x,y,z)

or

(X0,X1,X2,X3)

inournewnotation.Inotherwords,theircomponentstransformaccordingtotheLorentztransformation.Byanalogytoordinaryvelocities,4-velocitiesareassociatedwithsmallorinfinitesimalsegmentsalongapath,oraworldline,inspacetime.Eachlittlesegmenthasitsownassociated4-velocityvector.We’llwrite theordinary three-dimensionalvelocityas

or

andthe4-velocityas

What is the connection between 4-velocity and ordinary velocity? Ordinary non-relativistic velocity has, of course, only three components. This leads us to expect thatthere’s something funny about the fourth (whichwe label the zeroth) component. Let’sstartwithU0.Let’swriteitintheform

NowrecallthatX0isjustanotherwayofwritingt,sothatthefirstfactorontherightsideisjust1.Thuswecanwrite

or

Thenextstepistorecallthat sothat

or

where istheusual3-vectorvelocity.NowgoingbacktoEq.3.3andusingEq.3.4,wefindthat

and

Weseehereanewmeaningtotheubiquitousfactor

whichappearsinLorentztransformations,Lorentzcontraction,andtimedilationformulas.Weseethatit’sthetimecomponentofthe4-velocityofamovingobserver.

WhatshouldwemakeofthetimecomponentofU?WhatwouldNewtonhavemadeofit?Supposetheparticlemovesveryslowlywithrespect to thevelocityof light; inotherwords,v<<1.ThenitisclearthatU0isverycloseto1.IntheNewtonianlimititisjust1andcarriesnospecial interest. Itwouldhaveplayednorole inNewton’s thinking.Nowlet’sturntothespatialcomponentsofU.Inparticular,wecanwriteU1as

whichcanalsobewrittenas

Thefirstfactor,dx/dt,isjusttheordinaryxcomponentofvelocity,V1.ThesecondfactorisagaingivenbyEq.3.5.Puttingthemtogether,wehave

Again,let’saskwhatNewtonwouldhavethought.Forverysmallv,weknowthatis very close to 1. Therefore the space components of the relativistic velocity arepracticallythesameasthecomponentsoftheordinary3-velocity.

There’s one more thing to know about the 4-velocity: Only three of the fourcomponents Uμ are independent. They are connected by a single constraint. We canexpressthisintermsofaninvariant.Justasthequantity

(X0)2−(X1)2−(X2)2−(X3)2

is an invariant, the corresponding combination of velocity components is invariant,namely

(U0)2−(U1)2−(U2)2−(U3)2.

Is this quantity interesting? I’ll leave it as an exercise for you to show that it’s alwaysequalto1,

Here’sasummaryofourresultsabout4-velocity:

4-VelocitySummary:

Theseequationsshowyouhowtofindthecomponentsof4-velocity.Inthenonrelativisticlimitwherevisclosetozero,theexpression isverycloseto1,andthetwonotionsofvelocity,UiandVi, are the same.However, as the speedofanobject approaches thespeedoflight,UibecomesmuchbiggerthanVi.Everywherealongitstrajectory,aparticleischaracterizedbya4-vectorofpositionXμanda4-vectorofvelocityUμ.

Ourlistofingredientsisnearlycomplete.Wehaveonlyonemoreitemtoaddbeforewetacklethemechanicsofparticles.

Exercise3.1:Fromthedefinitionof(Δτ)2,verifyEq.3.7.

3.3MathematicalInterlude:AnApproximationToolA physicist’s tool kit is not complete without some good approximation methods. Themethod we describe is an old warhorse that’s indispensable despite its simplicity. Thebasis for the approximation is the binomial theorem.5 I won’t quote the theorem in itsgeneral form; just a couple of examples will do. What we will need is a goodapproximationtoexpressionslike

(1+a)p

that’s accuratewhena ismuch smaller than1. In this expression,p can be any power.Let’sconsidertheexamplep=2.Itsexactexpansionis

(1+a)2=1+2a+a2.

Whena issmall, thefirst term(in thiscase1) is fairlyclose to theexactvalue.Butwewanttodoalittlebitbetter.Thenextapproximationis

(1+a)2≈1+2a.

Let’s try thisout fora = .1The approximationgives (1+ .1)2 ≈ 1.2whereas the exactansweris1.21.Thesmallerwemakea,thelessimportantisthea2termandthebettertheapproximation.Let’sseewhathappenswhenp=3.Theexactexpansionis

(1+a)3=1+3a+3a2+a3

andthefirstapproximationwouldbe

(1+a)3≈1+3a.

Fora=.1theapproximationwouldgive1.3,whiletheexactansweris1.4641.Notbadbutnotgreat.Butnowlet’stryitfora=.01.Theapproximateansweris

(1.01)3≈1.03,

whiletheexactansweris

(1.01)3=1.030301,

whichismuchbetter.

Now, without any justification, I will write the general answer for the firstapproximationforanyvalueofp:

Ingeneral, ifp isnotan integer, theexactexpression isan infiniteseries.Nevertheless,Eq.3.11ishighlyaccurateforsmallaandgetsbetterandbetterasabecomessmaller.

We’lluseEq.3.11heretoderiveapproximationsfortwoexpressionsthatshowupallthetimeinrelativitytheory:

and

where v represents the velocity of a moving object or reference frame.We do this bywritingEqs.3.12and3.13intheform

Inthefirstcasetherolesofaandpareplayedbya=−v2andp=1/2.Inthesecondcasea=−v2butp=−1/2.Withthoseidentificationsourapproximationsbecome

We’vewrittentheseexpressionsinrelativisticunits,sothatv isadimensionlessfractionofthespeedoflight.Inconventionalunitstheytaketheform

Letmepauseforamomenttoexplainwhywe’redoingthis.Whyapproximatewhenit’snot difficult, especially with modern calculators, to calculate the exact expression toextremely high precision?We’re not doing it tomake calculations easy (though itmayhave that effect).We are constructing a new theory for describingmotion at very largevelocities.However,we are not free to do anythingwe like;we are constrained by thesuccessoftheoldertheory—Newtonianmechanics—fordescribingmotionmuchslower

thanthespeedoflight.OurrealpurposeinapproximationslikeEqs.3.16and3.17istoshow that the relativistic equations approach theNewtonian equationswhenv/c is verysmall.Foreasyreference,herearetheapproximationswewilluse.

Approximations:

Fromnowonwewilldispensewiththeapproximationsymbol≈andusetheapproximateformulasonlywhentheyareaccurateenoughtomeritanequalsign.

3.4ParticleMechanicsWithalltheseingredientsinplace,we’rereadytotalkaboutparticlemechanics.Thewordparticleoftenconjuresuptheimageofelementaryparticlessuchaselectrons.However,we’reusingthewordinamuchbroadersense.Aparticlecanbeanythingthatholdsitselftogether.Elementaryparticles certainlymeet this criterion, butmanyother thingsdo aswell:theSun,adoughnut,agolfball,ormyemailcorrespondent.Whenwespeakoftheposition or velocity of a particle,whatwe reallymean is the position or velocity of itscenterofmass.

BeforebeginningthenextsectionIurgeyoutorefreshyourknowledgeoftheprincipleofleastaction,Lagrangianmechanics,andHamiltonianmechanicsifyouhaveforgottenthem.VolumeIoftheTheoreticalMinimumseriesisoneplacetofindthem.

Figure3.3:TimelikeParticleTrajectory.

3.4.1PrincipleofLeastActionTheprincipleofleastactionanditsquantummechanicalgeneralizationmaybethemostcentralideainallofphysics.Allthelawsofphysics,fromNewton’slawsofmotion,toelectrodynamics,tothemodernso-calledgaugetheoriesoffundamentalinteractions,are

based on the action principle. Dowe know exactly why? I think the roots of it are inquantumtheory,butsufficeittosaythatitisdeeplyconnectedwithenergyconservationand momentum conservation. It guarantees the internal mathematical consistency ofequationsofmotion.Idiscussedactioningreatdetailinthefirstbookofthisseries.Thiswillbeaquickandabbreviatedreview.

Let’s briefly review how the action principle determines themotion of a particle inclassicalmechanics.Theactionisaquantitythatdependsonthetrajectoryoftheparticleasitmovesthroughspacetime.Youcanthinkofthistrajectoryasaworldline,suchasthatshowninFig.3.2,whichwe’vereproducedhereasFig.3.3forconvenience.Thisdiagramisagoodmodelforourdiscussionofaction.However,keepinmindthatwhenplottingthepositionofasystem,thexaxisrepresentstheentirespatialdescriptionofthesystem—xcouldstandforaone-dimensionalcoordinate,butitcouldalsostandforaspatial3-vector.(Itcouldevenrepresentall thespatialcoordinatesofalargenumberofparticles,buthereweconsideronlyasingleparticle).Wejustcallitspaceorcoordinatespace.Asusual, the vertical axis represents coordinate time, and the trajectory of a system is acurve.

ForaparticletheLagrangiandependsonthepositionandvelocityoftheparticleand,most important, it is built out of thekinetic andpotential energy.The curve inFig. 3.3represents thetimelikeworldlineofasingleparticlewithnonzeromass.We’llstudyitsbehaviorasitgoesfrompointatopointb.6

Our development of the least action principle will closely parallel what we did inclassicalmechanics.Theonlyrealdifferenceisthatwenowaddtherequirementofframeindependence; we want our laws of physics to be the same in every inertial referenceframe.Wecanachievethatbycastingourlawsintermsofquantitiesthatarethesameinevery reference frame. In other words, the action should be invariant, and that’s bestaccomplishedifitsconstituentsareinvariant.

Theleastactionprinciplesaysthatifasystemstartsoutatpointaandendsupatpointb, itwill“choose”aparticularkindofpathamongall thepossiblepaths.Specifically, itchooses the path that minimizes the quantity we call action.7 Action is built upincrementally as a sum over the trajectory. Each little segment of the trajectory has aquantityofactionassociatedwithit.Wecalculatetheactionovertheentirepathfromatobbyaddingalltheselittlechunksofactiontogether.Asweshrinkthesegmentsdowntoinfinitesimalsize,thesumbecomesanintegral.Thisideaofactionbeinganintegraloveratrajectorywithfixedendpointsissomethingwetakedirectlyfromprerelativityphysics.Thesameistruefortheideathatthesystemsomehowchoosesthepaththatminimizestheactionintegral.

3.4.2AQuickReviewofNonrelativisticActionRecalltheformulafortheactionofanonrelativisticparticlefromVolumeI.Theactionisan integral along the trajectoryof a system, and the integrand is called theLagrangian,denoted .Insymbols,

Asarule,theLagrangianisafunctionofthepositionandvelocityalongthetrajectory.Inthesimplestcaseofaparticlewithnoforcesactingonit,theLagrangianisjustthekineticenergy .Inotherwords,

wheremisthemassoftheparticleandvistheinstantaneousvelocity.8Theactionforthenonrelativisticparticleis

Noticethattheactionisproportionaltothemass:Foramarbleandabowlingballmovingonthesametrajectory,theactionofthebowlingballislargerthantheactionofthemarblebytheratiooftheirmasses.

3.4.3RelativisticActionThenonrelativisticdescriptionofparticlemotion ishighlyaccurateforparticlesmovingmuch slower than light, but breaks down badly for relativistic particles with largervelocities.Tounderstand relativisticparticlesweneed to startover from thegroundup,but one thing stays the same. The theory of relativistic motion is based on the actionprinciple.

Howthentocalculatetheactionofarelativisticparticleforeachlittlesegmentalongthetrajectory?Tomakesurethelawsofmotionarethesameineveryreferenceframe,theactionshouldbeinvariant.Butthere’sreallyonlyonethingthat’sinvariantwhenaparticlemovesfromsomepositiontoaneighboringposition:thepropertimeseparatingthetwo.Theproper timefromonepoint toanother isaquantity thatallobserverswillagreeon.TheywillnotagreeontheΔt’sorthe buttheywillagreeonΔτ.Soagoodguess,andit’stherightguess,istotaketheactiontobeproportionaltothesumofallthelittleΔτ’s.Thatsumissimplythetotalpropertimealongtheworldline.Inmathematicallanguage,

Action=−constant×∑Δτ

wherethesumisfromoneendofatrajectorytotheother—frompointatopointbinFig.3.3.We’llcomebacktotheconstantfactorandtheminussignshortly.

Once the action has been constructed, we do exactly the same thing as in classicalmechanics:Holdingthetwoendpointsfixed,wewiggletheconnectingpatharounduntilwefindthepaththatproducesthesmallestaction.BecausetheactionisbuiltupfromtheinvariantquantitiesΔτ,everyobserverwillagreeaboutwhichpathminimizesit.

Whatisthemeaningoftheconstantfactorintheaction?Tounderstandit,let’sgobackto the nonrelativistic case (Eq. 3.22) where we saw that the action for a given path is

proportional to themassof theparticle. Ifwewish toreproducestandardnonrelativisticphysics in the limit of small velocities, the relativistic action will also have to beproportionaltothemassoftheparticle.Thereasonfortheminussignwillbecomeclearasweproceed.Let’strydefiningtheactionas

Action=−m∑Δτ.

Nowlet’simagineeachlittlesegmentalongthetrajectoryshrinkingdowntoinfinitesimalsize.Mathematically,thismeansweconvertoursumtoanintegral,

andΔτhasbecomeitsinfinitesimalcounterpartdτ.We’veaddedthelimitsaandbtoshowthat theintegralrunsfromoneendof thetrajectorytotheother.WealreadyknowfromEq.3.4that

andwecanusethistoreplacedτintheactionintegralwith ,resultingin

Inournewnotation,v2becomes ,andtheactionintegralbecomes

I’musingthesymbol tomean ,wherethedotmeansderivativewithrespectto ordinary coordinate time. We might also write that with being theordinarythree-dimensionalvelocityvector.

We’ve converted the action integral to something almost familiar: an integral of afunction of velocity. It has the same general form as Eq. 3.20, but now instead of theLagrangianbeing thenonrelativistic kinetic energyofEq. 3.21, it has the slightlymorecomplicatedform

or

BeforegettingmorefamiliarwiththisLagrangian,let’sputbacktheappropriatefactorsofctorestorethecorrectdimensionsforconventionalunits.Tomaketheexpressiondimensionally consistent,wemust divide the velocity components by c. In addition, tomake theLagrangian have units of energy,we need tomultiply thewhole thing by c2.Thusinconventionalunits,

orinexplicitdetail,

Incaseyoudidn’tnotice,Eq.3.26marksthefirstappearanceoftheexpressionmc2.

3.4.4NonrelativisticLimitWewould like to show that in the limit of small velocities, the behavior of relativisticsystemsreducestoNewtonianphysics.SinceeverythingaboutthatmotionisencodedintheLagrangian,weonlyneedtoshowthatforsmallvelocitiestheLagrangianreducestoEq. 3.21. It was exactly for this and other similar purposes that we introduced theapproximationsofEqs.3.16and3.17,

IfweapplythefirstofthesetoEq.3.26,theresultis

whichwemayrewriteas

Thefirstterm, ,isgoodoldkineticenergyfromNewtonianmechanics:exactlywhatwe expect the nonrelativisticLagrangian to be. Incidentally, hadwenot put the overallminussignintheaction,wewouldnothavereproducedthistermwiththecorrectsign.

What about the additional term −mc2? Two questions come to mind. The first iswhetheritmakesanydifferencetothemotionofaparticle.Theansweristhattheaddition(orsubtraction)ofaconstanttoaLagrangian,foranysystem,makesnodifferenceatall.Wemayleaveitordeleteitwithoutanyconsequencesforthemotionofthesystem.Thesecondquestionis,whatdoesthattermhavetodowiththeequationE=mc2?We’llseethatshortly.

3.4.5RelativisticMomentumMomentumisanextremelyimportantconceptinmechanics,notleastofallbecauseitisconserved for a closed system.Moreover, if we divide a system into parts, the rate ofchangeofthemomentumofapartistheforceonthatpartduetotherestofthesystem.

Momentum,oftendenotedby ,isa3-vectorthatinNewtonianphysicsisgivenbythemasstimesthevelocity,

Relativisticphysicsisnodifferent;momentumisstillconserved.Buttherelationbetweenmomentumandvelocityismorecomplicated.Inhis1905paper,Einsteinworkedouttherelationinaclassicargumentthatwasnotonlybrilliantbutcharacteristicallysimple.Hebeganbyconsideringanobjectintherestframe,andthenimaginedtheobjectsplit intotwo lighter objects, each of which moved so slowly that the whole process could be

understoodbyNewtonianphysics.Thenhe imaginedobserving that sameprocess fromanotherframeinwhichtheinitialobjectwasmovingwithalargerelativisticvelocity.Thevelocity of the final objects could easily be determined by boosting (Lorentz-transforming)theknownvelocitiesintheoriginalframe.Then,puttingthepiecestogether,he was able to deduce the expression for their momenta by assuming momentumconservationinthemovingframe.

I will use a less elementary, perhaps less beautiful argument, but one that is moremodern and much more general. In classical mechanics (go back to Volume I) themomentumofasystem—sayaparticle—isthederivativeoftheLagrangianwithrespecttothevelocity.Intermsofcomponents,

Forreasonswehavenotyetexplained,weoftenprefertowriteequationsofthiskindas

wherePihasasubscriptinsteadofasuperscript;we’vewrittenithereonlyforreference.We’llexplainthemeaningofupperandlowerindicesinSection5.3.

Tofindtherelativisticexpressionformomentumofaparticle,allwehavetodoistoapplyEq.3.28totheLagrangianinEq.3.27.Forexample,thexcomponentofmomentumis

Carryingoutthederivative,weget

ormoregenerally,

Let’scomparethisformulawiththenonrelativisticversion

Pi=mVi.

The first interesting fact is that they are not so different. Go back to the definition ofrelativisticvelocity,

ComparingEq.3.9withEq.3.30weseethattherelativisticmomentumisjust themasstimestherelativisticvelocity,

We probably could have guessed Eq. 3.31 without doing all that work. However, it’s

important thatwecanderive it from thebasicprinciplesofmechanics: the fundamentaldefinitionofmomentumasthederivativeoftheLagrangianwithrespecttothevelocity.

Asyoumightexpect,therelativisticandnonrelativisticdefinitionsofmomentumcometogether when the velocity is much smaller than the speed of light, in which case theexpression

isverycloseto1.Butnoticewhathappensasthevelocityincreasesandgetsclosetoc.Inthatlimittheexpressionblowsup.Thusasthevelocityapproachesc,themomentumofamassiveobjectbecomesinfinite!

Let’s return to the email message that I began this lecture with and see if we cananswerthewriter.ThewholeargumentrestedonNewton’ssecondlaw:aswrittenbytheemailer,F=MA.It’swellknowntoreadersofthefirstvolumethatthiscanbeexpressedanotherway:namely,forceistherateofchangeofmomentum,

The two ways of expressing Newton’s second law are the same within the restricteddomain of Newtonian mechanics, where the momentum is given by the usualnonrelativisticformulaP=mV.However,themoregeneralprinciplesofmechanicsimplythat Eq. 3.32 is more fundamental and applies to relativistic problems as well asnonrelativisticones.

Whathappensif,astheemailersuggested,aconstantforceisappliedtoanobject?Theansweristhatthemomentumincreasesuniformlywithtime.Butsincegettingtothespeedoflightrequiresaninfinitemomentum,itwilltakeforevertogetthere.

3.5RelativisticEnergyLet’s now turn to themeaning of energy in relativistic dynamics.As I’m sure you areaware, it isanotherconservedquantity.Asyouprobablyalsoknow,at least ifyouhaveread Volume I, energy is the Hamiltonian of the system. If you need a refresher onHamiltonians,nowisthetimetotakeabreakandgobacktoVolumeI.

TheHamiltonianisaconservedquantity.It’soneofthekeyelementsofthesystematicapproach to mechanics developed by Lagrange, Hamilton, and others. The frameworktheyestablishedallowsustoreasonfrombasicprinciplesratherthanjustmakethingsupas we go along. The HamiltonianH is defined in terms of the Lagrangian. The mostgeneralwaytodefineitis

where theQi andPi are the coordinates and canonical momenta that define the phasespaceofthesysteminquestion.Foramovingparticle,thecoordinatesaresimplythethreecomponentsofposition,X1,X2,X3,andEq.3.33takestheform

WealreadyknowfromEq.3.31thatthemomentaare

Pi=mUi,

or

WealsoknowfromEq.3.24thattheLagrangianis

or

SubstitutingtheseequationsforPiand intoEq.3.34resultsin

This equation for theHamiltonian looks like amess, butwecan simplify it quite abit.First,noticethat(Xi)2isjustthevelocitysquared.Asaresult,thefirsttermdoesnotevenneedtobewrittenasasum;it’sjust .Ifwemultiplyanddividethesecondtermby ,itwillhavethesamedenominatorasthefirstterm.Theresultingnumeratorism(1−v2).Puttingthesethingstogethergivesus

Nowit’smuchsimpler,butwe’restillnotdone.Noticethatmv2 inthefirsttermcancelsmv2inthesecondterm,andthewholethingboilsdownto

That’stheHamiltonian—theenergy.Doyourecognizethefactor inthisequation?Ifnot,justreferbacktoEq.3.8.It’sU0.Nowwecanbesurethatthezerothcomponentofthe4-momentum,

istheenergy.Thisisactuallyabigdeal,soletmeshoutitoutloudandclear:

ThethreecomponentsofspatialmomentumPitogetherwiththeenergyP0forma4-vector.

ThishastheimportantimplicationthattheenergyandmomentumgetmixedupunderaLorentz transformation. For example, an object at rest in one frame has energy but nomomentum.Inanotherframethesameobjecthasbothenergyandmomentum.

Finally,theprerelativisticnotionofmomentumconservationbecomestheconservationof 4-momentum: the conservation of x-momentum, y-momentum, z-momentum, andenergy.

3.5.1SlowParticlesBeforegoingon,weshouldfigureouthowthisnewconceptofenergyisrelatedtotheoldconcept.We’ll change back to conventional units for a while and put c back into ourequations. Take Eq. 3.35. Recognizing that the Hamiltonian is the same thing as theenergy,wecanwrite

To restore theappropriate factorsofcwe first note that energyhasunitsofmass timesvelocitysquared(aneasywaytorememberthisisfromthenonrelativisticexpressionforkineticenergy ).Thereforetherightsideneedsafactorofc2.Inadditionthevelocityshouldbereplacedwithv/c,resultingin

Eq. 3.38 is the general formula for the energy of a particle of massm in terms of itsvelocity.Inthatsenseitissimilartothenonrelativisticformulaforkineticenergy.Infact,we should expect that when the velocity is much less than c, it should reduce to thenonrelativistic formula.We can check that by using the approximation inEq. 3.19. Forsmallv/cweget

ThesecondtermontherightsideofEq.3.39isthenonrelativistickineticenergy,butwhatisthefirstterm?Itis,ofcourse,afamiliarexpression,maybethemostfamiliarinallofphysics, namely mc2. How should we understand its presence in the expression forenergy?

Evenbefore theadventof relativity itwasunderstood that theenergyofanobject isnot justkineticenergy.Kineticenergy is theenergydue to themotionofanobject,butevenwhentheobjectisatrestitmaycontainenergy.Thatenergywasthoughtofastheenergyneeded to assemble the system.What is special about the energyof assembly isthat it does not depend on the velocity.Wemay think of it as “rest energy.” Eq. 3.39,whichisaconsequenceofEinstein’stheoryofrelativity,tellsustheexactvalueoftherestenergyofanyobject.Ittellsusthatwhenthevelocityofanobjectiszero,itsenergyis

I’msurethatthisisnotthefirsttimeanyofyouhaveseenthisequation,butitmaybethefirst timeyou’ve seen it derived from firstprinciples.Howgeneral is it?Theanswer isverygeneral.Itdoesnotmatterwhethertheobjectisanelementaryparticle,abarofsoap,astar,orablackhole.Inaframeinwhichtheobjectisatrest,itsenergyisitsmasstimesthesquareofthespeedoflight.

Terminology:MassandRestMassThe termrestmass is an anachronism,despite its continueduse inmanyundergraduatetextbooks.NobodyIknowwhodoesphysicsusesthetermrestmassanymore.Thenew

conventionisthatthewordmassmeanswhatthetermrestmassusedtomean.9Themassofaparticleisatagthatgoeswiththeparticleandcharacterizestheparticleitself,notthemotionof theparticle. Ifyou lookup themassofanelectron,youwon’tgetsomethingthat depends onwhether the electron ismoving or stationary.You’ll get a number thatcharacterizes the electron at rest.What about the thing that used to be calledmass, thethingthatdoesdependonparticlemotion?Wecallthatenergy,orperhapsenergydividedbythespeedoflightsquared.Energycharacterizesamovingparticle.Energyatrestisjustcalledmass.Wewillavoidthetermrestmassaltogether.

3.5.2MasslessParticlesSofarwehavediscussedthepropertiesofmassiveparticles—particlesthatwhenbroughtto rest have a nonzero rest energy. But not all particles have mass. The photon is anexample.Masslessparticlesarea littlestrange.Eq.3.37tellsus thatenergyis .Butwhatisthevelocityofamasslessparticle?It’s1!We’reintrouble!Ontheotherhand,perhaps the trouble isnotsobadbecause thenumeratoranddenominatorarebothzero.Thatdoesn’ttellustheanswer,butatleastitleavessomeroomfornegotiation.

Ourzero-over-zeroconundrumdoescontainonelittleseedofwisdom:It’sabadideato think about the energy of a massless particle in terms of its velocity, because allmasslessparticlesmovewithexactlythesamevelocity.Cantheyhavedifferentenergiesiftheyallmovewiththesamevelocity?Theanswerisyes,andthereasonisthatzerooverzeroisnotdetermined.

If tryingtodistinguishmasslessparticlesbytheirvelocities isadead-endroad,whatcanwedoinstead?Wecanwritetheirenergyasafunctionofmomentum.10Weactuallydo that quite often in prerelativity mechanics when we write the kinetic energy of aparticle.Anotherwaytowrite

is

Tofindtherelativisticexpressionforenergyintermsofmomentum,there’sasimpletrick:WeusethefactthatU0,Ux,Uy,andUzarenotcompletely independent.WeworkedouttheirrelationshipinSection3.2.ExpandingEq.3.10andsettingc=1forthemoment,wecanwrite

(U0)2−(Ux)2−(Uy)2−(Uz)2=1.

Thecomponentsofmomentumarethesameasthecomponentsof4-velocityexceptforafactorofmass,andwecanmultiplytheprecedingequationbym2toget

We recognize the first termas (P0)2.ButP0 itself is just the energy, and the remainingthreetermsontheleftsideofEq.3.41arethex,y,andzcomponentsofthe4-momentum.Inotherwords,wecanrewriteEq.3.41as

WecanseethatEqs.3.10and3.42areequivalenttoeachother.ThetermsinEq.3.10arecomponents of the 4-velocity, while the terms in Eq. 3.42 are the correspondingcomponentsofthe4-momentum.WecansolveEq.3.42forEtoget

Now let’s put the speed of light back into this equation and see what it looks like inconventionalunits.I’llleaveitasanexercisetoverifythattheenergyequationbecomes

Hereisourresult.Eq.3.44givesenergyintermsofmomentumandmass.Itdescribesallparticleswhethertheirmassesarezeroornonzero.Fromthisformulawecanimmediatelyseethelimitasthemassgoestozero.Wehadtroubledescribingtheenergyofaphotonintermsofitsvelocity,butwehavenotroubleatallwhenenergyisexpressedintermsofmomentum.WhatdoesEq.3.44sayaboutzero-massparticlessuchasphotons?Withm=0,thesecondtermofthesquarerootbecomeszero,andthesquarerootofP2c2isjustthemagnitudeofPtimesc.WhythemagnitudeofP?TheleftsideoftheequationisE,arealnumber.Thereforetherightsidemustalsobearealnumber.Puttingthisalltogether,wearriveatthesimpleequation

Energyforamasslessparticle isessentially themagnitudeof themomentumvector,butfordimensionalconsistency,wemultiplybythespeedoflight.Eq.3.45holdsforphotons.It’sapproximatelytrueforneutrinos,whichhaveatinymass.Itdoesnotholdforparticlesthatmovesignificantlyslowerthanthespeedoflight.

3.5.3AnExample:PositroniumDecayNowthatweknowhowtowritedowntheenergyofamasslessparticle,wecansolveasimplebut interestingproblem.There’s a particle calledpositronium that consists of anelectronandapositroninorbitaroundeachother.It’selectricallyneutralanditsmassisapproximatelythemassoftwoelectrons.11

Thepositronistheelectron’santiparticle,andifyouletapositroniumatomsitaroundforawhile,thesetwoantiparticleswillannihilateeachother,producingtwophotons.Thepositroniumwilldisappear,andthetwophotonswillgoflyingoffinoppositedirections.In other words, a neutral positronium particle with nonzero mass turns into pureelectromagnetic energy. Can we calculate the energy and momentum of those twophotons?

Thiswouldnotmakeanysenseatallinprerelativityphysics.Inprerelativityphysics,thesumofthemassesofparticlesisalwaysunchanged.Chemicalreactionshappen,somechemicalsturnintoothers,andsoforth.Butifyouweighthesystem—ifyoumeasureitsmass—thesumoftheordinarymassesneverchanges.However,whenpositroniumdecaysintophotons,thesumoftheordinarymassesdoeschange.Thepositroniumparticlehasafinitenonzeromass,andthetwophotonsthatreplaceitaremassless.Thecorrectruleisnot that the sum of masses is conserved; it’s that the energy and the momentum are

conserved.Let’sconsidermomentumconservationfirst.

Supposethepositroniumparticleisatrestinyourframeofreference.12Itsmomentuminthisframeiszero,bydefinition.Now,thepositroniumatomdecaystotwophotons.Ourfirstconclusionisthatthephotonsmustgooffbacktoback,inoppositedirections,withequalandoppositemomentum.Iftheydon’ttravelinoppositedirections,it’sclearthatthetotalmomentumwillnotbezero.The finalmomentummustbezerobecause the initialmomentumwaszero.Thismeansthattheright-movingphotongoesoffwithmomentumP,andtheleft-movingphotongoesoffwithmomentum−P.13

Nowwecanusetheprincipleofenergyconservation.Takeyourpositroniumatom,putitonascale,andmeasureitsmass.Initsrestframeithasenergyequaltomc2.Becauseenergy is conserved, this quantitymust equal the combined energy of the two photons.Eachofthesephotonsmusthavethesameenergyastheotherbecausetheirmomentahavethesamemagnitude.UsingEq.3.45,wecanequatethisenergytomc2asfollows:

mc2=2c|P|.

Solvingfor|P|,wefindthat

Eachphotonhasamomentumwhoseabsolutevalueismc/2.

This is the mechanism by which mass turns into energy. Of course, the mass wasalwaysenergy,butinthefrozenformofrestenergy.Whenthepositroniumatomdecays,itresultsintwophotonsgoingout.Thephotonsgoontocollidewiththings.Theymayheatuptheatmosphere;theycouldbeabsorbedbyelectrons,generateelectricalcurrents,andso forth. The thing that’s conserved is the total energy, not the individual masses ofparticles.

1Thisisanactualemailmessagereceived1/22/2007.

2Rememberthatwhenwetalkaboutfour-dimensionalspacetime—threespacecoordinatesandonetimecoordinate—

thesymbol isastand-inforall threedirectionsinspace.In thiscontext, refers to thesumoftheirsquares,whichwewouldnormallywriteas(Δx)2+(Δy)2+(Δz)2

3Aframewhosex′axisisparalleltothatconnectinglinewouldservejustaswell.

4Aframewhoset′axisisparalleltotheconnectinglinewouldservejustaswell.

5Lookitupifit’snotfamiliar.Hereisagoodreference:https://en.wikipedia.org/wiki/Binomialapproximation.

6Theconceptofaworldlineisjustasgoodinnonrelativisticphysicsasit isinrelativisticphysics.Butitacquiresacertainprimacyinrelativitybecauseoftheconnectionbetweenspaceandtime—thefactthatspaceandtimemorphintoeachotherunderaLorentztransformation.

7ThereisatechnicalpointthatIwillmentioninordertopreemptcomplaintsbysophisticatedreaders.Itisnotstrictlytruethattheactionneedstobeminimum.Itmaybeminimum,maximum,ormoregenerallystationary.Generallythisfinepointwillplaynoroleintherestofthisbook.Wewillthereforefollowtraditionandrefertotheprincipleofleast(minimum)action.

8Recallthatavariablewithadotontopofitmeans“derivativewithrespecttotime.”Forexample, isshorthandfor

dx/dt.

9Ibelievethis“new”conventionstartedaboutfortytofiftyyearsago.

10Infact,wecanthinkaboutanyparticleinthisway.

11 It’s interesting to note that the mass of a positronium particle is slightly less than the sum of the masses of itsconstituent electron and positron. Why? Because it’s bound. It has some kinetic energy due to the motion of itsconstituents, and that adds to itsmass.But it has an even greater amount of negative potential energy.The negativepotentialenergyoutweighsthepositivekineticenergy.

12Ifnot,justgetmovingandgetintotheframeofreferenceinwhichitisatrest.

13We don’t actually know what directions these back-to-back photons will take, except that they’ll be moving inopposite directions. The line connecting the two photons is oriented randomly, according to the rules of quantummechanics.However, there’snothing tostopusfromorientingourxaxis tocoincidewith their“chosen”directionofmotion.

Lecture4

ClassicalFieldTheory

It’sWorldSeriestime,andthebarinHermann’sHideawayispackedwithfanswatchingthegame.ArtarriveslateandpullsupastoolnexttoLenny.

Art:What’rethenamesoftheguysintheoutfield?

Lenny:Who’sinclassicalfield;What’sinquantumfield.

Art:Comeon,Lenny,Ijustaskedyou.Who’sinquantumfield?

Lenny:Who’sinclassicalfield.

Art:That’swhatI’maskingyou.Allright,lemmetryagain.Whataboutelectricfield?

Lenny:Youwanttoknowthenameoftheguyinelectricfield?

Art:Naturally.

Lenny:Naturally?No,Naturally’sinmagneticfield.

So far, we have focused on the relativistic motion of particles. In this lecture, we’llintroduce the theory of fields—not quantum field theory but relativistic classical fieldtheory.Theremaybesomeoccasionalpointsofcontactwithquantummechanics,andI’llpointthemout.Butforthemostpartwe’llsticktoclassicalfieldtheory.

The field theory that you probably know most about is the theory of electric andmagneticfields.Thesefieldsarevectorquantities.They’recharacterizedbyadirectioninspaceaswellasamagnitude.We’llstartwithsomethingalittleeasier:ascalarfield.Asyou know, scalars are numbers that have magnitude but no direction. The field we’llconsiderhere is similar toan importantscalar field inparticlephysics.Perhapsyoucanguesswhichoneitisaswegoalong.

4.1FieldsandSpacetimeLet’sstartwithspacetime.Spacetimealwayshasonetimecoordinateandsomenumberofspace coordinates. In principle, we could study physics with any number of timecoordinates and any number of space coordinates. But in the physical world, even fortheories in which spacetime may have ten dimensions, eleven dimensions, twenty-sixdimensions, there’s always exactly one time dimension. Nobody knows how to makelogicalsenseoutofmorethanonetimedimension.

Let’scallthespacecoordinatesXiand,forthemoment,thetimecoordinatet.Keepinmind that in field theory theXi are not degrees of freedom; they aremerely labels thatlabelthepointsofspace.Theeventsofspacetimearelabeled(t,Xi).Theindexirunsoverasmanyspacecoordinatesasthereare.

Not surprisingly, the degrees of freedom of field theory are fields. A field is ameasurablequantitythatdependsonpositioninspaceandmayalsovarywithtime.Thereareplentyof examples fromcommon,garden-varietyphysics.Atmospheric temperaturevariesfromplacetoplaceandtimetotime.Apossiblenotationfor itwouldbeT(t,Xi).Becauseithasonlyonecomponent—asinglenumber—it’sascalarfield.Windvelocityisavectorfieldbecausevelocityhasadirection,whichmayalsovaryoverspaceandtime.

Mathematically,we representa fieldasa functionof spaceand time.Weoften labelthisfunctionwiththeGreekletterϕ:

ϕ(t,Xi).

It’scommoninfieldtheorytosaythatspacetimeis(3+1)-dimensional,meaningthatthereare threedimensionsof space andonedimensionof time.Moregenerallywemight beinterested instudying fields inspacetimeswithothernumbersofspacedimensions. Ifaspacetimehasdspacedimensions,wewouldcallit(d+1)-dimensional.

4.2FieldsandActionAs I mentioned earlier, the principle of least action is one of the most fundamentalprinciplesofphysics,governingallknownlawsofphysics.Withoutitwewouldhaveno

reasontobelieveinenergyconservationoreventheexistenceofsolutionstotheequationswewritedown.Wewillalsobaseourstudyof fieldsonanactionprinciple.Theactionprinciples that govern fields are generalizations of those for particles. Our plan is toexamineinparalleltheactionprinciplesthatgovernfieldsandthosethatgovernparticles,comparing them aswe go. To simplify this comparison,wewill first restate the actionprinciplefornonrelativisticparticlesinthelanguageoffields.

4.2.1NonrelativisticParticlesReduxIwanttobrieflygobacktothetheoryofnonrelativisticparticles,notbecauseIamreallyinterested in slow particles, but because the mathematics has some similarity with thetheoryoffields.Infact,inacertainformalsenseitisafieldtheoryofasimplekind—afield theory inaworldwhosespacetimehaszero spacedimensions, andas always,onetimedimension.

Toseehowthisworks,let’sconsideraparticlethatmovesalongthexaxis.Ordinarilywewoulddescribethemotionoftheparticlebyatrajectoryx(t).However,withnochangein the content of the theory,wemight change thenotation and call the position of theparticleϕ.Insteadofx(t),thetrajectorywouldbedescribedbyϕ(t).

Ifweweretore-wirethemeaningofthesymbolϕ(t),thatis,ifweuseittorepresentascalarfield,itwouldbecomeaspecialcaseofϕ(t,Xi)—aspecialcaseinwhichtherearenodimensionsofspace.Inotherwords,aparticle theoryinonedimensionofspacehasthe same mathematical structure as a scalar field theory in zero dimensions of space.Physicists sometimes refer to the theory of a single particle as a field theory in (0+1)-dimensions,theonedimensionbeingtime.

Fig. 4.1 illustrates the motion of a nonrelativistic particle. Notice that we use thehorizontal axis for time, just to emphasize that t is the independent parameter. On thevertical axisweplot thepositionof theparticle at time t, calling itϕ(t).Thecurveϕ(t)represents thehistoryof theparticle’smotion. It tellsyouwhat thepositionϕ isateachmomentoftime.Asthediagramshows,ϕcanbenegativeorpositive.Wecharacterizethistrajectoryusingtheprincipleofleastaction.

Asyourecall,action isdefinedas the integralofsomeLagrangian , froman initialtimeatoafinaltimeb:

Figure4.1:NonrelativisticParticleTrajectory.

For nonrelativistic particles, the Lagrangian is simple; it’s the kinetic energyminus thepotentialenergy.Kineticenergyisusuallyexpressedas ,butinournewnotationwewould write or instead of v for velocity. With this notation, the kinetic energybecomes ,or .We’llsimplifythingsalittlebysettingthemassmequalto1.Thusthekineticenergyis

What about potential energy? In our example, potential energy is just a function ofposition—inotherwordsit’safunctionofϕ,whichwe’llcallV(ϕ).SubtractingV(ϕ)fromthekineticenergygivesustheLagrangian

andtheactionintegralbecomes

As we know from classical mechanics, the Euler-Lagrange equation tells us how tominimize the action integral and therefore provides the equation of motion for theparticle.1Forthisexample,theEuler-Lagrangeequationis

andourtaskistoapplythisequationtotheLagrangianofEq.4.1.Let’sstartbywritingdownthederivativeof withrespectto :

Next,theEuler-Lagrangeequationinstructsustotakethetimederivativeofthisresult:

This completes the left side of the Euler-Lagrange equation. Now for the right side.ReferringoncemoretoEq.4.1,wefindthat

Finally,settingtheleftsideequaltotherightsidegivesus

Thisequationshouldbefamiliar.It’sjustNewton’sequationforthemotionofaparticle.Therightside isforce,andthe leftside isacceleration.ThiswouldbeNewton’ssecondlawF=mahadwenotsetthemassto1.

The Euler-Lagrange equations provide the solution to the problem of finding thetrajectorythataparticlefollowsbetweentwofixedpointsaandb.They’reequivalenttofindingthetrajectoryofleastactionthatconnectsthetwofixedendpoints.2

Asyouknow,there’sanotherwaytothinkofthis.YoucandividethetimeaxisintoalotoflittlepiecesbydrawinglotsofcloselyspacedverticallinesonFig.4.1.Insteadofthinkingoftheactionasanintegral,justthinkofitasasumofterms.Whatdothosetermsdepend on?They depend on the value ofϕ(t) and its derivatives at each time. In otherwords,thetotalactionissimplyafunctionofmanyvaluesofϕ(t).Howdoyouminimizea function of ϕ? You differentiate with respect to ϕ. That’s what the Euler-Lagrangeequationsaccomplish.Anotherwaytosayitisthatthey’rethesolutiontotheproblemofmovingthesepointsarounduntilyoufindthetrajectorythatminimizestheaction.

4.3PrinciplesofFieldTheoryThusfarwe’vebeenstudyingafieldtheoryinaworldwithnospacedimensions.Basedonthisexample,let’strytodevelopsomeintuitionaboutatheoryforaworldmoreliketheonewelivein—aworldwithoneormorespacedimensions.Wetakeitasagiventhatfieldtheory,infactthewholeworld,isgovernedbyanactionprinciple.Stationaryactionisapowerfulprinciplethatencodesandsummarizesahugenumberofphysicslaws.

4.3.1TheActionPrincipleLet’sdefine theactionprincipleforfields.Foraparticlemovinginonedimension(Fig.4.1)wechosetwofixedendpoints,aandb,asboundariesalongthetimeaxis.Thenweconsideredallpossibletrajectoriesthatconnectthetwoboundarypointsandaskedfortheparticularcurvethatminimizestheaction.(Thisisalotlikefindingtheshortestdistancebetween twopoints.) In thisway the principle of least action tells us how to fill in thevalueofϕ(t)betweentheboundarypointsaandb.

Theproblemoffieldtheoryisageneralizationof this ideaoffillingintheboundarydata.We beginwith a spacetime regionwhichwemay visualize as a four-dimensionalbox.Toconstructitwetakeathree-dimensionalboxofspace—let’ssaythespaceinsidesomecube—andconsideritforsomeintervaloftime.Thatformsafour-dimensionalboxof spacetime. InFig.4.2 I’ve tried to illustrate sucha spacetimeboxbutwithonly twospacedirections.

The general problem of field theory can be stated in the following way: Given thevalues of ϕ everywhere on the boundary of the spacetime box, determine the fieldeverywhereinsidethebox.Therulesofthegamearesimilartotheparticlecase.Wewillneedanexpressionfortheaction—we’llcometothatsoon—butlet’sassumeweknowtheactionforeveryconfigurationofthefieldinthebox.Theprincipleofleastactiontellsustowiggle the field untilwe find the particular functionϕ(t,x, y, z) that gives the leastaction.

Figure4.2:BoundaryofaSpacetimeRegionforApplyingthePrincipleofLeastAction.Onlytwospacedimensionsareshown.

For particlemotion the actionwas constructed by adding up little bits of action foreachinfinitesimaltimesegment.Thisgaveanintegraloverthetimeintervalbetweentheboundary points a and b in Fig. 4.1. The natural generalization in field theory is toconstructanactionasasumover tinyspacetimecells—inotherwords,an integraloverthespacetimeboxinFig.4.2.

where isaLagrangianthatwestillhavenotspecified.Butinrelativitywe’velearnedtothink of these four coordinates on an equal footing, with each of them being part ofspacetime.Weblurthedistinctionbetweenspaceandtimebygivingthemsimilarnames—wejustcallthemXμ—wheretheindexμrunsoverallfourcoordinates,andit’sstandardpracticetowritetheprecedingintegralas

4.3.2StationaryActionforϕBecause theLagrangian for a field is integratedover space aswell as time, it’s oftencalledtheLagrangedensity.3

Whatvariablesdoes dependon?Returnforamomenttothenonrelativisticparticle.The Lagrangian depends on the coordinates of the particle, and the velocities. In thenotation we’ve been using, the Lagrangian depends on ϕ and . The naturalgeneralization,inspiredbyMinkowski’sideaofspacetime,isfor todependonϕandthepartialderivativesofϕwithrespecttoallcoordinates.Inotherwords, dependson

Thuswewrite

wheretheindexμrangesoverthetimecoordinateandallthreespacecoordinates.Inthisactionintegral,wedidnotwrite withexplicitdependenciesont,x,y,orz.Butforsome

problems, could indeed depend on these variables, just as theLagrangian for particlemotionmightdependexplicitlyontime.Foraclosedsystem—asystemwhereenergyandmomentumareconserved— doesnotdependexplicitlyontimeorspatialposition.

Asinordinaryclassicalmechanics,theequationsofmotionarederivedbywigglingϕandminimizingtheaction.InVolumeI,Ishowedthattheresultofthiswigglingprocesscan be achieved by applying a special set of equations called the Euler-Lagrangeequations.Forthemotionofaparticle,theEuler-Lagrangeequationstaketheform

How would the Euler-Lagrange equations change for the multidimensional spacetimecase?Let’slookcloselyateachtermontheleftside.Clearly,weneedtomodifyEq.4.4toincorporateall fourdirectionsofspacetime.Thefirst term,whichalreadyreferences thetimedirection,becomesasumofterms,oneforeachdirectionofspacetime.ThecorrectprescriptionistoreplaceEq.4.4with

Thefirstterminthesumisjust

whose similarity to the first term in Eq. 4.4 is obvious. The other three terms involveanalogousspatialderivativesthatfillouttheexpressionandgiveititsspacetimecharacter.

Eq.4.5istheEuler-Lagrangeequationforasinglescalarfield.Aswewillsee,theseequationsarecloselyrelatedtowaveequationsthatdescribewavelikeoscillationsofϕ.

Asistrueinparticlemechanics,theremaybemorethanasingledegreeoffreedom.Inthe case of field theory, that would mean more than one field. Let’s be explicit andsuppose there are two fields ϕ and χ. The action will depend on both fields and theirderivatives,

Iftherearetwofields,thentheremustbeanEuler-Lagrangeequationforeachfield:

Moregenerally, if there are several fields, therewill be anEuler-Lagrange equation foreveryoneofthem.

Incidentally,whileit’struethatwe’redevelopingthisexampleasascalarfield,sofarwe haven’t said anything that requires ϕ to be a scalar. The only hint of ϕ’s scalarcharacteristhefactthatwe’reworkinginonlyonecomponent;avectorfieldwouldhaveadditional components, and each new component would generate an additional Euler-Lagrangeequation.

4.3.3MoreAboutEuler-LagrangeEquationsThe Lagrangian in Eq. 4.1 for a nonrelativistic particle contains a kinetic energy termproportionalto

andapotentialenergyterm

Wemightguessthatthegeneralizationtoafieldtheorywouldlooksimilar,butwiththekineticenergytermhavingspaceaswellastimederivatives:

But there is somethingobviouslywrongwith thisguess:Spaceand timeenter into it inexactly the sameway.Even in relativity theory, time is not completely symmetricwithspace,andordinaryexperiencetellsustheyaredifferent.Onehint,comingfromrelativity,isthesigndifferencebetweenspaceandtimeintheexpressionforpropertime,

Laterwewillseethatthederivatives formthecomponentsofa4-vectorandthat

definesaLorentzinvariantquantity.Thissuggeststhatwereplacethesumofsquares(ofderivatives)withthedifferenceofsquares.Later,whenwebringLorentzinvariancebackintothepicture,we’llseewhythismakessense.We’lltakethismodifiedLagrangian,

tobe theLagrangian forour field theory.Wecanalsoview it as aprototype.Lateron,we’llbuildothertheoriesoutofLagrangiansthataresimilartothisone.

ThefunctionV(ϕ)iscalledthefieldpotential.Itisanalogoustothepotentialenergyofaparticle.Moreprecisely,itisanenergydensity(energyperunitvolume)ateachpointofspacethatdependsonthevalueofthefieldatthatpoint.ThefunctionV(ϕ)dependsonthecontext,and inat leastone importantcase it isdeduced fromexperiment.We’lldiscussthatcaseinSection4.5.1.Forthemoment,V(ϕ)canbeanyfunctionofϕ.

Let’s take this Lagrangian and work out its equations of motion step by step. TheEuler-LagrangeequationinEq.4.5tellsusexactlyhowtoproceed.Weallowtheindexμinthisequationtotakeonthevalues0,1,2,and3.Eachoftheseindexvaluesgeneratesaseparatetermintheresultingdifferentialequationforthefield.Here’showitworks:

Step1.Westartbysettingtheindexμequaltozero.Inotherwords,XμstartsoutasX0,which corresponds to the time coordinate. Calculating the derivative of withrespectto givesusthesimpleresult

Eq.4.5nowinstructsustotakeonemorederivative:

Thisterm, ,isanalogoustotheaccelerationofaparticle.

Step 2. Next, we set index μ equal to 1, andXμ now becomesX1, which is just the xcoordinate.Calculatingthederivativeof withrespectto givesus

and

Wegetsimilarresultsfortheyandzcoordinateswhenwesetμequalto2and3respectively.

Step3.RewriteEq.4.5usingtheprevioustwosteps:

Asusual,ifwewanttouseconventionalunitsweneedtorestorethefactorsofc.Thisiseasilydoneandtheequationofmotionbecomes

4.3.4WavesandWaveEquationsOfall thephenomenadescribedbyclassical field theory, themostcommonandeasy tounderstand is thepropagationofwaves.Soundwaves, lightwaves,waterwaves,wavesalong vibrating guitar strings: All are described by similar equations, which areunsurprisingly called wave equations. This connection between field theory and wavemotionisoneofthemostimportantinphysics.It’snowtimetoexploreit.

Eq.4.9isageneralizationofNewton’sequationofmotion

(Eq.4.3)forfields.Theterm representsakindofforceactingonthefield,pushingitawayfromsomenaturalunforcedmotion.Foraparticle,theunforcedmotionisuniformmotion with a constant velocity. For fields, the unforced motion is propagating wavessimilartosoundorelectromagneticwaves.Toseethis,let’sdotwothingstosimplifyEq.4.9. First we can drop the force term by setting V(ϕ) = 0. Second, instead of threedimensionsofspacewewillstudytheequationwithonlyonespacedirection,x.Eq.4.9nowtakesthemuchsimplerformofa(1+1)-dimensionalwaveequation,

I will show you a whole family of solutions. Let’s consider any function of thecombination(x+ct).We’llcallitF(x+ct).Nowconsideritsderivativeswithrespecttoxandt.Itiseasytoseethatthesederivativessatisfy

Ifweapplythesameruleasecondtimeweget

or

Eq.4.11isnothingbutthewaveequationofEq.4.10forthefunctionF.Wehavefoundalargeclassofsolutionstothewaveequation.Anyfunctionofthecombination(x+ct) issuchasolution.

WhatarethepropertiesofafunctionlikeF(x+ct)?Attimet=0itisjustthefunctionF(x).Astimegoeson,thefunctionchanges,butinasimpleway:Itmovesrigidlytotheleft(inthenegativexdirection)withvelocityc.Let’stakeaparticularexampleinwhichF(whichwenowidentifywithϕ)isasinefunctionwithwavenumberk:

Thisisaleft-movingsinewave,movingwithvelocityc.Therearealsocosinesolutionsaswellaswavepacketsandpulses.Aslongastheymoverigidlytotheleftwithvelocityctheyaresolutionsofthewaveequation,Eq.4.10.

Whataboutright-movingwaves?AretheyalsodescribedbyEq.4.10?Theansweris,yestheyare.AllyouhavetodotochangeF(x+ct)toaright-movingwaveisreplacex+ctwithx−ct.IwillleaveittoyoutoshowthatanyfunctionoftheformF(x−ct)movesrigidlytotherightandthatitalsosatisfiesEq.4.10.

4.4RelativisticFields

Whenwebuiltourtheoryofparticles,weusedtwoprinciplesinadditiontothestationaryactionprinciple.First,theactionisalwaysanintegral.Forparticles,it’sanintegralalongatrajectory.We’ve already dealt with this issue for fields by redefining the action as anintegral over spacetime. Second, the action needs to be invariant; it should be built upfromquantitiesinsuchawaythatithastheexactsameformineveryreferenceframe.

Howdidwemanagethatforthecaseofaparticle?Webuiltuptheactionbyslicingthetrajectory into lots of little segments, calculating the action for each segment, and thenaddingall these smallactionelements together.Then,we took the limitof thisprocess,where the size of each segment approaches zero and the sumbecomes an integral.Thecrucial point is this:Whenwe defined the action for one of the segments, we chose aquantity thatwas invariant—the proper time along the trajectory.Because all observersagree on the value of the proper time for each little segment, theywill agree about thenumberyougetwhenyouaddthemalltogether.Asaresult,theequationsofmotionyouderiveusingthestationaryactionprinciplehaveexactlythesameformineveryreferenceframe.Thelawsofparticlemechanicsareinvariant.IalreadyhintedonhowtodothisforfieldsinSection4.3.3,butnowwewanttogetseriousaboutLorentzinvarianceinfieldtheory.

Wewillneedtoknowhowtoforminvariantquantitiesfromfields,andthenusethemtoconstructactionintegralsthatarealsoinvariant.Todoso,we’llneedaclearconceptofthetransformationpropertiesoffields.

4.4.1FieldTransformationPropertiesLet’sgetbacktoArtintherailwaystationandLennymovingpasthiminthetrain.Theybothlookatthesameevent—Artcallsit(t,x,y,z)andLennycallsit(t′,x′,y′,z′).Theirspecial field detectors register numerical values for some fieldϕ. The simplest possiblekindoffieldisoneforwhichtheyobtainexactlythesameresult.IfwecallthefieldthatArt measures ϕ(t, x, y, z) and the field that Lenny measures ϕ′(t′, x′, y′, z′), then thesimplesttransformationlawwouldbe

Inotherwords, at anyparticularpoint of spacetimeArt andLenny (and everyone else)agreeaboutthevalueofthefieldϕatthatpoint.

Afieldwiththispropertyiscalledascalarfield.TheideaofascalarfieldisillustratedinFig.4.3.It’simportanttounderstandthatcoordinates(t′,x′)and(t,x)bothreferencethesame point in spacetime. Fig. 4.3 drives this point home. The unprimed coordinatesrepresentspacetimepositioninArt’sframe.TheprimedaxesstandforspacetimepositioninLenny’sframe.Thelabelϕ(t,x)isArt’snameforthefield;Lennycallsthesamethingϕ′(t′,x′)sinceheistheprimedobserver.Butit’sthesamefield,withthesamevalueatthesamepointinspacetime.Thevalueofascalarfieldataspacetimepointisinvariant.

Not all fields are scalars. Here’s an example from ordinary nonrelativistic physics:windvelocity.Windvelocityisavectorwiththreecomponents.Observerswhoseaxesareorienteddifferentlywillnotagreeon thevaluesof thecomponents.ArtandLennywill

certainlynotagree.TheairmightbestillinArt’sframe,butwhenLennystickshisheadoutthewindowhedetectsalargewindvelocity.

Figure4.3:Transformations.Fieldsφandφ′bothreferencethesamepointinspacetime.Bothfieldshavethesamevalueatthispoint.DisplacementsdXμand(dX′)μbothreferencethesamespacetimeinterval,buttheprimedcomponentsaredifferentfromtheunprimedcomponents.They’rerelatedbyLorentz

transformation.

Nextwecometo4-vectorfields.AgoodexampleisoneI’vealreadymentioned:windvelocity.HereishowwemightdefineitinArt’sframe.Atthespacetimepoint(t,x,y,z),Artmeasures the localcomponentsof thevelocityof theairmoleculesdXi/dtandmapsthemout.Theresultisa3-vectorfieldwithcomponents

Butyoumightalreadyguessthatinarelativistictheoryweshouldrepresentthemolecularvelocities relativistically: not by Vi = dXi/dt but by Uμ = dXμ/dτ. Mapping out therelativisticwindvelocity

or

woulddefinea4-vectorfield.

Given Art’s description of the relativistic wind velocity, we can ask: What are itscomponentsinLenny’sframe?Since4-velocityisa4-vector,theanswerfollowsfromtheLorentztransformationthatwewrotedownearlierinEq.2.16:

I didn’t include the dependence on the coordinates inEq. 4.13 as itwould have overlyclutteredtheequations,buttheruleissimple:TheUontherightsidearefunctionsof(t,x,y,z)andtheU′ontheleftsidearefunctionsof(t′,x′,y′,z′),butbothsetsofcoordinatesrefertothesamespacetimepoint.

Let’sconsideranothercomplexoffourquantitiesthatwe’llcallthespacetimegradientof the scalar field ϕ. Its components are the derivatives of ϕ.We’ll use the shorthandnotation∂μϕ,definedby

Forexample,

Inaslightlydifferentnotation,wecanwrite

Wemightexpectthatϕμ isa4-vectorandtransformsthesamewayasUμ;wewouldbemistaken,althoughnotbyalot.

4.4.2MathematicalInterlude:CovariantComponentsIwanttopausehereforacoupleofmathematicalpointsabouttransformationsfromoneset of coordinates to another. Let’s suppose we have a space described by two sets ofcoordinatesXμ and (X′)μ. These could beArt’s and Lenny’s spacetime coordinates, buttheyneedn’tbe.Let’salsoconsideran infinitesimal intervaldescribedbydXμord(X′)μ.Ordinarymultivariable calculus implies the following relation between the two sets ofdifferentials:

Einsteinwrotemanyequationsofthisform.Afterawhilehenoticedapattern:Wheneverhe had a repeated index in a single expression—the index ν on the right side of theequation issucharepeated index—itwasalwayssummedover. Inoneofhispapersongeneral relativity, after several pages, he apparently got tired ofwriting the summationsignandsimplysaidthatfromnowon,wheneveranexpressionhadarepeatedindexhewouldassumethat itwassummedover.Thatconventionbecameknownas theEinsteinsummation convention. It is completely ubiquitous today, to the point where no one inphysicsevenbotherstomentionit.I’malsotiredofwriting∑ν,sofromnowonwewilluseEinstein’scleverconvention,andEq.4.15becomes

IftheequationsrelatingXandx′arelinear,astheywouldbeforLorentztransformations,then the partial derivatives are constant coefficients. Let’s take the Lorentztransformations

asanexample.HereisalistofthefourconstantcoefficientsthatwouldresultfromEq.4.16:

IfweplugtheseintoEq.4.16wegettheexpectedresult,

This,ofcourse,istheperfectlyordinaryLorentztransformationofthecomponentsofa4-vector.

Let’s abstract from this exercise a general rule for the transformation of 4-vectors.GoingbacktoEq.4.16, let’sreplace4-vectorcomponentsd(X′)μanddXνwith (A′)μandAν.Theserepresent thecomponentsofany4-vectorA in frames relatedbyacoordinate

transformation.ThegeneralizationofEq.4.16becomes

where ν is a summation index. Eq. 4.19 is the general rule for transforming thecomponentsofa4-vector.FortheimportantspecialcaseofaLorentztransformation,thisbecomes

MyrealreasonfordoingthiswasnottoexplainhowdXμorAμtransforms,buttosetupthecalculationfortransforming∂μϕ.Theseobjects—therearefourofthem—alsoformthe components of a 4-vector, although of a slightly different kind than the dXμ. TheyobviouslyrefertothecoordinatesystemXbutcanbetransformedtotheX′frame.

The basic transformation rule derives from calculus, and it’s a multivariablegeneralizationofthechainruleforderivatives.Letmeremindyouoftheordinarychainrule.Letϕ(x)beafunctionofthecoordinateX,andlettheprimedcoordinatesX′alsobeafunctionofX.ThederivativeofϕwithrespecttoX′isgivenbythechainrule,

Themultivariable generalization involves a fieldϕ that dependson several independentcoordinatesXμandasecondsetofcoordinates(X′)ν.Thegeneralizedchainrulereads

orusingthesummationconvention,andtheshorthandnotationinEq.4.14,

Let’sbemoregeneralandreplace∂μϕwithAμsothatEq.4.21becomes

Takeamoment tocompareEqs.4.19and4.22. I’llwrite themagain tomake iteasy tocompare—firstEq.4.19andthenEq.4.22:

Therearetwodifferences.Thefirst is that inEq.4.19theGreekindicesonAappearassuperscriptswhileinEq.4.22theyappearassubscripts.Thathardlyseemsimportant,butitis.Theseconddifferenceisthecoefficients:InEq.4.19theyarederivativesofX′withrespecttoX,whileinEq.4.22theyarederivativesofXwithrespecttoX′.

Evidentlytherearetwodifferentkindsof4-vectors,whichtransformindifferentways:thekindthathavesuperscriptsandthekindwithsubscripts.Theyarecalledcontravariantcomponents (superscripts) and covariant components (subscripts), but since I alwaysforgetwhichiswhich,Ijustcallthemupperandlower4-vectors.Thusthe4-vectordXμisacontravariantorupper4-vector,whilethespacetimegradient∂μϕisacovariantorlower4-vector.

Let’sgobacktoLorentztransformations.InEq.4.18,Iwrotedownthecoefficientsforthetransformationofupper4-vectors.Heretheyareagain:

WecanmakeasimilarlistforthecoefficientsinEq.4.22forlower4-vectors.Infact,wedon’thavemuchworktodo.Wecangetthembyinterchangingtheprimedandunprimedcoordinates, which for Lorentz transformations just means interchanging the rest andmovingframes.Thisisespeciallyeasysinceitonlyrequiresustochangethesignofthevelocity(remember,ifLennymoveswithvelocityvinArt’sframe,thenArtmoveswithvelocity−vinLenny’sframe).Allwehavetodoistointerchangeprimedandunprimedcoordinatesandatthesametimereversethesignofv.

Herethenarethetransformationrulesforthecomponentsofcovariant(lower)4-vectors:

We’vealreadyseenexamplessuchasXμ,displacement from theorigin.Thedifferentialdisplacement between neighboring points, dXμ, is also a 4-vector. If you multiply 4-vectors by scalars (that is, by invariants), the result is also a 4-vector. That’s because

invariantsarecompletelypassivewhenyoutransform.We’vealreadyseenthatthepropertimedτisinvariant,andthereforethequantitydXμ/dτ,whichwecall4-velocity,isalsoa4-vector:

Whenwesay“Uμisa4-vector,”whatdoweactuallymean?WemeanthatitsbehaviorinotherreferenceframesisgovernedbytheLorentztransformation.Let’srecalltheLorentztransformation for the coordinates of two reference frameswhose relative velocity is valongthexaxis:

If a complex of four quantities (consisting of a time component and three spacecomponents) transforms in this way, we call it a 4-vector. As you know, differentialdisplacementsalsohavethisproperty:

Table4.1summarizes the transformationpropertiesofscalarsand4-vectors.Weuse theslightlyabstractnotationAμ torepresentanarbitrary4-vector.A0isthetimecomponent,andeachoftheothercomponentsrepresentsadirectioninspace.

Anexampleofafieldwiththesepropertieswouldbeafluidthatfillsallofspacetime.Ateverypointinthefluidtherewouldbea4-velocityaswellasanordinary3-velocity.Wecouldcallthis4-velocityUμ(t,x).Ifthefluidflows,thevelocitymightbedifferentindifferentplaces.The4-velocityofsuchafluidcanbethoughtofasafield.Becauseit’sa4-velocity, it’sautomaticallya4-vectorandwouldtransforminexactlythesamewayasourprototype4-vectorAμ.ThevaluesofU’scomponentsinyourframewouldbedifferentfromtheirvaluesinmyframe;theywouldberelatedbytheequationsinTable4.1.Therearelotsofotherexamplesof4-vectors,andwewon’ttrytolistthemhere.

Table4.1:FieldTransformations.Greekindexμtakesvalues0,1,2,3,whichcorrespondtot,x,y,zinordinary(3+1)-dimensionalspacetime.Innonrelativisticphysics,ordinaryEuclideandistanceisalsoconsideredascalar.

Ifyoutakethefourcomponentsofa4-vector,youcanmakeascalaroutofthem.Wealreadydidthiswhenweconstructedthescalar(dτ)2fromthe4-vectordXμ:

(dτ)2=(dt)2−(dx)2−(dy)2−(dz)2.

Wecanfollowthesameprocedurewithany4-vector.IfAμisa4-vector,thenthequantity

(A0)2−(Ax)2−(Ay)2−(Az)2

isascalarforexactlythesamereasons.OnceyouknowthattheAμcomponentstransformthe same way as t and x, you can see that the difference of the squares of the timecomponentandthespacecomponentwillnotchangeunderaLorentztransformation.Youcanshowthisusingthesamealgebraweusedwith(dτ)2.

We’veseenhowtoconstructascalarfroma4-vector.Nowwe’lldotheopposite,thatis,constructa4-vectorfromascalar.Wedothisbydifferentiatingthescalarwithrespecttoeachofthefourcomponentsofspaceandtime.Together,thosefourderivativesforma4-vector.Ifwehaveascalarϕ,thequantities

arethecomponentsofa(covariant,orlower)4-vector.

4.4.3BuildingaRelativisticLagrangianWenowhaveasetoftoolsforcreatingLagrangians.Weknowhowthingstransform,andweknowhowtoconstructscalarsfromvectorsandotherobjects.HowdoweconstructaLagrangian? It’s simple.TheLagrangian itself—the thing thatweaddupover all theselittle cells to form an action integral—must be the same in every coordinate frame. Inotherwords,itmustbeascalar!That’sallthereistoit.Youtakeafieldϕandconsiderall

thepossiblescalarsyoucanmakefromit.ThesescalarsarecandidatebuildingblocksforourLagrangian.

Let’slookatsomeexamples.Ofcourse,ϕ itself isascalar in thisexample,butso isanyfunctionofϕ.Ifeveryoneagreesonthevalueofϕ,theywillalsoagreeonthevaluesofϕ2,4ϕ3,sinh(ϕ),andsoon.Anyfunctionofϕ,forexampleapotentialenergyV(ϕ),isascalar and therefore is a candidate for inclusion in aLagrangian. In fact,we’ve alreadyseenplentyofLagrangiansthatincorporateV(ϕ).

Whatotheringredientscouldweuse?Certainlywewanttoincludederivativesofthefield.Withoutthem,ourfieldtheorywouldbetrivialanduninteresting.Wejustneedtobesuretoputthederivativestogetherinawaythatproducesascalar.Butthat’seasy!First,weusethederivativestobuildthe4-vector,

Next,weusethecomponentsofour4-vectortoconstructascalar.Theresultingscalaris

Here is a nontrivial expressionwe canput into aLagrangian.What else couldweuse?Certainly,wecanmultiplybyanumericalconstant.For thatmatter,wecanmultiplybyany function of any scalar. Multiplying two things that are invariant produces a thirdinvariant.Forexample,theexpression

wouldbea legalLagrangian. It’s somewhatcomplicated,andwewon’tdevelop ithere,butitqualifiesasaLorentzinvariantLagrangian.Wecoulddosomethingevenuglier,liketakingtheexpressioninsidethesquarebracketsandraisingittosomepower.Thenwe’dhave higher powers of derivatives. That would be ugly, but it would still be a legalLagrangian.

What about higher-order derivatives? In principle, we could use them if we turnedthem into scalars. But that would take us outside the confines of classical mechanics.Withinthescopeofclassicalmechanics,wecanusefunctionsofthecoordinatesandtheirfirstderivatives.Higherpowersoffirstderivativesareacceptable,buthigherderivativesarenot.

4.4.4UsingOurLagrangianDespitetherestrictionsofclassicalmechanicsandtherequirementofLorentzinvariance,we still have a tremendous amount of freedom in choosing aLagrangian toworkwith.Let’shaveacloselookatthisone:

This is essentially the same as the Lagrangian of Eq. 4.7, and now you can seewhy I

chose that one for our nonrelativistic example. The new factor in the first term justswitchesusbacktoconventionalunits.IalsoreplacedthegenericpotentialfunctionV(ϕ)withamoreexplicitfunction, .Thefactor isnothingmorethanaconventionwithnophysicalmeaning.It’sthesame thatappearsintheexpression forkineticenergy.Wecouldhave taken this tobemv2 instead; ifwedid, thenourmasswoulddiffer fromNewton’smassbyafactoroftwo.

BackinLecture1,weexplainedthatageneralLorentztransformationisequivalenttoacombinationofspacerotations,togetherwithasimpleLorentztransformationalongthexaxis.We’veshownthatEq.4.24isinvariantunderasimpleLorentztransformation.Isitalsoinvariantwithrespecttospacerotations?Theanswerisyes,becausethespatialpartoftheexpressionisthesumofthesquaresofthecomponentsofaspacevector.Itbehaveslike the expression x2 + y2 + z2 and is therefore rotationally invariant. Because theLagrangianofEq.4.24isinvariantnotonlywithrespecttoLorentztransformationsalongthexaxisbutalsowithrespecttorotationsofspace,it’sinvariantunderageneralLorentztransformation.

Eq.4.24isoneofthesimplestfieldtheories.4 Itgivesrise toawaveequationin thesamewayasEq.4.7does.Following thesamepatternwedid for thatexample, it’snothardtoseethatthewaveequationderivedfromEq.4.24is

It’s a particularly simplewave equation because it’s linear; the field and its derivativesonlyappeartothefirstpower.Therearenotermslikeϕ2orϕtimesaderivativeofϕ.Wegetanevensimplerversionwhenwesetμequaltozero:

4.4.5ClassicalFieldSummaryWenowhaveaprocess fordevelopingaclassical field theory.Our first examplewasascalarfield,butthesameprocesscouldapplytoavectororevenatensorfield.Startwiththefielditself.Next,figureoutallthescalarsyoucanmake,usingthefielditselfanditsderivatives.Onceyouhave listedorcharacterizedall thescalarsyoucancreate,buildaLagrangianfromfunctionsofthosescalars,forexample,sumsofterms.Next,applytheEuler-Lagrangeequations.Thatamounts towriting theequationsofmotion,or the fieldequations describing the propagation of waves, or whatever else the field theory issupposedtodescribe.Thenextstepistostudytheresultingwaveequation.

Classicalfieldsneedtobecontinuous.Afieldthat’snotcontinuouswouldhaveinfinitederivatives, and therefore infinite action. Such a field would also have infinite energy.We’llhavemoretosayaboutenergyinthenextlecture.

4.5FieldsandParticles—ATaste

Beforewrappingup,Iwant tosayafewthingsabout therelationbetweenparticlesandfields.5IfIhadworkedouttherulesforelectrodynamicsinsteadofasimplescalarfield,Imight tell youhowchargedparticles interactwith an electromagnetic field.Wehaven’tdone that yet. But howmight a particle interact with a scalar field ϕ? Howmight thepresenceofthescalarfieldaffectthemotionofaparticle?

Let’s think about the Lagrangian of a particle moving in the presence of apreestablishedfield.Supposesomeonehassolvedtheequationsofmotion,andweknowthatthefieldϕ(t,x)issomespecificfunctionoftimeandspace.Nowconsideraparticlemovinginthatfield.Theparticlemightbecoupledtothefield,insomewayanalogoustoachargedparticleinanelectromagneticfield.Howdoestheparticlemove?Toanswerthatquestion, we go back to particle mechanics in the presence of a field. We’ve alreadywrittendownaLagrangianforaparticle. Itwas−mdτ,becausedτ is justabout theonlyinvariantquantityavailable.Theactionintegralwas

Toget thecorrectnonrelativistic answers for lowvelocities,we found thatweneed theminus sign, and that the parameterm behaves like the nonrelativistic mass. Using therelationshipdτ2=dt2−dx2,werewrotethisintegralas

wheredx2standsforallthreedirectionsofspace.Then,wefactoredoutthedt2,andouractionintegralbecame

Noticingthatdx/dtisasynonymforvelocity,thisbecame

WethenexpandedtheLagrangian asapowerseriesandfound that itmatchesthefamiliarclassicalLagrangianatlowvelocities.ButthisnewLagrangianisrelativistic.

WhatcanwedotothisLagrangiantoallowtheparticletocoupletoafield?Forthefield toaffect theparticle, thefield itselfneeds toappearsomewhere in theLagrangian.WeneedtoinsertitinawaythatisLorentzinvariant.Inotherwords,wehavetoconstructascalarfromthefield.Aswesawbefore,therearemanywaystoaccomplishthis,butonesimpleactionwecouldtryis

or

ThiscorrespondstoaLagrangian,

Thisisoneofthesimplestthingsyoucando,buttherearelotsofotherpossibilities.Forexample, in theprecedingLagrangian,youcould replaceϕ(t,x)with its square,orwithany other function of ϕ(t, x). For now, we’ll just use this simple Lagrangian, and itscorrespondingactionintegral,Eq.4.28.

Eq. 4.29 is one possible Lagrangian for a particlemoving in a preestablished field.Nowwecanask:Howdoestheparticlemoveinthisfield?Thisissimilartoaskinghowaparticle moves in an electric or magnetic field. You write down a Lagrangian for theparticle in theelectricormagnetic field;youdon’tworryabouthowthe fieldgot there.Instead, you just write down the Lagrangian and then write out the Euler-Lagrangeequations.We’llworkoutsomeofthedetailsforthisexample.Butbeforewedo,IwanttopointoutaninterestingfeatureofEq.4.29.

4.5.1TheMysteryFieldSuppose,forsomereason,thefieldϕ(t,x)tendstomigratetosomespecificconstantvalueotherthanzero.Itjusthappensto“like”gettingstuckatthatparticularvalue.Inthatcase,ϕ(t,x)wouldbeconstantor approximately constant,despite its formaldependenceon tandx. Themotion of the particlewould then look exactly the same as themotion of aparticlewhosemass ism +ϕ. Letme say this again: The particlewithmassm wouldbehaveasthoughitsmassism+ϕ.Thisisthesimplestexampleofascalarfieldthatgivesrisetoashiftinthemassofaparticle.

At thebeginningof the lecture,Imentionedthat there’safield innature thatcloselyresemblestheonewe’relookingattoday,andIinvitedyoutotryguessingwhichoneitis.Haveyoufigureditout?Thefieldwe’relookingatbearsacloseresemblancetotheHiggsfield. Inourexample,ashift in thevalueofascalar fieldshifts themassesofparticles.Our example is not exactly the Higgsmechanism, but it’s closely related. If a particlestarts out with a mass of zero and is coupled to the Higgs field, this coupling caneffectively shift the particle’smass to a nonzero value. This shifting ofmass values isroughlywhatpeoplemeanwhentheysaythattheHiggsfieldgivesaparticleitsmass.TheHiggsfieldentersintotheequationsasifitwerepartofthemass.

4.5.2SomeNutsandBoltsWe’llwrapupthislecturewithaquickpeekattheEuler-Lagrangeequationsforourscalarfieldexample.We’renotgoingtofollowthemallthewaythroughbecausetheybecomeugly after the first few steps.We’ll just give you a taste of this process for now. Forsimplicity,we’llworkas if there’sonlyonedirectionofspace,with theparticlemovingonly in the x direction. I’ll copy our Lagrangian, Eq. 4.29, here for easy reference,replacingthevariablevwith :

Thefirststep inapplyingLagrange’sequations is tocalculate thepartialderivativeofwithrespectto .Rememberthatwhenwetakeapartialderivativewithrespecttosomevariable,wetemporarilyregardalltheothervariablesasconstants.Inthiscase,weregardtheexpressioninsquarebracketsasaconstantbecauseithasnoexplicitdependenceon .

Ontheotherhand, theexpression doesdependexplicitlyon .Taking thepartialderivativeofthisexpressionresultsin

This equation should look familiar. We obtained a nearly identical result in Lecture 3whenwedidasimilarcalculationtofindthemomentumofarelativisticparticle.Theonlythingthat’sdifferenthereistheextratermϕ(t,x)insidethesquarebrackets.Thissupportsthenotionthattheexpression[m+ϕ(t,x)]behaveslikeaposition-dependentmass.

ContinuingwiththeEuler-Lagrangeequations,thenextstepistodifferentiateEq.4.30withrespecttotime.We’lljustindicatethisoperationsymbolicallybywriting

That’stheleftsideoftheEuler-Lagrangeequation.Let’slookattherightside,whichis

Sincewe’redifferentiatingwithrespecttox,weaskwhether dependsexplicitlyonx.Itdoes,becauseϕ(t,x)dependsonx,andϕhappens tobe theonlyplacewherexappears.Theresultingpartialderivativeis

andthereforetheEuler-Lagrangeequationbecomes

That’stheequationofmotion.It’sadifferentialequationthatdescribesthemotionofthefield.Ifyoutrytoworkoutthetimederivativeontheleftside,you’llseethatit’squitecomplicated.We’llstopatthispoint,butyoumaywanttothinkabouthowtoincorporatethevelocityoflightc intothisequation,andhowthefieldbehavesinthenonrelativisticlimitoflowvelocities.We’llhavesomethingstosayaboutthatinthenextlecture.

1There’sonlyoneEuler-Lagrangeequationinthisexamplebecausetheonlyvariablesareϕand .

2Ioftensayleastorminimum,butyouknowverywellthatIreallymeanstationary:Theactioncouldbeamaximumoraminimum.

3This is justamatterofdimensionalconsistency.Unlikeparticlemotion, theaction integral forafield is takenoverdxdydz,aswellasdt.Fortheactiontohavethesameunitsforfieldsasitdoesforparticles,theLagrangianmustcarryunitsofenergydividedbyvolume.Hencethetermdensity.

4Wecouldmakeitevensimplerbydroppingthelastterm, .

5 Not the quantum mechanical relation between particles and fields, just the interaction between ordinary classicalparticlesandclassicalfields.

Lecture5

ParticlesandFields

Nov9,2016.

Thedayafterelectionday.Artismoroselystaringintohisbeer.Lennyisstaringintohisglassofmilk.Noone—notevenWolfgangPauli—canthinkofanythingfunnytosay.ButthenmightyJohnWheelerrisestohisfeetandraiseshishandatthebarwhereallcanseehim.SuddensilenceandJohnspeaks:

“Ladiesandgentlemen, in this timeof terribleuncertainty, Iwant toremindyouofonesure thing: SPACETIME TELLS MATTER HOW TO MOVE; MATTER TELLS SPACETIME HOW TO

CURVE”.

“Bravo!”

AcheerisheardinHermann’sHideawayandthingsbrighten.Pauliraiseshisglass:

“ToWheeler!Ithinkhe’shitthenailonthehead.Letmetrytosayitevenmoregenerally:FIELDSTELLCHARGESHOWTOMOVE;CHARGESTELLFIELDSHOWTOVARY.”

In quantum mechanics, fields and particles are the same thing. But our main topic isclassicalfieldtheory,wherefieldsarefields,andparticlesareparticles.Iwon’tsay“neverthetwainwillmeet”becausetheywillmeet;infact, they’llmeetverysoon.Butthey’renotthesamething.Thecentralquestionofthislectureisthis:

Ifafieldaffectsaparticle, forexamplebycreatingforcesonit,must theparticleaffectthefield?

IfA affectsB,whymustB necessarily affectA?We’ll see that the two-way nature ofinteractions, often called “action and reaction,” is built into the Lagrangian actionprinciple.Asasimpleexample,supposewehavetwocoordinates,xandy,alongwithanactionprinciple.1Generally,theLagrangianwilldependonxandy,andalsoon and .Onepossibility is that theLagrangian issimplyasumof two terms:aLagrangian forxand ,plusaseparateLagrangianforyand

whereI’velabeledtheLagrangiansontherightsidewithsubscriptstoshowthattheymaybedifferent.Let’slookattheEuler-Lagrangeequation(whichistheequationofmotion)forx,

Because has no dependency on x or on , the terms drop out, and the Euler-Lagrangeequationforthexcoordinatebecomes

The y variable and its time derivative do not appear at all. Likewise, the equation ofmotionfortheyvariablewillnotincludeanyreferencetoxor .Asaresult,xdoesnotaffecty,andydoesnotaffectx.Let’slookatanotherexample,

where the Vx and Vy terms are potential energy functions. Once again, the x and ycoordinatesofthisLagrangianarecompletelyseparated.TheLagrangianisasumoftermsthatonlyinvolvexoronlyinvolvey,andbyasimilarargumentthexandycoordinateswillnotaffecteachother.

But suppose we know that y does affect x. What would that tell us about theLagrangian?IttellsusthattheLagrangianmustbemorecomplicated;thattheremustbethingsinitthatsomehowaffectbothxandy.TowritesuchaLagrangian,weneedtoputinsomeadditionalingredientthatinvolvesbothxandy inawaythatyoucan’tunravel.Forexample,wecouldjustaddanxyterm:

Thisguaranteesthattheequationofmotionforxwillinvolvey,andviceversa.IfxandyappearintheLagrangiancoupledtogetherinthismanner,there’snowayforonetoaffecttheotherwithout theother affecting theone. It’s as simpleas that.That’s the reasonA

mustaffectBifBaffectsA.

Inthepreviouslecturewelookedatasimplefieldandaskedhowitaffectsaparticle.After a brief review of that example, we’ll ask the opposite question: How does theparticle affect the field? This is very much an analog of electromagnetic interactions,where electric and magnetic fields affect the motion of charged particles, and chargedparticles create and modify the electromagnetic field. The mere presence of a chargedparticle creates a Coulomb field. These two-way interactions are not two independentthings;theycomefromthesameLagrangian.

5.1FieldAffectsParticle(Review)Let’sstartoutwithagivenfield,

ϕ(t,x),

thatdependsontandx.Fornow,assumethatϕ issomeknownfunction.Itmayormaynotbeawave.We’renotgoingtoaskaboutthedynamicsofϕjustyet;firstwe’lllookattheLagrangianfortheparticle.Recallfrompreviouslectures(Eq.3.24,forexample)thatthisLagrangianis

I’velabeledit forclarity.Apartfromthefactor−m, theaction amounts toasum of all the proper times along each little segment of a path. In the limit, as thesegmentsbecomesmallerandsmaller,thesumbecomesanintegral

overthecoordinatetimet.Thisisthesameas−mtimestheintegralofthepropertimedτfromoneendofthetrajectorytotheother.ThisLagrangiandoesn’tcontainanythingthatcausesthefieldtoaffecttheparticle.Let’smodifyitinasimpleway,byaddingthefieldvalueϕ(t,x)tom.

Wecannowworkout theEuler-Lagrangeequationsandfindouthowϕ(t,x)affects themotionoftheparticle.Ratherthanworkoutthefullrelativisticequationsofmotion,we’lllookatthenonrelativisticlimitwheretheparticlemovesveryslowly.That’sthelimitinwhichthespeedoflightgoestoinfinity—whencisfarbiggerthananyothervelocityintheproblem.It’shelpfultorestoretheconstantcinourequationstoseehowthisworks.Themodifiedactionintegralis

Wecancheckthisfordimensionalconsistency.TheLagrangianhasunitsofenergy,andsodoes−mc2.I’vemultipliedϕ(t,x)byaconstantg,whichiscalledacouplingconstant. Itmeasures the strength bywhich the field affects themotionof the particle, andwe canselect its units to guarantee that g times ϕ(t, x) has units of energy. So far, g can be

anything;wedon’tknowitsvalue.Bothtermsinsidethesquarerootarepurenumbers.

Nowlet’sexpandthesquarerootusingtheapproximationformula

where isasmallnumber.Re-writingthesquarerootwithanexponentof ,andequating

with ,wecanseethat

Thehigher-order terms are far smaller because they involvehigher powersof the ratioWecannowusethisapproximateexpressiontoreplacethesquarerootintheaction

integral,resultingin

Let’s lookat this integraland find thebiggest terms—the terms thataremost importantwhenthespeedof lightgets large.Thefirst term,mc2, is justanumber.Whenyoutakederivatives of theLagrangian it just “comes along for the ride” and has nomeaningfulimpactontheequationsofmotion.We’llignoreit.Inthenextterm,

thespeedof lightcancels itselfoutaltogether.Therefore this termispartof the limit inwhich the speed of light goes to infinity, and it survives that limit. This term is quitefamiliar;it’souroldfriend,thenonrelativistickineticenergy.Theterm

becomes zero in the limit of largec becauseof thec2 in its denominator;we ignore it.Finally,thetermgϕ(t,x)containsnospeedsoflight.Thereforeitsurvives,andwekeepitintheLagrangian,whichnowbecomes

That’sall there iswhen theparticlemovesslowly.Wecannowcompare itwith theoldfashionednonrelativisticLagrangian,kineticenergyminuspotentialenergy,

T−V.

We’ve already recognized the first term of Eq. 5.3 as kinetic energy, and nowwe canidentifygϕ(t,x)asthepotentialenergyofaparticleinthisfield.Theconstantgindicatesthe strength of coupling between the particle and the field. In electromagnetism, forexample,thestrengthofcouplingofthefieldtoaparticleisjusttheelectriccharge.Thebiggertheelectriccharge,thebiggertheforceonaparticleinagivenfield.We’llcomebacktothisidea.

5.2ParticleAffectsField

Howdoes theparticle affect the field?The important thing tounderstand is that there’sonlyoneaction,the“total”action.Thetotalactionincludesactionforthefieldandactionfortheparticle.Ican’temphasizethisenough:Whatwe’restudyingisacombinedsystemthatconsistsofa)afieldandb)aparticlemovingthroughthefield.

Fig. 5.1 illustrates the physics problem we’re trying to solve. It shows a region ofspacetime,representedasacube.2Timepointsupwardandthexaxispointstotheright.Insidethisregion,there’saparticlethattravelsfromonespacetimepointtoanother.Thetwodotsaretheendpointsofitstrajectory.Wealsohaveafieldϕ(t,x)insidetheregion.Iwish I could think of a clever way to draw this field without cluttering the diagram,becauseit’severybitasphysicalastheparticle;it’spartofthesystem.3

Figure5.1:Particlemovingthrougharegionof

spacetimefilledwith“redmush”—thescalarfieldϕ(t,

x).

Tofindouthow this field-particle systembehaves,weneed toknow theLagrangianandminimize theaction. Inprinciple, this is simple;we justvary theparametersof theproblemuntilwefindasetofparametersthatresultsinthesmallestpossibleaction.Wewigglethefieldaroundindifferentways,andwewiggletheparticletrajectorybetweenitstwoendpointsuntil theaction integral isassmallaswecanmake it.Thatgivesus thetrajectoryandthefieldthatsatisfytheprincipleofleastaction.

Let’s write down the whole action, the action that includes both the field and theparticle.First,weneedanactionforthefield,

wherethesymbol means“Lagrangianforthefield.”Thisactionintegralistakenovertheentirespacetimeregion,t,x,y,andz.Thesymbold4x isshorthandfordtdxdydz.Forthis example,we’ll baseour fieldLagrangianonEq.4.7, aLagrangian thatweused inLecture4.We’lluseasimplifiedversionthatonlyreferencesthexdirectioninspaceandhasnopotentialfunctionV(ϕ).4TheLagrangian

leadstotheactionintegral

Thisistheactionforthefield.5Itdoesn’tinvolvetheparticleatall.Nowlet’sincorporatetheactionfortheparticle,Eq.5.2withthespeedsoflightremoved.Thisisjust

or

AlthoughActionparticleistheparticleaction,italsodependsonthefield.Thisisimportant;whenwewiggle thefield, thisactionvaries. Infact, it iswrongto thinkofActionparticlepurelyasaparticleaction:Theterm

Figure5.2:ParticleatRestinImaginaryRedMush(blackinkonpaper).

isaninteractiontermthat tells theparticlehowtomoveinthefield,but italsotells thefieldhowtovaryinthepresenceoftheparticle.FromnowonIwillcallthistermandceasethinkingofitashavingtodoonlywiththeparticle.

We’ll consider the simple special case where the particle is at rest at x = 0. It’sreasonable to assume that there’s some solution where the particle is at rest—classicalparticlesdorestsometimes.Fig.5.2showsthespacetimetrajectoryofarestingparticle.It’sjustaverticalline.

How do we modify Eq. 5.5 to show that the particle is at rest?We just set , thevelocity,equaltozero.Thesimplifiedactionintegralis

Becausetheparticlesitsatthefixedpositionx=0,wecanreplaceϕ(t,x)withϕ(t,0)andwrite

Let’stakeacloserlookat .Noticethatitonlydependsonthevalueofthefieldatthe origin. More generally it depends on the value of the field at the location of theparticle.Neverthelesswhenthefieldiswiggled,ϕ(t,0)willwiggle,affectingtheaction.Aswewillsee,thisaffectstheequationofmotionforthefield.

Theactionforafieldisnormallywrittenasanintegraloverspaceandtime,butEq.5.6isanintegralthatonlyrunsovertime.Thereisnothingwrongwithit,butit’sconvenienttorewriteitasanintegraloverspaceandtime.We’lluseatrickthatinvolvestheideaofasource function.Letρ(x) be a fixed definite function of space, but for themoment nottime.(I’llgiveyouahint:ρ(x)issomethinglikeachargedensity.)Let’sforgettheparticlebutreplaceitwithatermintheLagrangianthatI’llcontinuetocall ,

Thecorrespondingtermintheactionforthefieldis

ThislooksquitedifferentfromtheactioninEq.5.2.Tomakethemthesame,weusethetrickthatDiracinvented—theDiracdeltafunctionδ(x).Thedeltafunctionisafunctionofthe space coordinates that has a peculiar property. It is zero everywhere but at x = 0.Neverthelessithasanonzerointegral,

Let’simaginegraphingthedeltafunction.It iszeroeverywhereexceptintheimmediatevicinityofx=0.Butitissolargeinthatvicinitythatithasatotalareaequalto1.Itisaveryhighandnarrowfunction,sonarrowthatwemaythinkof itasconcentratedat theorigin,butsohighthatithasafinitearea.

Norealfunctionbehavesthatway,buttheDiracfunctionisnotanordinaryfunction.Itis really a mathematical rule. Whenever it appears in an integral multiplying anotherfunctionF(x),itpicksoutthevalueofFattheorigin.Hereisitsmathematicaldefinition:

whereF(x)isan“arbitrary”function.6IfyouhavesomefunctionF(x)andintegrateitasshown,thedeltafunctionpicksoutthevalueofF(x)atx=0.Itfiltersoutallothervalues.It’s the analog of the Kronecker delta, but operates on continuous functions. You canvisualizeδ(x)asafunctionwhosevalueiszeroeverywhere,exceptwhenxgetsveryclosetozero.Whenx is close tozero,δ(x)hasavery tall spike.Forourcurrentproblemweneedathree-dimensionaldeltafunctionthatwecallδ3(x,y,z).Wedefineδ3(x,y,z)astheproductofthreeone-dimensionaldeltafunctions,

δ3(x,y,z)=δ(x)δ(y)δ(z).

As we do elsewhere, we often use the shorthand δ3(x), where x represents all threedirectionsofspace.Whereisδ3(x,y,z)nonzero?It’snonzerowhereall threefactorsontherightsidearenonzero,andthatonlyhappensinoneplace:thepointx=0,y=0,z=0.Atthatpointofspaceit’senormous.

The trick inwriting the interaction termasan integralshouldnowbefairlyobvious.We simply replace the particle with a source function ρ that we choose to be a deltafunction,

TheactioninEq.5.8thentakestheform

Thepointisthatifweintegratethisoverthespacecoordinates,thedeltafunctionruletellsustosimplyevaluategϕ(x)attheorigin,andwegetsomethingwe’veseenbefore:

Nowlet’scombine the fieldactionwith the interaction term.Wedo that in thesimplestpossibleway,byjustaddingthetwoofthemtogether.CombiningtheactiontermsofEqs.5.4and5.12resultsin

orreplacingthesourcefunctionwiththedeltafunction,

Hereitis,thetotalactionasanintegraloverspaceandtime.TheexpressioninsidethebigsquarebracketsistheLagrangian.LikeanygoodfieldLagrangian,ithasthefieldaction(representedherebypartialderivatives)alongwithadelta function term that representstheeffectoftheparticleonthefield.Thisparticulardeltafunctionrepresentsthespecialsituationwhere the particle is at rest. But we could also jazz it up so that the particlemoves.Wecoulddothatbymakingthedeltafunctionmovearoundwithtime.Butthat’snot important right now. Instead, let’s work out the equations ofmotion, based on theLagrangian

5.2.1EquationsofMotionForconvenience,I’llrewriteEq.4.5,theEuler-Lagrangeequation,righthere.

Eq.5.15tellsuswhattodowith inordertofindtheequationsofmotion.Theindexμruns through the values 0, 1, 2, and 3. The first value it takes is 0, which is the time

component.Thereforeourfirststepistofindthederivative

Let’sunwindthiscalculationstepbystep.First,whatisthepartialof withrespectto

?There’sonlyonetermin thatinvolves .Straightforwarddifferentiationshowsthattheresultis

Applying toeachsideresultsin

That’s the first term in the equation of motion. It looks a lot like an acceleration.Westartedwithakineticenergytermthatamountsto .Whenweworkedoutthederivative,wegotsomethingthatlookslikeanaccelerationofϕ.

Next,weletμ take thevalue1andcalculate thefirstspacecomponent.Theformofthese termsisexactly thesameas theformof the timecomponentexceptfor theminussign.Sothefirsttwotermsoftheequationofmotionare

Becausetheyandzcomponentshavethesameformasthexcomponent,wecanaddtheminaswell,

Ifthoseweretheonlyterms,Iwouldsetthisexpressionequaltozero,andwewouldhavea good old-fashioned wave equation for ϕ. However, the Lagrangian depends on ϕ inanotherwaybecauseoftheinteractionterm.Thecompleteequationofmotionis

andwecanseethatthelasttermis

Addingthisfinalpiecetotheequationofmotiongivesus

Thusweseethatthesourcefunctionappearsinthefieldequationforϕasanadditiontothewaveequation.Withoutthesourceterm,ϕ=0isaperfectlygoodsolution,butthat’snottruewhenthesourcetermispresent.Thesourceisliterallythat:asourceoffieldthatpreventsthetrivialϕ=0frombeingasolution.

Intheactualcasewherethesourceisaparticleatrest,ρ(x)canbereplacedbythedeltafunction,

Let’ssupposeforamomentthatwe’relookingforstaticsolutionstoEq.5.16.Afterall,the particle is standing still, and theremay be a solutionwhere the field itself doesn’tchangewith time. It seems plausible that a standing-still particle can create a field thatalsostandsstill—afieldthatdoesnotvarywithtime.Wemightthenlookforasolutionin

whichϕistime-independent.Thatmeansthe terminEq.5.16wouldbecomezero.Wecanthenchangethesignsoftheremainingtermstoplusandwrite

Perhaps you recognize this as Poisson’s equation. It describes, among other things, theelectrostaticpotentialofapointparticle.It’softenwrittenbysetting∇2ϕequaltoasource(a charge density) on the right side.7 In our example, the charge density is just a deltafunction.Inotherwords,it’sahigh,sharpspike,

Of course, our current example is not an electric or magnetic field. Electrodynamicsinvolvesvectorfields,andwe’relookingatascalarfield.Butthesimilaritiesarestriking.

ThethirdtermoftheLagrangian(Eq.5.14)tieseverythingtogether.Ittellstheparticletomoveasiftherewereapotentialenergy−gϕ(t,x).Inotherwords,itexertsaforceontheparticle.Thissameterm,whenusedfortheequationofmotionoftheϕfield,tellsusthattheϕfieldhasasource.

Thesearenotindependentthings.Thefactthatthefieldaffectstheparticletellsusthattheparticleaffectsthefield.Foraparticleatrestinastaticfield,Eq.5.18tellsusexactlyhow. The parameterg determines how strongly the particle affects the field. The sameparameter also tells us how strongly the field affects the particle. Itmakes a nice littlestory:FieldsandparticlesaffecteachotherthroughacommontermintheLagrangian.

5.2.2TimeDependenceWhathappensifweallowtheparticletomoveandthefieldtochangewithtime?We’llconfineourselvestoasingle

dimensionofspace.Eq.5.18becomes

∇2ϕ=gδ(x).Inonedimension,theleftsideofthisequationisjustthesecondderivativeofϕ.SupposeIwanted to put the particle someplace else. Instead of putting it at the origin, suppose Iwanttoputitatx=a?AllIneedtodoischangexintheprecedingequationtox−a,andtheequationwouldbecome

∇2ϕ=gδ(x−a).

Thedelta functionδ(x−a)has itsspikewherex is equal toa.Suppose further that theparticleismoving,anditspositionisafunctionoftime,a(t).Wecanwritethisas

∇2ϕ=gδ(x−a(t)).Thiswouldtellyouthatthefieldhasasource,andthatthesourceismoving.Atanygiventimethesourceisatpositiona(t).Inthiswaywecanaccommodateamovingparticle.Butwestillhaveonelittlewrinkletodealwith.Iftheparticleismoving,wewouldnotexpectthe field to be time independent. If the particlemoves around, then the fieldmust also

dependontime.Rememberthe termthatwezeroedoutintheequationofmotion(Eq.5.16)?Foratime-dependentfield,wehavetorestorethatterm,resultingin

Iftherightsidedependsontime,there’snowaytofindasolutionwhereϕitselfistime-independent.Theonlywaytomakeitconsistentistorestorethetermthatinvolvestime.

Amovingparticle,forexampleaparticlethatacceleratesorvibrates,willgivethefieldatimedependence.Youprobablyknowwhatitwilldo:Itwillradiatewaves.Butat themoment,we’renotgoingtosolvewaveequations.Instead,we’llspendalittletimetalkingaboutthenotationofrelativity.

5.3UpstairsandDownstairsIndicesNotationisfarmoreimportantthanmostpeoplerealize.Itbecomespartofourlanguageandshapesthewaywethink—forbetterorforworse.Ifyou’reskepticalaboutthis,justtryswitchingtoRomannumeralsthenexttimeyoudoyourtaxes.

Themathematical notationwe introduce heremakes our equations look simple andpretty.Itsavesusfromhavingtowritethingslike

everysingletimewewriteanequation.8 In theprevious lecture,wespentsometimeonstandardnotationforrelativisticvectors,4-vectors,andscalars.We’llrevisitthatmaterialbriefly,thenexplainthenewcondensednotation.

ThesymbolXμstandsforthefourcoordinatesofspacetime,whichwecanwriteas

Xμ=(t,x,y,z),

where the index μ runs over the four values 0 through 3. I’m going to start payingattention towhere I put this index.Here, I’ve put the index up on top.That carries nomeaningfornow,butsoonitwill.

ThequantityXμ,ifthoughtofasadisplacementfromtheorigin,isa4-vector.Callingit a4-vector is a statement about theway it behavesunderLorentz transformation; anycomplexoffourquantitiesthattransformsthesamewayastandxisa4-vector.

The three space components of a 4-vectormay equal zero in your reference frame.

You,inyourframe,wouldsaythatthisdisplacementispurelytimelike.Butthisisnotaninvariant statement. Inmy frame, the spacecomponentswouldnotall equalzero,and Iwould say that the object does move in space. However, if all four components of adisplacement4-vectorarezeroinyourframe, theywillalsobezeroinmyframeandinevery other frame. A statement that all four components of a 4-vector are zero is aninvariantstatement.

Differencesof4-vectorsarealso4-vectors.Soaredifferentialdisplacementssuchas

dXμ.

Startingwitha4-vector,wecanmakeascalar,aquantitythatremainsthesameineveryframe.Forexample,wecanconstructaproper timedτ2 fromadisplacement.Youhavealreadyseenhow.Thequantity

(dτ)2=dt2−dx2−dy2−dz2

anditscounterpart

(ds)2=−dt2+dx2+dy2+dz2

arethesameineveryreferenceframe.They’rescalarsandhaveonlyonecomponent.Ifascalarisequaltozeroinoneframeofreference,it’szeroineveryframe.Indeed,thisisthedefinitionofascalar;it’sathingthat’sthesameineveryframe.

ThepatternwefollowedforcombiningthecomponentsofdXμtoformthespacetimeintervalds2(andthepropertimedτ)isverygeneral.Wecanapplyit toany4-vectorAμ,andtheresultwillalwaysbeascalar.We’llusethispatternoverandoveragain,andwedon’twanttowritethelongexpression

−(At)2+(Ax)2+(Ay)2+(Az)2

every timeweneed it. Instead,we’llcreatesomenewnotation tomake thingseasier.Amatrix called themetric figures heavily in this new notation. In special relativity, themetric isoftencalledη (Greek lettereta),with indicesμandν. It’s a simplematrix. Infact, it’s almost the same as the identitymatrix. It has three diagonal elements that areequalto1, just liketheunitmatrix.Butthefourthdiagonalelementis−1.Thiselementcorrespondstotime.Here’swhattheentirematrixlookslike:

In this notation, we represent a 4-vector as a column. For example, the 4-vectorAν iswrittenas

Let’stakethematrixημνandmultiplyitbythevectorAν,whereνrunsfrom0to3.Usingthesummationsign,wecanwritethatas

Thisisanewkindofobject,whichwe’llcallAμ,withalowerindex,

Let’s figure outwhat this newobject “A-with-a-lower-index” is. It’s not the original 4-vectorAν.Ifηreallyweretheidentitymatrix,multiplyingavectorbyitwouldgiveyoutheexactsamevector.Butit’snottheidentitymatrix.The−1inthediagonalmeansthatwhenyouformtheproduct∑νημνAν,thesignofthefirstcomponent,At,getsflipped,andeverythingelsestaysthesame.WecanwriteimmediatelythatAνis

In other words, when I perform this operation, I simply change the sign of the timecomponent.Ingeneralrelativity,themetrichasadeepergeometricmeaning.Butforourpurposes,it’sjustaconvenientnotation.9

5.4EinsteinSumConventionIf necessity is the mother of invention, laziness is the father. The Einstein summationconventionisanoffspringofthishappymarriage.WeintroduceditinSection4.4.2,andnowweexploreitsusealittlefurther.

Wheneveryouseethesameindexbothdownstairsandupstairs inasingle term,youautomatically sum over that index. Summation is implied, and you don’t need asummationsymbol.Forexample,theterm

means

A0A0+A1A1+A2A2+A3A3becausethesameindexμappearsbothupstairsanddownstairsinthesameterm.Ontheotherhand,theterm

AνAμ

doesnotimplysummation,becausetheupstairsanddownstairsindicesarenotthesame.Likewise,

AνAνdoesnotimplysummationeventhoughtheindexν isrepeated,becausebothindicesaredownstairs.

Youmay recall that some of the equations in Section 3.4.3 used the symbol tosignifythesumofsquaresofspacecomponents.Byusingupstairsanddownstairsindicesalongwiththesummationconvention,wecouldhavewritten

whichismoreelegantandprecise.

The operation of Expression 5.19 has the effect of changing the sign of the timecomponent.Ishouldwarnyouthatsomeauthorsfollowtheconvention(+1,−1,−1,−1)fortheplacementoftheseminussigns.Iprefertheconvention(−1,1,1,1),typicallyusedbythosewhostudygeneralrelativity.

An index that triggers the summation convention, like ν in the following example,doesn’thaveaspecificvalue.It’scalledasummationindexoradummyindex;it’sathingyousumover.Bycontrast,an indexthat isnot summedover iscalleda free index.Theexpression

dependsonμ(whichisafreeindex),butitdoesn’tdependonthesummationindexν.Ifwe replace ν with any other Greek letter, the expression would have exactly the samemeaning. Ishouldalsomention that the termsupstairs indexanddownstairs indexhaveformal names. An upper index is called contravariant, and a lower index is calledcovariant.Ioftenusethesimplerwordsupperandlower,butyoushouldlearntheformalterms as well.We can haveA with an upper (contravariant) index, orA with a lower(covariant)index,andweusethematrixηtoconvertonetotheother.Convertingonekindofindextotheotherkindiscalledraisingtheindexorloweringtheindex,dependingonwhichwaywego.

Exercise5.1:ShowthatAνAνhasthesamemeaningasAμAμ.

Exercise5.2:Writeanexpression thatundoes theeffectofEq.5.20. Inotherwords,howdowe“gobackwards”?

Let’shaveanotherlookattheExpression5.19,

AμAμ.

Thisexpressionissummedoverbecauseitcontainsarepeatedindex,oneupperandonelower.Previously,weexpanded itusing the indices0 through3.Wecanwrite thesameexpressionusingthelabelst,x,y,andz:

AμAμ=AtAt+AxAx+AyAy+AzAz.

For thethreespacecomponents, thecovariantandcontravariantversionsareexactly thesame.Thefirstspacecomponent is just(Ax)2,anditdoesn’tmatterwhetheryouput theindexupstairsordownstairs.Thesameis truefor theyandzcomponents.But the timecomponentbecomes−(At)2,

AμAμ=−(At)2+(Ax)2+(Ay)2+(Az)2.

The timecomponenthasaminus signbecause theoperationof loweringor raising thatindex changes its sign.The contravariant and covariant time components haveopposite

signs,andAttimesAtis−(At)2.Ontheotherhand,thecontravariantandcovariantspacecomponentshavethesamesigns.

ThequantityAμAμ is exactlywhatwe think of as a scalar. It’s the difference of thesquareofthetimecomponentandthesquareofthespacecomponent.IfAμhappenstobeadisplacement such asXμ, then it’s the same as the quantity τ2, exceptwith anoverallminussign;inotherwords,it’s−τ2.Butwhateversignithas,thissumisclearlyascalar.

Thisprocessiscalledcontractingtheindices,andit’sverygeneral.AslongasAμisa4-vector,thequantityAμAμisascalar.Wecantakeany4-vectoratallandmakeascalarbycontracting its indices.WecanalsowriteAμAμ a littledifferentlyby referring toEq.5.20andreplacingAμwithημνAν.Inotherwords,wecanwrite

On the right side, we use the metric η and sum over μ and ν. Both sides of Eq. 5.21representthesamescalar.Nowlet’slookatanexampleinvolvingtwodifferent4-vectors,AandB.Considertheexpression

AμBμ.

Isthisascalar?Itcertainlylookslikeone.Ithasnoindicesbecausealltheindiceshavebeensummedover.

Toprovethatit’sascalar,we’llneedtorelyonthefactthatthesumsanddifferencesofscalarsarealsoscalars.Ifwehavetwoscalarquantities,thenbydefinitionyouandIwillagreeabout theirvalueseven thoughourreferenceframesaredifferent.But ifweagreeabouttheirvalues,wemustalsoagreeaboutthevalueoftheirsumandthevalueoftheirdifference.Therefore,thesumoftwoscalarsisascalar,andthedifferenceoftwoscalarsisalsoascalar.Ifwekeepthisinmind,theproofiseasy.Juststartwithtwo4-vectorsAμandBμandwritetheexpression

(A+B)μ(A+B)μ.

Thisexpressionmustbeascalar.Whyisthat?BecausebothAμandBμare4-vectors,theirsum(A +B)μ is also a 4-vector. If you contract any 4-vectorwith itself, the result is ascalar.Now,let’smodifythisexpressionbysubtracting(A−B)μ(A−B)μ.Thisbecomes

Thismodifiedexpressionisstillascalarbecauseit’sthedifferenceoftwoscalars.Ifweexpandtheexpression,wefindthattheAμAμtermscancel,andsodotheBμBμterms.TheonlyremainingtermsareAμBμandAμBμ,andtheresultis

I’llleaveitasanexercisetoprovethat

AμBμ=AμBμ.

It doesn’t matter if you put the ups down and the downs up; the result is the same.

Therefore,theexpressionevaluatesto

BecauseweknowthattheoriginalExpression5.22isascalar,theresultAμBμmustalsobeascalar.

YoumayhavenoticedthattheexpressionAμBμlooksalotliketheordinarydotproductoftwospacevectors.YoucanthinkofAμBμastheLorentzorMinkowskiversionofthedot product. The only real difference is the change of sign for the time component,facilitatedbythemetricη.

5.5ScalarFieldConventionsNext,we’llsetupsomeconventionsforascalarfield,ϕ(x).Inthisdiscussion,xrepresentsallfourcomponentsofspacetime,includingtime.Beforewegetrolling,Ineedtostateatheorem.Theproofisnothard,andI’llleaveitasanexercise.

Supposeyouhaveaknown4-vectorAμ.TosayAμisa4-vectorisnotjustastatementthatithasfourcomponents.ItmeansthatAμtransformsinaparticularwayunderLorentztransformations.SupposealsothatwehaveanotherquantityBμ.Wedon’tknowwhetherBμisa4-vector.Butwearetoldthatwhenweformtheexpression

AμBμ,

the result is a scalar.Given theseconditions, it’spossible toprove thatBμmustbea4-vector.With this result inmind, let’s consider the change invalueofϕ(x) between twoneighboringpoints. Ifϕ(x) is a scalar, you and Iwill agree on its value at each of twoneighboringpoints.Therefore,we’llalsoagreeonthedifferenceofitsvaluesatthesetwopoints: Ifϕ(x) is a scalar, the change inϕ(x) between two neighboring points is also ascalar.

Whatifthetwoneighboringpointsareinfinitesimallyseparated?Howdoweexpressthe difference inϕ(x) between these neighboring points?The answer comes frombasiccalculus: Differentiate ϕ(x) with respect to each of the coordinates, and multiply thatderivativebythedifferentialofthatcoordinate,

Followingthesummationconvention,therightsideisthederivativeofϕ(x)withrespecttottimesdt,plusthederivativeofϕ(x)withrespecttoxtimesdx,andsoon.It’sthesmallchangeinϕ(x)whengoingfromonepointtoanother.Wealreadyknowthatbothϕ(x)anddϕ(x)arescalars.Clearly,dXμ itself isa4-vector.Infact, it’s thebasicprototypeofa4-vector.Tosummarize:WeknowthattheleftsideofEq.5.24isascalar,andthatdXμontherightsideisa4-vector.Whatdoesthattellusaboutthepartialderivativeontherightside?According to our theorem, itmust be a 4-vector. It stands for a complex of fourquantities,

ForEq.5.24tomakesenseasaproduct,thequantity mustcorrespondtoacovariantvector,becausethedifferentialdXμ iscontravariant.Wehavediscoveredthatderivativesofascalarϕ(x)with respect to thecoordinatesare thecovariant componentsAμ of a4-vector.This isworthemphasizing:Thederivativesofascalarwithrespect toXμ formacovariant vector. They’re sometimes written with the shorthand symbol ∂μϕ, whichwenowdefinetobe

Thesymbol∂μϕhasalowerindextoindicatethatitscomponentsarecovariant.Isthereacontravariant version of this symbol?Youbet.The contravariant version has nearly thesamemeaningexcept that its timecomponenthas theoppositesign.Let’swrite thisoutexplicitly:

5.6ANewScalarWenowhavethetoolsweneedtoconstructanewscalar.Thenewscalaris

∂μϕ∂μϕ.

Ifweexpandthisusingthesummationconvention,weget

Whatdoesthisstandfor?It’ssimilartoafieldLagrangianthatwewrotedownearlier.Eq.4.7 contains the same expression with the signs reversed.10 In our new notation, theLagrangianis

ThismakesiteasytoseethattheLagrangianforthatscalarfieldisitselfascalar.Asweexplained before, having a scalar-valued Lagrangian is critical because an invariantLagrangianleadstoequationsofmotionthatareinvariantinform.Muchoffieldtheoryisabout theconstructionof invariantLagrangians.So far, scalarsand4-vectorshavebeenourmain ingredients.Goingforward,wewillneed toconstructscalarLagrangiansfromotherobjectsaswell:thingslikespinorsandtensors.Thenotationwe’vedevelopedherewillmakethattaskmucheasier.

5.7TransformingCovariantComponentsThe familiar Lorentz transformation equations, as presented, apply to contravariant

components.Theequationsareslightlydifferentforcovariantcomponents.Let’sseehowthisworks.Thefamiliarcontravarianttransformationsfortandxare

Thecovariantcomponentsare thesame,exceptfor the timecomponent.Inotherwords,wecanreplaceAxwithAx,and(A′)xwith(A′)x.However,thecovarianttimecomponentAtisthenegativeofthecontravarianttimecomponent.SowemustreplaceAtwith−(A)t,and(A′)twith−(A′)t.Makingthesesubstitutionsinthefirstequationresultsin

whichsimplifiesto

Applyingthemtothesecondequationgivesus

Theseequationsarealmostthesameasthecontravariantversions,exceptthatthesignofvhasbeenreversed.

5.8MathematicalInterlude:UsingExponentialstoSolveWaveEquationsStartingwithaknownLagrangian, theEuler-Lagrangeequationsprovidea template forwriting the equations of motion. The equations of motion are themselves differentialequations. For some purposes, knowing the form of these equations is good enough.However,sometimeswewouldliketosolvethem.

Findingsolutionstodifferentialequationsisahugetopic.Nevertheless,strippeddowntoitsbarebones,thebasicapproachisasfollows:

1.Propose(okay,guess)afunctionthatmightsatisfythedifferentialequation.

2.Plugthefunctionintothedifferentialequation.Ifitworks,you’redone.Otherwise,returntoStep1.

Insteadof rackingour brains over a solution,we’ll just provideone that happens towork.Itturnsoutthatexponentialfunctionsoftheform

arethemainbuildingblocksforwaveequations.Youmayfinditpuzzlingthatwechooseacomplexvaluedfunctionasasolution,whenourproblemsassumethatϕisarealvaluedscalarfield.Tomakesenseofthis,rememberthat

wherekx−ωtisreal.Eq.5.25highlightsthefactthatacomplexfunctionisthesumofareal function and i times another real function. Once we’ve worked out our solution

,weregardthesetworealfunctionsastwosolutionsandignorethei.Thisiseasytoseeifthecomplexfunctionissettozero.Inthatcase,itsrealandimaginarypartsmustseparatelyequalzero,andbothpartsaresolutions.11InEq.5.25,

cos(kx−ωt)

istherealpart,and

sin(kx−ωt)

istheimaginarypart.

If we ultimately extract real functions as our solutions, why bother with complexfunctionsatall?Thereasonis that it’seasytomanipulate thederivativesofexponentialfunctions.

5.9WavesLet’slookatthewaveequationandsolveit.WealreadyhavetheLagrangianforϕ,

However,Iwanttoextenditslightlybyaddingonemoreterm.Theadditionalterm,, is also a scalar. It’s a simple function ofϕ and does not contain any derivatives. Theparameterμ2isaconstant.OurmodifiedfieldLagrangianis

ThisLagrangianrepresentsthefieldtheoryanalogoftheharmonicoscillator.Ifwewerediscussing a harmonic oscillator and we called the coordinate of the oscillator ϕ, thekineticenergywouldbe

The potential energy would be , where μ2 represents a spring constant. TheLagrangianwouldbe

ThisLagrangianwouldrepresentthegoodoldharmonicoscillator.It’ssimilartoEq.5.26,our field Lagrangian. The only difference is that the field Lagrangian has some spacederivatives.Let’sworkouttheequationsofmotionthatcorrespondtoEq.5.26andthensolvethem.We’llstartwiththetimecomponent.TheEuler-Lagrangeequationstellustocalculate

forEq.5.26.Itshouldbeeasytoseethattheresultis

Thisistheanalogoftheaccelerationtermfortheharmonicoscillator.Wegetadditionaltermsby takingderivatives of the space components ofEq. 5.26.With these additionalterms,theleftsideoftheequationofmotionbecomes

Tofindtherightside,wecalculate .Theresultofthatcalculationis

Gathering our results for the left and right sides of the Euler-Lagrange equations, theequationofmotionforϕis

Nowlet’sputeverythingontheleftside,givingus

This is a nice simple equation.Do you recognize it? It’s theKlein-Gordon equation. ItprecededtheSchrödingerequationandwasanattempttodescribeaquantummechanicalparticle. The Schrödinger equation is similar.12 Klein andGordonmade themistake oftryingtoberelativistic.Hadtheynottriedtoberelativistic,theywouldhavewrittentheSchrödingerequationandbecomeveryfamous.Instead,theywrotearelativisticequationand became much less famous. The Klein-Gordon equation’s connection to quantummechanicsisnotimportantfornow.Whatwewanttodoissolveit.

Therearemanysolutions,allofthembuiltupfromplanewaves.Whenworkingwithoscillatingsystems,it’susefultopretendthatthecoordinateiscomplex.Then,attheendofthecalculation,welookattherealpartsandignorethei.WeexplainedthisideaintheprecedingMathematicalInterlude.

The solutions that interest us are the ones that oscillate with time and have acomponentoftheform

e−iωt.

This function oscillates with frequency ω. But we’re interested in solutions that alsooscillateinspace,whichhavetheformeikx.Inthreedimensions,wecanwritethisas

wherethethreenumberskx,ky,andkzarecalledthewavenumbers.13Theproductofthese

twofunctions,

isafunctionthatoscillatesinspaceandintime.We’lllookforsolutionsofthisform.

Incidentally,there’saslickwaytoexpresstherightsideofEq.5.28.Wecanwriteitas

Wheredoesthatexpressioncomefrom?Ifyouthinkofkasa4-vector,withcomponents(−ω,kx,ky,kz),thentheexpressionkμXμontherightsideisjust−ωt+kxx+kyy+kzz.14

Thisnotationiselegant,butfornowwe’llsticktotheoriginalform.

Let’s see what happens if we try to plug our proposed solution (Eq. 5.28) into theequationofmotion(Eq.5.27).We’llbetakingvariousderivativesofϕ.Eq.5.27tellsuswhichderivatives to take.We start by taking the secondderivativeofϕwith respect totime.DifferentiatingEq.5.28twicewithrespecttotimegivesus

Differentiatingtwicewithrespecttoxresultsin

Weget similar resultswhenwe differentiatewith respect to y and z. So far, theKlein-Gordonequationhasgeneratedtheterms

basedonourproposedsolution.Butwe’renot finished.Eq.5.27alsocontains the term+μ2ϕ.Thishastobeaddedtotheotherterms.Theresultis

Atthispoint,it’seasytofindasolution.Wejustsetthefactorinsidetheparenthesesequaltozeroandfindthat

Thistellsusthefrequencyintermsofthewavenumbers.Either+ωor−ωwillsatisfythisequation. Also, notice that each term under the square root is itself a square. So if aparticularvalueof(say)kx ispartofasolution,thenitsnegativewillalsobeapartofasolution.

Notice the parallel between these solutions and the energy equation, Eq. 3.43, fromLecture3,repeatedhere:

Eq. 5.30 represents the classical field version of an equation that describes a quantummechanical particlewithmassμ, energyω, andmomentum k.15We’ll come back to itagainandagain.

1Ofcoursewecouldhavemanymorethantwo,andtheydon’tneedtobeorthogonalCartesiancoordinates.

2Withtoofewdimensions,ofcourse.

3Inthevideo,Leonardshowsthefieldbyusingacoloredmarkertofilltheregionwith“redmush.”Ourdiagramsdon’tusecolor,sowehavetosettlefor“imaginaryredmush.”Justthinkofyourleastfavoriteschoolcafeteriaentrée.–AF

4Or,equivalently,V(ϕ)=0.

5Toavoidclutter,I’musingonlyonespacecoordinate.

6F(x)isnotstrictlyarbitrary.However,itrepresentsabroadclassoffunctions,andwecanthinkofitasarbitraryforourpurposes.

7Thesymbol∇2ϕisshorthandfor .SeeAppendixBforaquicksummaryofthemeaningof andothervector notation. You could also refer to Lecture 11 in Volume I of the Theoretical Minimum series. Many otherreferencesareavailableaswell.

8Substituteyourfavoriteexpletiveforthewordsingle.

9Itmightbebetter toexpressAνasa rowmatrix.However, thesummationconventiondescribed in thenextsectionminimizestheneedtowriteoutmatricesincomponentform.

10Eq.4.7alsocontainsapotentialenergyterm−V(ϕ),whichwe’reignoringfornow.

11Confusingly,theimaginarypartofacomplexfunctionistherealfunctionthatmultipliesi.

12TheSchrödingerequationonlyhasafirstderivativewithrespecttotime,andincludesthevaluei.

13Youcanthinkofthemasthreecomponentsofawavevector,where

14Itturnsoutthat(−ω,kx,ky,kz)reallyisa4-vector,butwehaven’tprovedit.

15TheequationshouldalsoincludesomePlanck’sconstantsthatI’veignored.

Interlude:CrazyUnits

“Hi,Art,areyouupforsometalkaboutunits?”

“Electromagneticunits?Oyvey,I’drathereatwormholes.Dowehaveto?”

“Well,takeyourpick:unitsorawormholedinner.”

“Okay,Lenny,youwin—units.”

When I first started learning about physics, something bothered me: Why are all thenumbers—theso-calledconstantsofnature—sobigor so small?Newton’sgravitationalconstantis6.7×10−11,Avogadro’snumberis6.02×1023, thespeedoflightis3×108,Planck’sconstantis6.6×10−34,andthesizeofanatomis10−10.NothinglikethiseverhappenedwhenIwaslearningmathematics.True,π≈3.14159ande≈2.718.Thesenaturalmathematical numbers were neither big nor small, and although they had their owntranscendentaloddness,IcouldusethemathIknewtoworkouttheirvalues.Iunderstandwhy biologymight have nasty numbers—it’s amessy subject—but physics?Why suchuglinessinthefundamentallawsofnature?

I.1UnitsandScaleTheanswerturnedouttobethatthenumericalvaluesoftheso-calledconstantsofnatureactuallyhavemoretodowithbiologythanphysics.1Asanexample, takethesizeofanatom,about10−10meters.Butwhydowemeasureinmeters?Wheredidthemetercomefromandwhyisitsomuchbiggerthananatom?

Whenasked thisway, theanswerstarts tocome intofocus.Ameter issimplyaunitthat’s convenient for measuring ordinary human-scale lengths. It seems that the meteraroseasaunit formeasuringropeorclothand itwassimply thedistancefromaman’snose(supposedlytheking’snose)tohisoutstretchedfingertips.

Butthatraisesthequestion,whyisaman’sarmsolong—1010—inatomicradii?Heretheanswerisobvious:Ittakesalotofatomstomakeanintelligentcreaturethatcanevenbothertomeasurerope.Thesmallnessofanatomisreallyallaboutbiology,notphysics.Youwithme,Art?

Andwhat about the speed of light;why so large?Here again the answermay havemore todowith life thanwithphysics.Therearecertainlyplaces in theuniversewherethings—evenlarge,massivethings—moverelativetoeachotherwithspeedsclosetothespeedoflight.Onlyrecently,twoblackholeswerediscoveredtobeorbitingeachotheratanappreciablefractionofthespeedoflight.Theycrashedintoeachother,butthat’stheway it goes;moving that fast can be dangerous. In fact, an environment full of objectswhizzingaroundatnearly thespeedof lightwouldbe lethal foroursoftbodies.So thefact that lightmovesvery fast on the scaleof ordinaryhumanexperience is, at least inpart,biology.Wecanonlylivewherethingswithappreciablemassmoveslowly.

Avogadro’snumber?Again, intelligentcreaturesarenecessarilybigonthemolecularscale,andtheobjectsthatwecaneasilyhandle,likebeakersandtesttubes,arealsobig.Thequantitiesofgasand fluid that fill abeaker are large (innumberofmolecules) forreasonsofconveniencetoourlargesoftselves.

Aretherebetterunitsmoresuitedtothefundamentalprinciplesofphysics?Yesindeed,butlet’sfirstrecallthatinstandardtextbookswearetoldthattherearethreefundamentalunits:length,mass,andtime.If,forexample,wechosetomeasurelengthinunitsoftheradiusofahydrogenatom,insteadofthelengthofaman’sarm,therewouldbenolargeorsmallconstantsintheequationsofatomicphysicsorchemistry.

But there isnothinguniversal about the radiusof an atom.Nuclearphysicistsmightstillcomplainaboutthesmallsizeoftheprotonor,evenworse,thesizeofthequark.Theobvious fixwouldbe touse thequark radiusas the standardof length.Butaquantum-gravity theorist would complain: “Look here, my equations are still ugly; the Plancklength is 10−19 in your stupid nuclear physics units. Furthermore, the Planck length ismuchmorefundamentalthanthesizeofaquark.”

I.2PlanckUnitsAsthebookssay,therearethreeunits:length,mass,andtime.Isthereamostnaturalsetof units?To put it anotherway, are there three phenomena that are so fundamental, souniversal, thatwecanuse them todefine themost fundamentalchoiceofunits? I thinkthere are, and so did Planck in 1900. The idea is to pick aspects of physics that arecompletely universal,meaning that they applywith equal force to all physical systems.Withsomeminorhistoricaldistortion,herewasPlanck’sreasoning:

The first universal fact is that there is a speed limit that allmattermust respect.Noobject—noobject—can exceed the speed of light,whether it be a photon or a bowlingball.Thatgivesthespeedoflightauniversalaspectthatthespeedofsound,oranyotherspeed,doesnothave.SoPlancksaid,let’schoosethemostfundamentalunitssothatthespeedoflightis1,c=1.

Nexthesaidthatgravityprovidesuswithsomethinguniversal,Newton’suniversallawofgravitation:EveryobjectintheuniverseattractseveryotherobjectwithaforceequaltoNewton’sconstanttimestheproductofthemasses,dividedbythesquareofthedistancebetween them. There are no exceptions; nothing is immune to gravity. Again, Planckrecognized something universal about gravity that is not true of other forces. HeconcludedthatthemostfundamentalchoiceofunitsshouldbedefinedsothatNewton’sgravitationalconstantissettounity:G=1.

Finally,athirduniversalfactofnature—onethatPlanckcouldnotfullyappreciatein1900—is the Heisenberg Uncertainty Principle.Without too much explanation, what itsays is that allobjects innatureare subject to the same limitationon theaccuracywithwhichtheycanbeknown:Theproductoftheuncertaintyinposition,andtheuncertaintyofmomentum,isatleastasbigasPlanck’sconstantdividedby2,

Again,thisisauniversalpropertythatappliestoeveryobject,nomatterhowbigorsmall—humans, atoms, quarks, and everything else. Planck’s conclusion: The mostfundamentalunitsshouldbesuchthathisownconstant issetto1.This,asitturnsout,isenoughtofixthethreebasicunitsoflength,mass,andtime.Today,theresultingunitsarecalledPlanckunits.

So then why don’t all physicists use Planck units? There is no doubt that thefundamental lawsofphysicswouldbemost simplyexpressed. In fact,many theoreticalphysicistsdowork inPlanckunits,but theywouldnotbeatallconvenient forordinary

purposes.ImagineifweusedPlanckunitsindailylife.Thesignsonfreewayswouldread

FigureI.1:TrafficSigninPlanckUnits.

Thedistancetothenextexitwouldbe1038,andthetimeinasingledaywouldbe8.6×1046. Perhaps more important for physics, ordinary laboratory units would haveinconveniently largeandsmallvalues.Soforconvenience’ssakewelivewithunits thataretailoredtoourbiologicallimitations.(Bytheway,noneofthisexplainstheincrediblefact that in our countrywe stillmeasure in inches, feet, yards, slugs, pints, quarts, andteaspoons.)

I.3ElectromagneticUnitsArt:Okay,Lenny,Igetwhatyouaresaying.Butwhataboutelectromagneticunits?Theyseem to be especially annoying.What’s that thing 0 in all the equations, the thing thetextbooks call the dielectric constant of the vacuum?2Why does the vacuum have adielectricconstantanyway,andwhyisitequalto8.85×10−12?Thatseemsreallyweird.

Art isright;electromagneticunitsareanuisanceallof theirown.Andheisright that itdoesn’t make sense to think of the vacuum as a dielectric—not in classical physics,anyway.Thelanguageisaholdoverfromtheoldethertheory.

Therealquestionis:Whywasitnecessarytointroduceanewunitforelectriccharge—theso-calledcoulomb?Thehistoryisinterestingandactuallybasedonsomephysicalfacts,butprobablynottheonesyouimagine.I’llstartbytellingyouhowIwouldhavesetthingsup,andwhyitwouldhavefailed.

WhatIwouldhavedoneistostartbytryingtoaccuratelymeasuretheforcebetweentwo electric charges, let’s say by rubbing two pith balls with cat’s fur until they werecharged.PresumablyIwouldhavefoundthattheforcewasgovernedbyCoulomb’slaw,

Then Iwouldhavedeclared that a unit charge—one forwhichq=1—is an amountofcharge such that two of them, separated by 1 meter, have a force between them of 1newton.(Thenewtonisaunitofforceneededtoacceleratea1-kilogrammassby1meterpersecondpersecond.)Inthatwaytherewouldbenoneedforanewindependentunitofcharge,andCoulomb’slawwouldbesimple,justlikeIwroteearlier.

MaybeifIhadbeenparticularlycleverandhadabitofforesight,Imighthaveputafactorof4πinthedenominatoroftheCoulomblaw:

Butthat’sadetail.

NowwhywouldIhavefailed,oratleastnothadgoodaccuracy?Thereasonisthatitis difficult to work with charges; they are hard to control. Putting a decent amount ofchargeonapithballishardbecausetheelectronsrepelandtendtojumpofftheball.Sohistoricallyadifferentstrategywasused.

FigureI.2:ParallelWires.Nocurrentflowsbecausetheswitchisopen.

FigureI.3:ParallelWireswithCurrentsFlowinginOppositeDirections.The

switchisclosed.

By contrast with charge, working with electric current in wires is easy. Current ischargeinmotion,butbecausethenegativechargeofthemovingelectronsinawireisheldin place by the positive charges of the nuclei, they are easy to control. So instead ofmeasuringtheforcebetweentwostaticcharges,weinsteadmeasuretheforcebetweentwocurrent-carryingwires.Figures I.2and I.3 illustratehowsuchanapparatusmightwork.Westartwithacircuitcontainingabattery,aswitch,andtwolongparallelwiresstretchedtight and separated by a known distance. For simplicity the distance could be ameter,althoughinpracticewemaywantittobeagooddealsmaller.

Now we close the switch and let the current flow. The wires repel each other forreasonsthatwillbeexplainedlaterinthisbook.Whatweseeisthatthewiresbellyoutinresponsetotheforce.Infact,wecanusetheamountofbellyingtomeasuretheforce(perunitlength).Thisallowsustodefineaunitofelectriccurrentcalledanampereoramp.

One amp is the current needed to cause parallel wires separated by 1meter, torepelwithaforceof1newtonpermeteroflength.

Noticethatinthisway,wedefineaunitofcurrent,notaunitofelectriccharge.Currentmeasurestheamountofchargepassinganypointonthecircuitperunittime.Forexample,itistheamountofchargethatpassesthroughthebatteryin1second.

Art:Butwait,Lenny.Doesn’t thatalsoallowustodefineaunitofcharge?Can’twesay that our unit of charge—call it 1 coulombof charge—is the amount of charge thatpassesthroughthebatteryinonesecond,giventhatthecurrentisoneamp?

Lenny: Very good! That’s exactly right. Let me say it again: The coulomb is by

definition the amount of charge passing through the circuit in one second, when thecurrentisoneamp;thatis,whentheforceonthewiresisonenewtonpermeteroflength(assumingthewiresareseparatedbyonemeter).

Thedisadvantageisthatthedefinitionofacoulombisindirect.TheadvantageisthattheexperimentissoeasythatevenIdiditinthelab.Theproblem,however,isthattheunitofchargedefined thiswayisnot thesameunit thatwouldresult frommeasuring theforcebetweenstaticcharges.

How do the units compare? To answer that,wemight try to collect two buckets ofcharge,eachacoulomb,andmeasure theforcebetweenthem.Thiswouldbedangerousevenifitwerepossible;acoulombisareallyhugeamountofcharge.Thebucketwouldexplodeand thechargewould just flyapart.So thequestionbecomes,whydoes it takesuchahugeamountofchargeflowinginthewirestoproduceamodestforceof1newton?

Art:Why is there a force between the wires anyway? Even though the wires havemovingelectrons,thenetchargeonthewireiszero.Idon’tseewhythereisanyforce.

Lenny:Yes,youarerightthatthenetchargeiszero.Theforceisnotelectrostatic.It’sactuallyduetothemagneticfieldcausedbythemotionofthecharges.Thepositivenucleiareatrestanddon’tcauseamagneticfield,butthemovingelectronsdo.

Art:Okay,butyoustillhaven’t toldmewhy it takessuchawhoppingbigamountofmoving charge to create a mere newton of force between the wires. Am I missingsomething?

Lenny:Onlyonething.Thechargesmoveveryslowly.

Theelectronsinatypicalcurrent-carryingwiredoindeedmoveveryslowly.Theybouncearoundveryquickly,butlikeadrunkensailor,theymostlygetnowhere;ontheaverageittakes an electron about an hour to move 1 meter along a wire. That seems slow, butcomparedtowhat?Theansweristhattheymoveveryslowlycomparedtotheonlynaturalphysicalunitofvelocity,thespeedoflight.Intheendthat’swhyittakesahugeamountofcharge,movingthroughthewiresofacircuit,toproduceasignificantforce.

Now thatweknow that the standardunit of charge, the coulomb, is a tremendouslylargeamountofcharge,let’sgobacktoCoulomb’slaw.Theforcebetweentwocoulomb-sizechargesisenormous.Toaccountforthiswehavetoputahugeconstantintotheforcelaw.Insteadof

wewrite

where 0isthesmallnumber8.85×10−12.

Art:Soultimatelytheweirddielectricconstantof thevacuumhasnothingtodowithdielectrics.Ithasmoretodowith theslowmolasses-likemotionofelectrons inmetallicwires.Whydon’twejustgetridof 0andsetitequalto1?

Lenny:Good idea, Art. Let’s do that from now on. But don’t forget that wewill beworkingwithaunitofchargethatisaboutonethree-hundred-thousandthofaCoulomb.Forgettingabouttheconversionfactorcouldleadtoanastyexplosion.

1 Ifyou’ve readourpreviousbookonquantummechanics, you’veheard this sermonbefore. It’s interesting that thesameissuesofscale thataffectourchoiceofunitsalso limitourability todirectlyperceivequantumeffectswithoursenses.

2Alsocalledvacuumpermittivity,orpermittivityoffreespace.

Lecture6

TheLorentzForceLaw1

Art:Lenny,whoisthatdignifiedgentlemanwiththebeardandwire-frameglasses?

Lenny:Ah,theDutchuncle.That’sHendrik.Wouldyouliketomeethim?

Art:Sure,isheafriendofyours?

Lenny:Art,they’reallfriendsofmine.Comeon,I’llintroduceyou.

ArtFriedman,meetmyfriendHendrikLorentz.

PoorArt,he’snotquitepreparedforthis.

Art:Lorentz?DidyousayLorentz?OhmyGod!Areyou?Ishe?Areyoureally,areyoureallythe…”

Dignifiedasalways,HLbowsdeeply.

Lorentz:HendrikAntoonLorentz,atyourservice.

Later a star-struck Art quietly asks, Lenny, is that really the Lorentz? The one whodiscoveredLorentztransformations?

Lenny:Sureheis,andalotmorethanthat.BringmeanapkinandapenandI’lltellyouabouthisforcelaw.

Of all the fundamental forces in nature, and there aremany of them, fewwere knownbefore the 1930s.Most are deeply hidden in themicroscopic quantumworld and onlybecameobservablewith the advent ofmodern elementary particle physics.Most of thefundamentalforcesarewhatphysicistscallshort-range.Thismeansthattheyactonlyonobjects that are separated by very small distances.The influence of a short-range forcedecreases so rapidlywhen theobjects are separated that, for themostpart, theyarenotnoticedintheordinaryworld.Anexampleistheso-callednuclearforcebetweennucleons(protons and neutrons). It’s a powerful force whose role is to bind these particles intonuclei.Butaspowerfulasthatforceis,wedon’tordinarilynoticeit.Thereasonisthatitseffectsdisappearexponentiallywhenthenucleonsareseparatedbymorethanabout10−15meters.Theforcesthatwedonoticearethelong-rangeforceswhoseeffectsfadeslowlywithdistance.

Ofalltheforcesofnature,onlythreewereknowntotheancients—electric,magnetic,andgravitational.ThalesofMiletos(600BC)wassaidtohavemovedfeatherswithamberthat had been rubbed with cat fur. At about the same time he mentioned lodestone, anaturallyoccurringmagneticmaterial.Aristotle,whowasprobablylateonthescene,hadatheoryofgravity,evenifitwascompletelywrong.Thesethreeweretheonlyforcesthatwereknownuntilthe1930s.

Whatmakes these easily observed forces special is that they are long-range. Long-range forces fade slowlywith distance and can be seen between objectswhen they arewellseparated.

Gravitationalforceisbyfarthemostobviousofthethree,butsurprisinglyitismuchweakerthanelectromagneticforce.Thereasonisinterestingandworthashortdigression.It goes back to Newton’s universal law of gravitational attraction: Everything attractseverythingelse.EveryelementaryparticleinyourbodyisattractedbyeveryparticleintheEarth.That’salotofparticles,allattractingoneanother,andtheresultisasignificantandnoticeable gravitational attraction, but in fact the gravitational attraction betweenindividualparticlesisfartoosmalltomeasure.

The electric forces between charged particles ismany orders ofmagnitude strongerthan the gravitational force. But unlike gravity, electric force can be either attractive(betweenoppositecharges)or repulsive (between likecharges).Bothyouand theEarthare composed of an equal number of positive charges (protons) and negative charges(electrons), and the result is that the forcescancel. Ifwe imaginedgetting ridofall theelectronsinbothyouandtheEarth,therepulsiveelectricforceswouldeasilyoverwhelmgravity and blast you from theEarth’s surface. In fact, itwould be enough to blast theEarthandyouintosmithereens.

Inanycase,gravityisnotthesubjectoftheselectures,andtheonlyotherlong-rangeforcesareelectromagnetic.Electricandmagneticforcesarecloselyrelatedtoeachother;ina sense theyarea single thing,and theunifying link is relativity.Aswewill see, anelectricforceinoneframeofreferencebecomesamagneticforceinanotherframe,andviceversa.Toput itanotherway,electricandmagneticforcestransformintoeachotherunderLorentztransformation.Therestoftheselecturesareaboutelectromagneticforces

and how they are unified into a single phenomenon through relativity. Going back toPauli’s(fictitious)paraphraseofJohnWheeler’s(real)slogan,

Fieldstellchargeshowtomove;chargestellfieldshowtovary.

We’ll begin with the first half—fields tell charges how to move. Or to put it moreprosaically,fieldsdeterminetheforcesonchargedparticles.

Anexamplethatmaybefamiliartoyou—ifnot,itsoonwillbe—istheelectricfield .Unlike the scalar field thatwediscussed in the last lecture, theelectric field isavectorfield—a3-vector, tobeprecise. Ithas threecomponentsandpoints insomedirection inspace.Itcontrolstheelectricforceonachargedparticleaccordingtotheequation

In this equation the symbol e represents the electric charge of the particle. It can bepositive, inwhich case the force is in the samedirection as the electric field; it can benegative,inwhichcasetheforceandthefieldareinoppositedirections;or,asinthecaseofaneutralatom,theforcecanbezero.

Magneticforceswerefirstdiscoveredbytheiractiononmagnetsorbitsoflodestone,but they also act on electrically charged particles if the particles are in motion. Theformula involves a magnetic field (also a 3-vector), the electric charge e, and thevelocityoftheparticle We’llderiveitlaterfromanactionprinciple,butjumpingahead,theforceonachargedparticleduetoamagneticfieldis

Thesymbol×representstheordinarycross-productofvectoralgebra,whichIassumeyouhaveseenbefore.2One interestingpropertyofmagnetic forces is that theyvanish for aparticleatrestandincreaseastheparticlevelocityincreases.Iftherehappenstobebothanelectricandamagneticfield,thefullforceisthesum

Eq.6.1wasdiscoveredbyLorentzandiscalledtheLorentzforcelaw.

We’ve already discussed scalar fields and the way they interact with particles. Weshowed how the same Lagrangian (and the same action) that tells the field how toinfluence the particle also tells the particle how to influence the field. Going forward,we’lldothesamethingforchargedparticlesandtheelectromagneticfield.Butbeforewedo,Iwanttobrieflyreviewournotationalschemeandextendittoincludeanewkindofobject: tensors. Tensors are a generalization of vectors and scalars and include them asspecialcases.Aswewillsee,theelectricandmagneticfieldsarenotseparateentitiesbutcombinetogethertoformarelativistictensor.

6.1ExtendingOurNotationOurbasicbuildingblocks are4-vectorswithupper and lower indices. In the contextofspecial relativity, there’s little difference between the two types of indices. The onlydifferenceoccurs in the timecomponent (thecomponentwith indexzero)ofa4-vector.

Foragiven4-vector,thetimecomponentwithanupperindexhastheoppositesignofthetimecomponentwithlowerindex.Insymbols,

A0=−A0.

Itmayseemlikeoverkilltodefineanotationwhosesolepurposeistokeeptrackofsignchanges for time components. However, this simple relationship is a special case of amuchbroadergeometricrelationshipbasedonthemetrictensor.Whenwestudygeneralrelativity, the relationship between upper and lower indices will become far moreinteresting.Fornow,upperand lower indices simplyprovideaconvenient, elegant,andcompactwaytowriteequations.

6.1.14-VectorSummaryHereisaquicksummaryofconceptsfromLecture5foreasyreference.

4-vectorshavethreespacecomponentsandonetimecomponent.AGreekindexsuchasμ refers toanyorallof thesecomponentsandmay take thevalues (0, ).The firstofthese (the component labeled zero) is the time component. Components 1, 2, and 3correspond to the x, y, and z directions of space. For example,A0 represents the timecomponentofthe4-vectorA.A2representsthespacecomponentintheydirection.Whenwe focus only on the three space components of a 4-vector, we label themwith Latinindicessuchasmorp.Latinindicesmaytakethevalues(),butnotzero.Symbolically,wecanwrite

Sofar,I’velabeledmy4-vectorswithupstairs,orcontravariantindices.Byconvention,acontravariant index is the sort of thingyouwould attach to the coordinates themselves,suchasXμ,ortoacoordinatedisplacementsuchasdXμ.Thingsthattransforminthesamewayascoordinatesordisplacementscarryupstairsindices.

ThecovariantcounterparttoAμiswrittenwithadownstairsindex,Aμ.Itdescribesthesame 4-vector using a different notation. To switch from contravariant notation tocovariantnotation,weusethe4×4metric

Theformula

convertsanupstairsindextoadownstairsindex.Therepeatedindexνontherightsideisasummationindex,andthereforeEq.6.2isshorthandfor

The covariant and contravariant components of any 4-vector A are exactly the same,except for the time components. The upstairs and downstairs time components haveoppositesigns.Eq.6.2isequivalenttothetwoequations

Eq.6.2givesthissameresultbecauseofthe−1intheupper-leftpositionofημν.

6.1.2FormingScalarsForanytwo4-vectorsAandB,wecanformaproductAνBνusingtheupstairscomponentsofoneandthedownstairscomponentsoftheother.3Theresultisascalar,whichmeansithasthesamevalueineveryreferenceframe.Insymbols,

AνBν=scalar.

The repeated index ν indicates a summation over four values. The long form of thisexpressionis

6.1.3DerivativesCoordinates and their displacements are the prototypes for contravariant (upstairs)components. In the same way, derivatives are the prototype for covariant (downstairs)components.Thesymbol∂μstandsfor

InLecture5,weexplainedwhythesefourderivativesarethecovariantcomponentsofa4-vector.Wecanalsowritethemincontravariantform.Tosummarize:

CovariantComponents:

ContravariantComponents:

Asusual,theonlydifferencebetweenthemisthesignofthetimecomponent.

Thesymbol∂μdoesn’tmeanmuchallbyitself;ithastoactonsomeobject.Whenitdoes,itaddsanewindexμtothatobject.Forexample,if∂μoperatesonascalar,itcreatesanewobjectwithcovariant indexμ.Takinga scalar fieldϕ as a concrete example,wecouldwrite

Therightsideisacollectionofderivativesthatformsthecovariantvector

Thesymbol∂μalsoprovidesanewwaytoconstructascalarfromavector.Supposewehavea4-vectorBμ(t,x)thatdependsontimeandposition.InotherwordsBisa4-vector

field.IfBisdifferentiable,itmakessensetoconsiderthequantity

∂μBμ(t,x).

Underthesummationconvention,thisexpressiontellsustodifferentiateBμwithrespecttoeachofthefourcomponentsofspacetimeandadduptheresults:

Theresultisascalar.

The summing process we’ve illustrated here is very general; it’s called indexcontraction. Indexcontractionmeans identifyinganupper indexwithan identical lowerindexwithinasingleterm,andthensumming.

6.1.4GeneralLorentzTransformationBackinLecture1,weintroducedthegeneralLorentztransformation.Here,wereturntothatideaandaddsomedetails.

Lorentztransformationsmakejustasmuchsensealongtheyaxisorthezaxisastheydoalong thex axis.There’scertainlynothing special about thex directionor anyotherdirection. In Lecture 1, we explained that there’s another class of transformations—rotations of space—that are also considered members of the family of Lorentztransformations.Rotationsofspacedonotaffecttimecomponentsinanyway.

Once you accept this broader definition of Lorentz invariance, you can say that aLorentztransformationalongtheyaxisissimplyarotationoftheLorentztransformationalong the x axis. You can combine rotations together with the “normal” LorentztransformationstomakeaLorentztransformationinanydirectionorarotationaboutanyaxis.Thisisthegeneralsetoftransformationsunderwhichphysicsisinvariant.Theproofofthisresultisnotimportanttousrightnow.Whatisimportantisthatphysicsisinvariantnot only under simple Lorentz transformations but also under a broader category oftransformationsthatincludesrotationsofspace.

HowcanwefoldLorentztransformationsintoourindex-basednotationscheme?Let’sconsiderthetransformationofacontravariantvector

Aμ.

Bydefinition, this vector transforms in the sameway as the contravariant displacementvectorXμ.Forexample,thetransformationequationforthetimecomponentA0is

ThisisthefamiliarLorentztransformationalongthexaxisexceptthatI’vecalledthetimecomponent A0 and I’ve called the x component A1. We can always write thesetransformationsintheformofamatrixactingonthecomponentsofavector.Forexample,Icanwrite

(A′)μ

torepresentthecomponentsofthe4-vectorAμinmyframeofreference.Toexpressthesecomponentsasfunctionsofthecomponentsinyourframe,we’lldefineamatrix withupperindexμandlowerindexν,

isamatrixbecauseithastwoindices;it’sa4×4matrixthatmultipliesthe4-vectorAν.4

Let’smakesurethatEq.6.3isproperlyformed.Theleftsidehasafreeindexμ,whichcantakeanyofthevalues(0,).Therightsidehastwoindices,μandν.Thesummationindexνisnotanexplicitvariableintheequation.Theonlyfreeindexontherightsideisμ.Inotherwords,eachsideoftheequationhasafreecontravariantindexμ.Therefore,theequationisproperlyformed;ithasthesamenumberoffreeindicesontheleftsideasitdoesontherightside,andtheircontravariantcharactersmatch.

Eq.6.4givesanexampleofhowwewoulduseLμνinpractice.WehavefilledinmatrixelementsthatcorrespondtoaLorentztransformationalongthexaxis.5

What does the equation say? Following the rules of matrix multiplication, Eq. 6.4 isequivalent tofoursimpleequations.Thefirstequationspecifies thevalueof t′,which isthefirstelementofthevectorontheleftside.Wesett′equaltothedotproductofthefirstrowofthematrixwiththecolumnvectorontherightside.Inotherwords,theequationfort′becomes

orsimply

CarryingoutthesameprocessforthesecondrowofLμνgivestheequationforx′

Thethirdandfourthrowsproducetheequations

It’seasytorecognizetheseequationsasthestandardLorentztransformationalongthexaxis.Ifwewantedtotransformalongtheyaxisinstead,wewouldjustshufflethesematrixelementsaround.I’llleaveittoyoutofigureouthowtodothat.

Nowlet’sconsideradifferentoperation:arotationinthey,zplane,wherethevariables

tandxplaynopartatall.Arotationcanalsoberepresentedasamatrix,buttheelementswouldbedifferent fromourfirstexample.Tobeginwith, theupper-leftquadrantwouldlook like a 2 × 2 unit matrix. That assures that t and x are not affected by thetransformation.Whataboutthelower-rightquadrant?Youprobablyknowtheanswer.Torotate the coordinates by angle θ, the matrix elements would be the sines and cosinesshowninEq.6.5.

Followingtherulesofmatrixmultiplication,Eq.6.5isequivalenttothefourequations

Inasimilarway,wecanwritematricesrepresentingrotationsinthex,yorx,zplanesbyshufflingthesematrixelementstodifferentlocationswithinthematrix.

Oncewedefineasetof transformationmatrices forsimple linearmotionandspatialrotations, we can multiply these matrices together to make more complicatedtransformations.Inthisway,wecanrepresentacomplicatedtransformationusingasinglematrix. The simple transformation matrices shown here, along with their y and zcounterparts,arethebasicbuildingblocks.

6.1.5CovariantTransformationsSofar,we’veexplainedhowtotransforma4-vectorwithacontravariant (upper) index.Howdowetransforma4-vectorwithacovariant(lower)index?

Suppose you have a 4-vector with covariant components. You know what thesecomponentslooklikeinyourframe,andyouwanttofindoutwhattheylooklikeinmyframe.WeneedtodefineanewmatrixMμ

νsuchthat

Thisnewmatrixneedstohavealowerindexμ,becausetheresulting4-vectorontheleftsidehasalowerindexμ.

Remember thatM represents the same Lorentz transformation as our contravarianttransformation matrix L; the two matrices represent the same physical transformationbetween coordinate frames. ThereforeM andLmust be connected. Their connection isgivenbyasimplematrixformula:

M=ηLη.

I’llletyouprovethisonyourown.We’renotgoingtouseMverymuch,butit’snicetoknowhowLandMareconnectedforagivenLorentztransformation.

It turns out thatη is its own inverse, just like the unitmatrix is its own inverse. Insymbols,

η−1=η.

The reason is that eachdiagonal entry is itsown inverse; the inverseof1 is1, and theinverseof−1is−1.

Exercise 6.1: Given the transformation equation (Eq. 6.3) for the contravariantcomponentsofa4-vectorAν,whereLμν isaLorentztransformationmatrix,showthattheLorentztransformationforA’scovariantcomponentsis

(A′)μ=MμνAν,

where

M=ηLη.

6.2TensorsAtensorisamathematicalobjectthatgeneralizesthenotionsofascalarandavector;infact,scalarsandvectorsareexamplesoftensors.Wewillusetensorsextensively.

6.2.1Rank2TensorsA simple way to approach tensors is think of them as “a thing with some number ofindices.” The number of indices a tensor has—the rank of the tensor—is an importantcharacteristic.Forexample,ascalarisatensorofrankzero(ithaszeroindices),anda4-vectorisatensorofrankone.Atensorofranktwoisathingwithtwoindices.

Let’s look at a simple example. Consider two 4-vectors, A and B. As we’ve seenbefore,wecanformaproductfromthembycontraction,AμBμ,wheretheresultisascalar.But now let’s consider amore general kindof product, a productwhose result has twoindices,μandν.We’llstartwiththecontravariantversion.WejustmultiplyAμandBνforeachμandν:

AμBν.

Howmanycomponentsdoesthisobjecthave?Eachindexcantakefourdifferentvalues,soAμBν can take4×4or sixteendifferentvalues.Wecan list them:A0B0,A0B1,A0B2,A0B3, A1B0, and so forth. The symbol AμBν stands for a complex of sixteen differentnumbers.It’sjustthesetofnumbersyougetbymultiplyingtheμcomponentofAwiththeν component ofB. This object is a tensor of rank two. I’ll use the generic labelT fortensor:

Tμν=AμBν.

Notalltensorsareconstructedfromtwovectorsinthisway,buttwovectorscanalwaysdefineatensor.HowdoesTμνtransform?IfweknowhowAtransformsandweknowhow

B transforms (which is the same way) we can immediately figure out the transformedcomponentsofAμBν,whichwecall

(T′)μν=(A′)μ(B′)ν.

But,ofcourse,wedoknowhowAandBtransform;Eq.6.3tellsushow.SubstitutingEq.6.3intotherightsideforbothA′andB′givestheresult

Thisrequiressomeexplanation.InEq.6.3,therepeatedindexwascalledν.However,intheprecedingequation,wereplacedνwithσfortheAfactor,andwithτfortheBfactor.Remember,thesearesummationindicesanditdoesn’tmatterwhatwecallthemaslongaswe’reconsistent.Toavoidconfusion,wewanttokeepthesummationindicesdistinctfromotherindicesanddistinctfromeachother.

Each of the four symbols on the right side stands for a number. Therefore, we’reallowedtoreorderthem.Let’sgroupthematrixelementstogether:

Notice that we can now identifyAσBτ on the right side with the untransformed tensorelementTστandrewritetheequationas

Eq.6.9givesusanewtransformationruleaboutanewkindofobjectwithtwoindices.TμνtransformswiththeactionoftheLorentzmatrixoneachoneoftheindices.

6.2.2TensorsofHigherRankWe can invent more complicated kinds of tensors—tensors with three indices, forexample,suchas

Tμνλ.

Howwould this transform?Youcould thinkof it as theproductof three4-vectorswithindicesμ,ν,andλ.Thetransformationformulaisstraightforward:

There’satransformationmatrixforeachindex,andwecangeneralizethispatterntoanynumberofindices.

At the beginning of the section, I said that a tensor is a thing that has a bunch ofindices.That’strue,butit’snotthewholestory;noteveryobjectwithabunchofindicesisconsidered a tensor. To qualify as a tensor the indexed object has to transform like theexamples we’ve shown. The transformation formula may be modified for differentnumbersofindices,ortoaccommodatecovariantindices,butitmustfollowthisgeneralpattern.

Although tensors are defined by these transformation properties, not every tensor isconstructedbytakingaproductoftwo4-vectors.Forexample,supposewehavetwoother4-vectors,CandD.WecouldtaketheproductAμBνandaddittotheproductCμDν,

AμBν+CμDν.

Adding tensorsproducesother tensors, and thepreceding expression is indeed a tensor.But it cannot ingeneralbewrittenasaproductof two4-vectors. Its tensorcharacter isdetermined by its transformation properties, not by the fact that it may ormay not beconstructedfromapairof4-vectors.

6.2.3InvarianceofTensorEquationsTensor notation is elegant and compact. But the real power behind it is that tensorequationsareframeinvariant.Iftwotensorsareequalinoneframe,they’reequalineveryframe.That’seasytoprove,butkeepinmindthatfortwotensorstobeequal,alloftheircorrespondingcomponentsmustbeequal.Ifeverycomponentofatensorisequaltothecorrespondingcomponentofsomeothertensor,thenofcoursethey’rethesametensor.

Anotherwaytosayitisthatifallthecomponentsofatensorarezeroinonereferenceframe,thenthey’rezeroineveryframe.Ifasinglecomponentiszeroinoneframe,itmaywellbenonzeroinadifferentframe.Butifallthecomponentsarezero,theymustbezeroineveryframe.

6.2.4RaisingandLoweringIndicesI’ve explained how to transform a tensor with all of its indices upstairs (contravariantcomponents). I couldwritedown the rules for transforming tensorswithmixedupstairsand downstairs indices. Instead I’ll just tell you that once you know how a tensortransforms, you can immediately deduce how its other variants transform. By othervariantsImeandifferentversionsofthesametensor—thesamegeometricquantity—butwithsomeofitsupstairsindicespusheddownstairsandviceversa.Forexample,considerthetensor

Tμν=AμBν.

Thisisatensorwithoneindexupstairsandoneindexdownstairs.Howdoesittransform?Nevermind!Youdon’tneedtoworryaboutitbecauseyoualreadyknowhowtotransformTμσ(withbothindicesupstairs),andyoualsoknowhowtoraiseandlowerindicesusingthematrixη.Recallthat

Tμν=Tμσησν.

Youcanloweratensorindexinexactlythesamewayasyoulowera4-vectorindex.Justmultiplybyηasabove,usingthesummationconvention.TheresultingobjectisTμν.

Butthere’saneveneasierwaytothinkaboutit.Givenatensorwithallofitsindicesupstairs,howdoyoupullsomeofthemdownstairs?It’ssimple.If theindexthatyou’reloweringisatimeindex,youmultiplyby−1.Ifit’saspaceindex,youdon’tmultiplyatall.That’swhatηdoes.Forexample,thetensorcomponentT00isexactlythesameasT00becauseI’veloweredtwotimeindices,

T00=T00.

Lowering two time indicesmeansmultiplying the component by −1 twice. It’s like therelationshipbetween

A0B0

and

A0B0.

TherearetwominussignsingoingfromA0toA0,andfromB0toB0.Eachloweredindexintroducesaminussign,andthefirstproductisequaltothesecond.Buthowdoes

A0B1

comparewith

B1andB1arethesame,becauseindex1referstoaspacecomponent.ButA0andA0haveoppositesignsbecauseindex0isatimecomponent.ThesamerelationshipholdsbetweentensorcomponentsT01andT01:

T01=−T01,

because only one time component was lowered. Whenever you lower or raise a timecomponent,youchangesigns.That’sallthereistoit.

6.2.5SymmetricandAntisymmetricTensorsIngeneral,thetensorcomponent

Tμν

isnotthesameas

Tνμ,

wheretheindicesμandνhavebeenreversed.Changingtheorderoftheindicesmakesadifference.Toillustratethis,considertheproductoftwo4-vectorsAandB.Itshouldbeclearthat

Forexample,ifwechoosespecificcomponentssuchas0and1,wecanseethat

Clearly,thesearenotalwaysthesame:AandBaretwodifferent4-vectorsandthere’snobuilt-inreasonforanyofthecomponentsA0,A1,B0,orB1tomatchupwitheachother.

Whiletensorsarenotgenerallyinvariantwhenchangingtheorderofindices,therearespecial situations where they are. Tensors that have this special property are calledsymmetrictensors.Insymbols,asymmetrictensorhastheproperty

Tμν=Tνμ.

Letmeconstructoneforyou.IfAandBare4-vectors,then

AμBν+AνBμ

isasymmetrictensor.Ifyouinterchangetheindicesμandν,thevalueofthisexpressionremains the same.Goaheadand try it.Whenyou rewrite the first term, itbecomes thesameastheoriginalsecondterm;whenyourewritethesecondterm,itbecomesthesameas theoriginal first term.The sumof the rewritten terms is the sameas the sumof theoriginal terms. If you startwith any tensorof rank two, you can construct a symmetrictensorjustaswedidhere.

Symmetrictensorshaveaspecialplaceingeneralrelativity.They’relessimportantinspecialrelativity,buttheydocomeup.Forspecialrelativity, theantisymmetric tensor ismoreimportant.Antisymmetrictensorshavetheproperty

Fμν=−Fνμ.

Inotherwords,whenyoureversetheindices,eachcomponenthasthesameabsolutevaluebut changes its sign. To construct an antisymmetric tensor from two 4-vectors,we canwrite

AμBν−AνBμ.

Thisisalmostthesameasourtrickforconstructingasymmetrictensor,exceptwehaveaminus sign between the two terms instead of a plus sign. The result is a tensorwhosecomponentschangesignwhenyouinterchangeμandν.Goaheadandtryit.

Antisymmetric tensors have fewer independent components than symmetric tensors.Thereasonisthattheirdiagonalcomponentsmustbezero.6IfFμνisantisymmetric,eachdiagonalcomponentmustequalitsownnegative.Theonlywaythatcanhappenisforthediagonalcomponentstoequalzero.Forexample,

F00=−F00=0.

Thetwoindicesareequal(whichiswhatitmeanstobeadiagonalcomponent),andzerois theonlynumber that equals its ownnegative. If you thinkof a rank two tensor as amatrix(atwo-dimensionalarray),thenamatrixrepresentinganantisymmetrictensorhaszerosallalongthediagonal.

6.2.6AnAntisymmetricTensorImentionedearlierthattheelectricandmagneticfieldscombinetoformatensor.Infact,they form an antisymmetric tensor.We will eventually derive the tensor nature of thefields and ,butfornowjustaccepttheidentificationasanillustration.

The names of the elements of the antisymmetric tensor are suggestive, but for nowthey’re just names. The diagonal elements are all zero. We’ll call it Fμν. It will beconvenienttowriteoutthedownstairs(covariant)components,Fμν:

Thistensorplaysakeyroleinelectromagnetism,where standsforelectricfield,andstandsformagneticfield.We’llseethattheelectricandmagneticfieldscombinetoformanantisymmetrictensor.Thefields and arenotindependentofeachother.Inthesamewaythatxcangetmixedupwith tunderaLorentz transformation, cangetmixedupwith . What you see as a pure electric field, I might see as having some magneticcomponent.We haven’t shown this yet, butwewill soon. This iswhat Imeant earlierwhenIsaidthatelectricandmagneticforcestransformintooneanother.

6.3ElectromagneticFieldsLet’sdosomephysics!WecouldbeginbystudyingMaxwell’sequations,theequationsofmotionthatgovernelectromagneticfields.We’lldothatinLecture8.Fornow,we’llstudythemotionofachargedparticleinanelectromagneticfield.TheequationthatgovernsthismotioniscalledtheLorentzforcelaw.Laterwe’llderivethislawfromanactionprinciple,combinedwithrelativisticideas.Thenonrelativistic(low-velocity)versionoftheLorentzforcelawis

whereeistheparticle’schargeandthesymbols , , ,and areordinary3-vectors.7Theleftsideismasstimesacceleration;therightsidemustthereforebeforce.Therearetwocontributions to the force, electric and magnetic. Both terms are proportional to theelectricchargee.

In thefirst termtheelectricchargeofaparticleemultiplies theelectric field .Theother termis themagneticforce, inwhichthechargemultiplies thecrossproductof theparticle’svelocity withthesurroundingmagneticfield .Ifyoudon’tknowwhatacrossproductis,pleasetakethetimetolearn.TheAppendixcontainsabriefdefinitionasdoesthe first book in this series, Classical Mechanics (Lecture 11). There are many otherreferencesaswell.

We’lldomuchofourworkwithordinary3-vectors.8Thenwe’llextendourresultsto4-vectors;wewillderivethefull-blownrelativisticversionoftheLorentzforcelaw.We’regoingtofindoutthatthetwotermsontherightsideofEq.6.12arereallypartofthesametermwhenwritteninaLorentzinvariantform.

6.3.1TheActionIntegralandtheVectorPotentialBackinLecture4,IshowedyouhowtoconstructaLorentzinvariantaction(Eq.4.28)foraparticlemovinginascalarfield.Let’squicklyreviewtheprocedure.WebeganwiththeactionforafreeparticleinEq.4.27,

andaddedatermrepresentingtheeffectofthefieldontheparticle,

Thismakesafinetheoryofaparticleinteractingwithascalarfield,butit’snotthetheoryof a particle in an electromagnetic field. What options do we have for replacing thisinteractionwithsomethingthatmightyieldtheLorentzforcelaw?Thatisthegoalfortherest of this lecture: to derive the Lorentz force law, Eq. 6.12, from an invariant actionprinciple.

How can we modify this Lagrangian to describe the effects of an electromagneticfield?Here there is abit of a surprise.Onemight think that the correctprocedure is toconstruct a Lagrangian that involves the particle’s coordinate and velocity components,andthatdependsonthe and fieldsinamannersimilartotheactionforaparticleinthepresence of a scalar field. Then, if all goes well, the Euler-Lagrange equation for themotionwouldinvolveaforcegivenbytheLorentzforcelaw.Surprisingly,thisturnsouttobe impossible.Tomakeprogresswehave to introduceanother fieldcalled thevectorpotential.Insomesense,theelectricandmagneticfieldsarederivedquantitiesconstructedfromthemorebasicvectorpotential,whichitselfisa4-vectorcalledAμ(t,x).Whya4-vector?All I can do at this point is ask you towait a little; youwill see that the endjustifiesthemeans.

HowcanweuseAμ (t,x) to construct an action for a particle in an electromagneticfield?ThefieldAμ(t,x)isa4-vectorwithalowerindex.Itseemsnaturaltotakeasmallsegment of the trajectory, the 4-vector dXμ, and combine it withAμ (t, x) to make aninfinitesimalscalarquantityassociatedwiththatsegment.Foreachtrajectorysegment,weformthequantity

dXμAμ(t,x)

Thenweaddthemup;thatis,weintegratefromoneendtotheother,frompointatopointb:

Because the integrand is a scalar, every observer agrees about its value on each littlesegment.Ifweaddquantitiesthatweagreeabout,we’llcontinuetoagreeaboutthem,andwe’llgetthesameanswerfortheaction.Toconformtostandardconventions,I’llmultiplythisactionbyaconstant,e:

I’msureyou’vealreadyguessedthateistheelectriccharge.Thisistheothertermintheaction for a particlemoving in an electromagnetic field. Let’s collect both parts of theactionintegral:

6.3.2TheLagrangianBothtermsinEq.6.13werederivedfromLorentzinvariantconstructions:thefirsttermisproportionaltothepropertimealongthetrajectoryandthesecondisconstructedfromtheinvariantdXμAμ.WhateverfollowsfromthisactionmustbeLorentz invarianteven if itmaynotbeobvious.

Thefirsttermisanoldfriendandcorrespondstothefree-particleLagrangian

ThesecondtermisnewandhopefullywillgiverisetotheLorentzforcelaw,althoughatthemomentitlooksquiteunrelated.Initscurrentform,theterm

isnotexpressedintermsofaLagrangian,becausetheintegralisnottakenovercoordinatetimedt.Thatiseasilyremedied;wejustmultiplyanddividebydttorewriteitas

Inthisform,thenewtermisnowtheintegralofaLagrangian.CallingtheintegrandLint,whereintstandsforinteraction,theLagrangianbecomes

Nowlet’snoticethatthequantities

comeintwodifferentflavors.Thefirstis

RecallingthatX0andtarethesamething,wecanwrite

Theotherflavorcorrespondstotheindexμtakingoneofthevalues(1,2,3),thatis,oneofthethreespacedirections.Inthatcasewerecognize

(withLatinindexp)tobeacomponentoftheordinaryvelocity,

Ifwecombinethetwotypesofterms,theinteractionLagrangianbecomes

where the repeated Latin indexpmeans thatwe sum from p=1 to p = 3. Because thequantities arethecomponentsofvelocityvp,theexpression isnothingbutthedotproductofthevelocitywiththespatialpartofthevectorpotential.ThereforewecouldwriteEq.6.16inaformthatmaybemorefamiliar,

Summarizing all these results, the action integral for a charged particle now looks likethis:

Onceagain,inmorefamiliarnotation,it’s

Now that the entire action is expressed as an integral over coordinate time dt, we caneasilyidentifytheLagrangianas

Exactlyaswedidforscalarfields,I’mgoingtoimaginethatApisaknownfunctionoftandx,andthatwe’rejustexploringthemotionoftheparticleinthatknownfield.WhatcanwedowithEq.6.18?WritetheEuler-Lagrangeequations,ofcourse!

6.3.3Euler-LagrangeEquationsHerearetheEuler-Lagrangeequationsforparticlemotiononceagain,foreasyreference:

Remember that this is shorthand for threeequations,oneequation for eachvalueofp.9Ourgoal is towrite theEuler-LagrangeequationsbasedonEq.6.18andshowthat theylook like the Lorentz force law. This amounts to substituting fromEq. 6.18 into Eq.6.19.Thecalculationisabitlongerthanforthescalarfieldcase.I’llholdyourhandandtakeyouthroughthesteps.Let’sstartbyevaluating

alsoknownasthecanonicalmomentum.TheremustbeacontributionfromthefirstterminEq.6.18,becausethattermcontains .Infact,wealreadyevaluatedthisderivativebackinLecture3.Eq.3.30showstheresult(withdifferentvariablenamesandinconventionalunits).Inrelativisticunits,therightsideofEq.3.30isequivalentto

That’sthederivativeofthefirsttermof (Eq.6.18)withrespecttothepthcomponentofvelocity.Thesecondtermof doesnotcontain explicitly,soitspartialderivativewithrespectto iszero.However,thethirdtermdoescontain ,anditspartialderivativeis

eAp(t,x)

Puttingthesetermstogethergivesusthecanonicalmomentum,

We sneaked a small change of notation intoEq. 6.20 bywriting instead of . Thismakes no difference because space components such as p can be moved upstairs ordownstairsatwillwithoutchanging theirvalue.Thedownstairs index isconsistentwiththe left side of the equation and will be convenient for what follows. Eq. 6.19 nowinstructsustotakethetimederivative,whichis

That’s the full left sideof theEuler-Lagrange equation.What about the right side?Therightsideis

Inwhatwaydoes dependonXp?Thesecondterm,eA0(t,x),clearlydependsonXp. Itsderivativeis

There’sonlyonemoretermtoaccountfor: .Thistermismixedinthatitdependsonbothvelocityandposition.Butwealreadyaccountedfor thevelocitydependenceonthe left sideof theEuler-Lagrangeequation.Now, for the rightside,we take thepartialderivativewithrespecttoXp,whichis

NowwecanwritethefullrightsideoftheEuler-Lagrangeequation,

Settingthisequaltotheleftsidegivesustheresult

Doesthislookfamiliar?Thefirsttermontheleftsideresemblesmasstimesacceleration,exceptforthesquarerootinthedenominator.Forlowvelocities,thesquarerootisverycloseto1,andthetermliterallybecomesmasstimesacceleration.We’llhavemoretosay

aboutthesecondterm, ,lateron.

Youmay not recognize the first term on the right side of Eq. 6.21 but it’s actuallyfamiliar to many of you. However, to see that we have to use different notation.Historically,thetimecomponentofthevectorpotentialA0(t,x)wascalled−ϕ(t,x)andϕwascalledtheelectrostaticpotential.Sowecanwrite

andthecorrespondingterminEq.6.21isjustminustheelectricchargetimesthegradientoftheelectrostaticpotential.Inelementaryaccountsofelectromagnetism,theelectricfieldisminusthegradientofϕ.

Let’sreviewthestructureofEq.6.21.Theleftsidehasafree(nonsummed)covariantindexp.Therightsidealsohasafreecovariantindexp,aswellasasummationindexn.This is theEuler-Lagrange equation for each component of position,X1,X2, andX3. Itdoesn’tyetlooklikesomethingwemightrecognize,butitsoonwill.

Our nextmove is to evaluate the time derivatives on the left side. The first term iseasy;we’llsimplymovethemoutsidethederivative,whichgives

Howdowedifferentiatethesecondterm,eAp(t,x),withrespect to time?This isa littletricky.Ofcourse,Ap(t,x)maydependexplicitlyontime.Butevenifitdoesn’t,wecan’tassume that it’sconstant,because theposition of theparticle isnot constant; it changeswithtimebecausetheparticlemoves.EvenifAp(t,x)doesnotdependexplicitlyontime,its value changes over time as it tracks the motion of the particle. As a result, thederivativegeneratestwoterms.ThefirsttermistheexplicitderivativeofApwithrespecttot,

wheretheconstantejustcomesalongfortheride.ThesecondtermaccountsforthefactthatAp(t,x)alsodepends implicitlyontime.Thistermis thechangeinAp(t,x)whenXn

changes,times .Puttingthesetermstogetherresultsin

Collectingall threetermsofthetimederivative, theleftsideofEq.6.21nowlookslikethis:

andwecanrewritetheequationas

To avoid clutter, I’ve decided towriteAp instead ofAp(t, x). Just remember that everycomponentofAdependsonallfourspacetimecoordinates.Theequationiswellformedbecauseeachsidehasasummationindexnandacovariantfreeindexp.10

Thisisalottoassimilate.Let’stakeabreathandlookatwhatwehavesofar.ThefirstterminEq.6.22istherelativisticgeneralizationof“masstimesacceleration.”We’llleavethistermontheleftside.Aftermovingalltheothertermstotherightside,here’swhatweget:

If the left side is the relativisticgeneralizationofmass timesacceleration, the rightsidecanonlybetherelativisticforceontheparticle.

Therearetwokindsoftermsontherightside:termsthatareproportionaltovelocity(because they contain as a factor), and terms that are not. Grouping similar termstogetherontherightsideresultsin

Art:Ouch,myheadhurts.Idon’trecognizeanyofthis.IthoughtyousaidweweregoingtogettheLorentzforcelaw.Youknow,

AllIseeincommonwithEq.6.24istheelectricchargee.

Lenny:Hangon,Art,we’regettingthere.

LookattheleftsideofEq.6.24.Supposethevelocityissmallsothatwecanignorethesquareroot in thedenominator.Thenthe leftside is justmass timesacceleration,whichNewtonwouldcalltheforce.Sowehaveour .

On the right sidewe have two terms, one ofwhich contains a velocity ,which Icouldcallvn.Theothertermdoesn’tdependonthevelocity.

Art:IthinkIseethelight,Lenny!Isthetermwithoutthevelocity ?

Lenny:Nowyou’recooking,Art.Goforit;what’sthetermwiththevelocity?

Art:Holycow!Itmustbe…Yes!Itmustbe

Art’sright.Considerthetermthatdoesn’tdependonvelocity,namely

IfwedefinetheelectricfieldcomponentEptobe

wegetthefirsttermoftheLorentzlaw, .

Thevelocity-dependenttermisabitharderunlessyouareamasterofcrossproducts,inwhichcaseyoucanseethatit’s incomponentform.Incaseyou’renotamasterofcrossproducts,we’llworkitoutnext.11

Let’sfirstconsiderthezcomponentof ,

Wewanttocomparethiswiththezcomponentof

whichwegetbyidentifyingtheindexpwiththedirectionz.Theindexnisasummationindex,whichmeansweneedtosubstitutethevalues(1,2,3),orequivalently(x,y,z),andthenadduptheresults.There’snoshortcut;justpluginthesevaluesandyou’llget

Exercise 6.2: Expression 6.28 was derived by identifying the index p with the zcomponentof space, and then summingovern for thevalues (1, 2, 3).Whydoesn’tExpression6.28containavzterm?

NowallwehavetodoistomatchExpression6.28withtherightsideofEq.6.26.Wecandothisbymakingtheidentifications

and

TheminussigninthesecondequationisduetotheminussigninthesecondtermofEq.6.26.Wecandoexactlythesamethingfortheothercomponents,andwefindthat

Thereisashorthandnotationfortheseequationsthatyoushouldbefamiliarwith.12Eqs.6.29aresummarizedbysayingthat isthecurlof ,

HowshouldonethinkaboutEqs.6.25and6.30?Onewayistosaythatthefundamentalquantitydefininganelectromagnetic field is thevectorpotential,and thatEqs.6.25and6.30 define derived objects that we recognize as electric andmagnetic fields. But onemightrespondthatthequantitiesthataremostphysical,theonesdirectlymeasuredinthelaboratory,areEandB,andthevectorpotentialisjustatrickfordescribingthem.Eitherwayisfine:it’saMonday-Wednesday-FridayversusTuesday-Thursday-Saturdaykindofthing.ButwhateveryourphilosophyabouttheprimacyofAversus(E,B)happenstobe,theexperimentalfactisthatchargedparticlesmoveaccordingtotheLorentzforcelaw.Infact,wecanwriteEq.6.24inthefullyrelativisticallyinvariantform,

6.3.4LorentzInvariantEquationsEq.6.31istheLorentzinvariantformoftheequationofmotionforachargedparticle;theleftsideisarelativisticformofmasstimesacceleration,andtherightsideistheLorentzforcelaw.WeknowthisequationisLorentzinvariantbecauseofthewaywedefinedtheLagrangian.However,itisnotmanifestlyinvariant;itisnotobviouslyinvariantbasedonthe structure of the equation. When a physicist says that an equation is manifestlyinvariantunderLorentztransformations,shemeansthatitiswritteninafour-dimensionalformwithupstairsanddownstairsindicesmatchingupintherightway.Inotherwords,itiswrittenasatensorequationwithbothsidestransformingthesameway.That’sourgoalfortherestofthislecture.

Let’sgoall thewayback to theprevious formof theEuler-Lagrangeequations (Eq.6.24),whichI’llrewritehereforconvenience.

BackinLectures2and3(seeEqs.2.16and3.9),wefoundthatthequantities

arethespacecomponentsofthevelocity4-vector.Inotherwords,

BecausepisaLatinindex,wecanalsowrite

Therefore(ignoringthefactormfornow),theleftsideofEq.6.24canbewrittenas

Thesearenotcomponentsofa4-vector.However,theycanbeputinto4-vectorformifwemultiplythembydt/dτ,

Theseare the space components of a 4-vector—the relativistic acceleration.The full 4-vector

has four components. Let’s therefore multiply both sides of Eq. 6.24 by dt/dτ. We’llreplacetwithX0,andwe’llwritedX0/dτinsteadofdt/dτ.Theresultis

Recallingourconventionsforindexranges,wecouldtrytorewritetherightsideofthisequationasasingletermbyreplacingtheLatinindiceswithGreekindicesthatrangefrom

0through4:

However,thiswouldcreateaproblem.Eachsideoftheequationwouldhaveadownstairsfreeindex,whichisfine.ButtheleftsidewouldhaveaLatinfreeindexp,whiletherightsidewouldhaveaGreek free indexμ. Indices canbe renamed, of course, butwe can’tmatchupaLatinfreeindexononesidewithaGreekfreeindexontheotherbecausetheyhavedifferentranges.Towrite thisequationproperly,weneedtoreplaceXpwithXμontheleftside.Theproperlyformedequationis

Ifweset theindexμ tobethespaceindexp, thenEq.6.33justreproducesEq.6.24.Infact,Eq. 6.33 is themanifestly invariant equation ofmotion for a charged particle. It’smanifestly invariant because all the objects in the equations are 4-vectors and all therepeatedindicesareproperlycontracted.

Thereisoneimportantsubtlety:WestartedwithEq.6.24,whichrepresentsonlythreeequationsforthespacecomponents,labeledbytheindexp.WhenweallowtheindexμtorangeoverallfourdirectionsofMinkowskispace,Eq.6.33givesoneextraequationforthetimecomponent.

Butisthisnewequation(thezerothequation)correct?Bymakingsureatthebeginningthattheactionwasascalar,weguaranteedthatourequationsofmotionwouldbeLorentzinvariant. If the equations are Lorentz invariant, and the three space components of acertain4-vectorareequaltothethreespacecomponentsofsomeother4-vector,thenweknowautomaticallythattheirzerothcomponents(thetimecomponents)mustalsomatch.Because we’ve built a Lorentz invariant theory where the three space equations arecorrect,thetimeequationmustalsobecorrect.

Thislogicpervadeseverythinginmodernphysics:MakesureyourLagrangianrespectsthe symmetries of the problem. If the symmetry of the problem is Lorentz symmetry,make sure your Lagrangian is Lorentz invariant. This guarantees that the equations ofmotionwillalsobeLorentzinvariant.

6.3.5Equationswith4-VelocityLet’smassage Eq. 6.33 into a slightly different form thatwill come in handy later on.Recallthatthe4-vectorUμ,definedas

iscalledthe4-velocity.MakingthissubstitutioninEq.6.33(withaloweredμindexontheleftside)resultsin

Thederivative

thatappearsontheleftsideiscalledthe4-acceleration.ThespatialpartofthisequationintermsofUis

6.3.6RelationshipofAμto and

Let’ssummarizetherelationbetweenthevectorpotentialAμandthefamiliarelectricandmagneticfields and .

Historically, and were discovered through experiments on charges and electriccurrents (electric currents are just charges inmotion). Those discoveries culminated inLorentz’s synthesis of the Lorentz force law. But try as youmay, when formulated intermsof and thereisnowaytoexpressthedynamicsofchargedparticlesasanactionprinciple.Forthat,thevectorpotentialisessential.WritingtheEuler-LagrangeequationsandcomparingthemwiththeLorentzforcelawprovidedthefollowingequationsrelatingthe and fieldstoAμ:

Theseequationshaveacertainsymmetrythatbegsustoputthemintotensorform.We’lldothatinthenextlecture.

6.3.7TheMeaningofUμ

AttheendofthislectureastudentraisedhishandandremarkedthatwehaddoneagooddealofheavymathematicalliftingtogettoEq.6.34.Butcouldwecomebacktoearthanddiscuss the meaning of the relativistic 4-velocityUμ? In particular, what about zerothequation—theequationforthetimecomponentoftherelativisticacceleration?

Sure,let’sdoit.Firstthespacecomponents,showninEq.6.35.Theleftsideisjustarelativisticgeneralizationofmasstimesacceleration—themaofNewton’sF=ma.Ifthe

particleismovingslowly,thenit’sexactlythat.Buttheequationalsoworksiftheparticleismovingfast.TherightsideistherelativisticversionoftheLorentzforce;thefirsttermis the electric force, and the second term is the magnetic force that acts on a movingcharge.Thezerothequationistheonethatneedsexplanation.Let’swriteitas

Noticethatweomittedthefirsttermontherightside(whereμ=0)because

Let’s start with the left side of Eq. 6.37. It’s something a little unfamiliar: the timecomponentof the relativistic acceleration.To interpret itwe recall (Eqs.3.36and3.37)thattherelativistickineticenergyisjust

Usingconventionalunitstomakethespeedoflightexplicit,thisbecomes

Justtokeepthebookkeepingstraight,mU0isminus thekineticenergybecauseloweringthetimeindexchangesthesign.Therefore,apartfromtheminussign,theleftsideofEq.6.37isthetimerateofchangeofthekineticenergy.Therateofchangeofkineticenergyistheworkdoneontheparticleperunittimebywhateverforceisactingonit.

Next,weconsidertherightsideofEq.6.37;ithastheform

Usingthefactthat istheelectricforceontheparticle,wecanexpresstherightsideoftheequationas

Fromelementarymechanics thedot productof the force and thevelocity is exactly thework (per unit time) done by force on a moving particle. So we see that the fourthequationisjusttheenergybalanceequation,expressingthefactthatthechangeofkineticenergyistheworkdoneonasystem.

What happened to themagnetic force?How comewe did not have to include it incalculatingthework?Oneanswerthatyouprobablylearnedinhighschoolisthis:

Magneticforcesdonowork.

But there is a simple explanation: The magnetic force is always perpendicular to thevelocity,soitdoesnotcontributeto

6.4InterludeontheFieldTensorThe first question a modern physicist might ask about a new quantity is, how does ittransform?Inparticular,howdoesittransformunderLorentztransformations?Theusual

answeristhatthenewquantity—whateveritrefersto—isatensorwithsomanyupstairsindicesandsomanydownstairsindices.Theelectricandmagneticfieldsarenoexception.

Art:Lenny, Ihaveabigproblem.Theelectric fieldhas threespacecomponentsandnotime component. How the devil can it be any kind of four-dimensional tensor? Sameproblemwiththemagneticfield.

Lenny:You’reforgetting,Art:3+3=6.

Art:Youmean there’s somekindof tensor that has six components? I don’t get it; a4-vectorhasfourcomponents,atensorwithtwoindiceshas4×4=16components.What’sthetrick?

The answer to Art’s question is that while it’s true that tensors with two indices havesixteencomponents,antisymmetric tensorsonlyhavesix independentcomponents.Let’sjustreviewtheideaofanantisymmetrictensor.

Supposewehavetwo4-vectorsCandB.Therearetwowayswecanputthemtogethertomakeatwo-indexedtensor:CμBνandBμCν.Byaddingandsubtracting these two,wecanbuildasymmetricandanantisymmetrictensor,

Sμν=CμBν+BμCνand

Aμν=CμBν−BμCν.

In particular, let’s count the number of independent components of an antisymmetrictensor. First of all, the diagonal elements all vanish. That means of the sixteencomponents,onlytwelvesurvive.

But there are additional constraints; the off-diagonal elements come in equal andoppositepairs.Forexample,A02=−A20.Onlyhalfofthetwelveareindependent,sothatleavessix.YesArt,therearetensorsthathaveonlysixindependentcomponents.

NowlookatEq.6.36.Alloftherightsideshavetheform

or,equivalently,

Fμν is obviously a tensor andwhat ismore, it is antisymmetric. It has six independentcomponentsandtheyarepreciselythecomponentsof and ComparingwithEq.6.36,wecanlistthepositivecomponents:

Youcanusetheantisymmetrytofillintheothercomponents.NowyoucanseewhereEq.

6.11comesfrom.I’llwriteitagainherealongwithallitsupstairs-downstairsversions.

Youcanthinkoftherowsofthesematricesbeinglabeledbyindexμ,withitsvalues(0,1,2,3)runningalongtheleftedgefromtoptobottom.Likewise,thecolumnsarelabeledbyindexνwiththesame(unwritten)valuesrunningacrossthetop.

LookatthetoprowofFμν.WecanthinkoftheseelementsasF0ν,becauseμ=0foreachofthem,whileνtakesonitsfullrangeofvalues.Theseareelectricfieldcomponents.Wethinkofthemas“mixed”timeandspacecomponentsbecauseoneoftheirindicesisatimeindexwhiletheotherisaspaceindex.Thesameistruefortheleftmostcolumn,andforthesamereasons.

Nowlookatthe3×3submatrixthatexcludesthetopmostrowandleftmostcolumn.Thissubmatrixispopulatedwithmagneticfieldcomponents.

Whatdidweaccomplishbyorganizing theelectricandmagnetic fields intoa singletensor?Alot.Wenowknowhowtoanswerthefollowingquestion:SupposeLenny,inhisframeon themoving train, sees a certain electric andmagnetic field.What electric andmagneticfielddoesArtsee in theframeof therailstation?Achargeat rest in the trainwouldhaveafamiliarelectricfield(Coulombfield)inLenny’sframe.FromArt’spointofview that same charge ismoving, and to compute the field thatArt sees is now just amatterofLorentz-transformingthefieldtensor.We’llexplorethisideafurtherinSection8.1.1.

1Sincethesedialoguestakeplaceinanalternateuniverse,ArtgetstomeetLorentzforthefirsttime—again!

2There’sabriefreviewinAppendixB.

3It’sokayifAandBhappentobethesamevector.

4Thisequationexposesaslightconflictofnotationalconventions.OurconventionfortheuseofGreekindicesstatesthattheyrangeinvaluefrom0to3.However,standardmatrixnotationassumesthatindexvaluesrunfrom1to4.Wewillfavorthe4-vectorconvention(0to3).Thisdoesnoharmaslongaswesticktotheordering(t,x,y,z).

5Wecouldhavelabeledthecomponentsofthecolumnvector(A0;A1;A3;A3)insteadof(t;x;y;z),andditto for thevectorwithprimedcomponents.Thesearejustdifferentlabelsthatrepresentthesamequantities.

6Diagonalcomponentsarecomponentswhosetwoindicesareequaltoeachother.Forexample,A00,A11,A22,andA33

arediagonalcomponents.

7Eq.6.12isoftenwrittenwiththespeedoflightshownexplicitly.You’llsometimesseeitwrittenusing insteadofbyitself.Herewe’lluseunitsforwhichthespeedoflightis1.

8Butwe’llintroducea4-vectorpotentialAμearlyinthegame.

9Whenanupstairs indexappears in thedenominatorof aderivative, it adds adownstairs index to the resultof thatderivative.Any timewe have a Latin index in an expression,we canmove that index upstairs or downstairs asweplease.That’sbecauseLatinindicesrepresentspacecomponents.

10Itwouldbeokayifeachsideoftheequationhadadifferentsummationindex.

11Thesummaryof3-vectoroperatorsinAppendixBshouldbehelpfulfornavigatingtherestofthissection.

12It’sdescribedinAppendixB.

Lecture7

FundamentalPrinciplesandGaugeInvariance

Art:Whyarephysicists sohungupon fundamentalprinciples like leastaction, locality,Lorentzinvariance,and—what’stheotherone?

Lenny:Gauge invariance. The principles help us evaluate new theories. A theory thatviolatesthemisprobablyflawed.Butsometimeswe’reforcedtorethinkwhatwemeanbyfundamental.

Art:Okay,thespeedingrailroadcar:It’sLorentzinvariant.It’sracingdownthetrack,soithasalltheactionitneeds(butnomore).Itstopsateverystation,soit’slocal.

Lenny:Ugh.

Art:Andit’sgaugeinvariant;otherwiseitfallsoffthetracks!

Lenny:Ithinkyoujustfelloffthetracks.Let’sslowthingsdownalittle.

Suppose a theoretical physicist wants to construct a theory to explain some newlydiscovered phenomenon. The new theory is expected to follow certain rules orfundamental principles.There are fourprinciples that seem togovern all physical laws.Briefly,theyare:

•Theactionprinciple

•Locality

•Lorentzinvariance

•Gaugeinvariance

These principles pervade all of physics. Every known theory, whether it’s generalrelativity,quantumelectrodynamics,thestandardmodelofparticlephysics,orYang-Millstheory, conforms to them. The first three principles should be familiar, but gaugeinvarianceisnew;wehaven’tseenitbefore.Themaingoalofthislectureistointroducethisnewidea.We’llstartwithasummaryofallfourfundamentalprinciples.

7.1SummaryofPrinciples

ActionPrincipleThefirst rule is thatphysicalphenomenaaredescribedbyanactionprinciple.Wedon’tknowofanyexceptions to thispattern.Theconceptof energyconservation, tocite justone example, is derived entirely through the action principle. The same is true formomentumconservationandfor therelationbetweenconservationlawsandsymmetriesingeneral.Ifyoujustwriteequationsofmotion,theymaymakeperfectlygoodsense.Butifthey’renotderivedfromanactionprinciple,wewouldgiveuptheguaranteeofenergyandmomentumconservation.Energyconservation,inparticular,isaconsequenceoftheactionprinciple,togetherwiththeassumptionthatthingsareinvariantundershiftingthetimebyafixedamount—atransformationthatwecallatimetranslation.

So that’s our first principle: Look for an action such that the resulting equations ofmotiondescribe thephenomenathatarediscovered in the laboratory.Wehaveseen twokindsofaction.Theactionforparticlemotionis

where representstheLagrangian.Theactionforfieldtheoriesis

For field theories, represents the Lagrangian density. We’ve seen how the Euler-Lagrangeequationsgovernbothofthesecases.

LocalityLocalitymeansthatthingshappeningatoneplaceonlydirectlyaffectconditionsnearbyin

space and time. If you poke a system at some point in time and space, the onlydirecteffectisonthingsinitsimmediatevicinity.Forexample,ifyoustrikeaviolinstringatitsend point, only its nearest neighbor would feel the effect immediately. Of course, theneighbor’smotionwould affect its neighbor, and so forth down the chain. In time, theeffectwouldbefeltalongtheentirelengthofthestring.Buttheshort-timeeffectislocal.

Howdoweguaranteethatatheoryrespectslocality?Onceagain,ithappensthroughtheaction.Forexample,supposewe’retalkingaboutaparticle.Inthatcase,theactionisanintegralovertime(dt)alongtheparticletrajectory.Toguaranteelocality,theintegrand—theLagrangian —mustdependonlyon thecoordinatesof thesystem.Foraparticle,that means the position components of the particle, and their first time derivatives.Neighboringtimepointscomeintoplaythroughthetimederivatives.Afterall,derivativesarethingsthatcapturerelationshipsbetweennearneighbors.However,higherderivativesareruledoutbecausethey’re“lesslocal”thanfirstderivatives.1

Fieldtheoriesdescribeafieldcontainedinavolumeofspaceandtime(Figs.4.2and5.1).Theactionisanintegralnotonlyovertime,butalsooverspace(d4x).Inthiscase,localitysaysthattheLagrangiandependsonthefieldΦandonitspartialderivativeswithrespect toXμ.Wecancall thesederivativesΦμ.This is enough to guarantee that thingsonlyaffecttheirnearbyneighborsdirectly.

You could imagine a world in which poking something in one place has aninstantaneous effect in someother place. In that case theLagrangianwouldnot dependonlyonnearestneighborsthroughderivativesbutonmorecomplicatedthingsthatallow“actionatadistance.”Localityforbidsthis.

Aquickwordaboutquantummechanics:Quantummechanicsisoutsidethescopeofthis book. Nevertheless, many readers may wonder how—or whether—the localityprincipleappliestoquantummechanics.Let’sbeasclearaspossible:Itdoes.

Quantummechanicalentanglementisoftenreferredtomisleadinglyasnonlocality.Butentanglementisnotthesameasnonlocality.Inourpreviousbook,QuantumMechanics,we explain this in great detail. Entanglement does not mean you can send signalsinstantaneouslyfromoneplacetoanother.Localityisfundamental.

LorentzInvarianceAtheorymustbeLorentzinvariant.Inotherwords,theequationsofmotionshouldbethesameineveryreferenceframe.We’vealreadyseenhowthisworks.IfwemakesuretheLagrangianisascalar,weguaranteethatthetheoryisLorentzinvariant.Insymbols,

=Scalar

Lorentzinvarianceincludestheideaofinvarianceunderrotationsofspace.2

GaugeInvarianceThe last rule is somewhatmysterious and takes some time to fully understand.Briefly,gaugeinvariancehastodowithchangesthatyoucanmaketothevectorpotentialwithout

affectingthephysics.We’llspendtherestofthislectureintroducingit.

7.2GaugeInvarianceAn invariance, also called a symmetry, is a change in a system that doesn’t affect theactionortheequationsofmotion.Let’slookatsomefamiliarexamples.

7.2.1SymmetryExamplesThe equationF=m is perhaps the best-known equation ofmotion. It has exactly thesameformifyoutranslatetheoriginofcoordinatesfromonepointtoanother.Thesameistrueforrotationsofthecoordinates.Thislawisinvariantundertranslationsandrotations.

Asanotherexample,considerourbasic field theoryfromLecture4.TheLagrangianforthistheory(Eq.4.7)was

whichwealsoexpressedas

For now, we’ll consider a simplified version, with V(Φ) set to zero and all the spacecoordinatesfoldedintothesinglevariablex:

The equation of motion (Eq. 4.10) we derived from this Lagrangian was (slightlysimplifiedhere)

I’veignoredthefactor inthefirsttermbecauseit’snotimportantforthisexample.Thisequation has a number of invariances, includingLorentz invariance.To discover a newinvariance,wetrytofindthingsabouttheequationthatwecanchangewithoutchangingitscontentormeaning.Supposeweaddaconstanttotheunderlyingfield:

Inotherwords, supposewe take a fieldΦ that is already a solution to the equationofmotion and just add a constant.Does the result still satisfy the equation ofmotion?Ofcourseitdoes,becausethederivativesofaconstantarezero.IfweknowthatΦ satisfiestheequationofmotion,then(Φ+c)alsosatisfiesit:

YoucanalsoseethisintheLagrangian,

−∂μΦ∂μΦ

where once again I’ve ignored the factor 1/2 for simplicity. What happens to this

Lagrangian (and consequently to the action) if we add a constant toΦ?Nothing! Thederivativeofaconstantiszero.Ifwehaveaparticularactionandafieldconfigurationthatminimizes the action, adding a constant to the field makes no difference; it will stillminimizetheaction.Inotherwords,addingaconstanttosuchafieldisasymmetry,oraninvariance.It’sasomewhatdifferentkindofinvariancethanwe’veseenbefore,butit’saninvarianceallthesame.

Nowlet’s recall theslightlymorecomplicatedversionof this theory,where the termV(Φ)isnotzero.BackinLecture4,weconsideredthecase

whosederivativewithrespecttoΦis

Withthesechanges,theLagrangianbecomes

andtheequationofmotionbecomes

WhathappenstoEq.7.3ifweaddaconstanttoΦ?IfΦ isasolution, is(Φ+c)stillasolution?It isnot.There’snochange in thefirst two terms.Butaddingaconstant toΦclearlychangesthethirdterm.WhatabouttheLagrangian,Eq.7.2?AddingaconstanttoΦhasnoimpactonthetermsinsidethesquarebrackets.Butitdoesaffecttherightmostterm;Φ2isnotthesameas(Φ+c)2.Withtheextraterm

intheLagrangian,addingaconstanttoΦisnotaninvariance.

7.2.2ANewKindofInvarianceLet’sreturntotheactionintegral

whichweintroducedinLecture6.SupposewemodifyAμbyaddingthefour-dimensionalgradientofascalarS.Insymbols,

Theprecedingsummakessensebecausebothtermsarecovariantintheindexμ.Butdoesthis replacement change the equations of motion? Does it change the orbits that theparticlewillfollow?Doesitmakeanychangewhateverinthedynamicsoftheparticle?Thereplacementchangestheactioninastraightforwardway;

Figure7.1:SpacetimeTrajectory.Thesolidlineisthestationary-actionpath.Thedashedlineisthevariedpathwiththesamefixedend

points.

becomes

Whatdoesthenewintegral—therightmostintegralinEq.7.5—represent?InFig.7.1,theparticletrajectoryisbrokenintolittlesegments,asusual.ConsiderthechangeinSalongoneoftheselittlesegments.Straightforwardcalculustellsusthat

is the change in S in going from one end of the segment to the other. If we add (orintegrate)thesechangesforallthesegments,itgivesusthechangeinSingoingfromtheinitialpointofthetrajectorytothefinalpoint.Inotherwords,it’sequaltoSevaluatedattheendpointbminusSatthestartingpointa.Insymbols,

SitselfisanarbitraryscalarfunctionthatIjustthrewin.Doesaddingthisnewtermtothevectorpotentialchangethedynamicsoftheparticle?3

We’ve changed the action in a way that only depends on the end points of thetrajectory.Buttheactionprincipletellsustosearchfortheminimumactionbywigglingthetrajectorysubject totheconstraint that theendpointsremainfixed.Because theendpoints remainfixed, theS(b)−S(a) term is the same for every trajectory, including thetrajectorywithstationaryaction.Therefore,ifyoufindatrajectorywithstationaryaction,itwill remainstationarywhenyoumake thischange to thevectorpotential.Adding thefour-dimensionalgradientofascalarhasnoeffectonthemotionoftheparticlebecauseit

only affects the action at the end points. In fact, adding any derivative to the actiontypicallydoesn’tchangeanything.Wedon’tevencarewhatSis;ourreasoningappliestoanyscalarfunctionS.

This is the concept of gauge invariance. The vector potential can be changed, in acertainway,withoutanyeffecton thebehaviorof thechargedparticle.Adding toavectorpotential iscalledagaugetransformation.Thewordgauge isahistoricalartifactthathaslittletodowiththeconcept.

7.2.3EquationsofMotionGaugeinvariancemakesaboldclaim:

Dreamupanyscalarfunctionyoulike,additsgradienttothevectorpotential,andtheequationsofmotionstayexactlythesame.

Can we be sure about that? To find out, just go back to the equations of motion: theLorentzforcelawofEq.6.33,

Wecanrewritethisas

where

andthe4-velocityUνis

The equations ofmotion do not directly involve the vector potential. They involve thefield tensorFμν, whose elements are the electric and magnetic field components. Anychange to the vector potential that does not affectFμνwill not affect themotion of theparticle.Our task is toverify that the field tensorFμν isgauge invariant.Let’s seewhathappenswhenweaddthegradientofascalartothevectorpotentialsofEq.7.6:

Thislookscomplicated,butitsimplifiesquiteeasily.Thederivativeofasumisthesumofthederivatives,sowecanwrite

or

Butweknowthattheorderinwhichpartialderivativesaretakendoesn’tmatter.Inotherwords,

Thereforethetwosecond-orderderivativesinEq.7.8areequalandcanceleachotherout.4Theresultis

whichisexactlythesameasEq.7.6.Thisclosesthecircle.Addingthegradientofascalartothe4-vectorpotentialhasnoimpactontheactionorontheequationsofmotion.

7.2.4PerspectiveIfourgoalwastowriteequationsofmotionforparticlesandelectromagneticfields,whydidwebotheraddingavectorpotential,especially ifwecanchangeitwithoutaffectingtheelectricandmagneticfields?

The answer is that there’s no way to write an action principle for the motion of aparticlethatdoesnotinvolvethevectorpotential.Yetthevalueofthevectorpotentialatagivenpoint isnotphysicallymeaningfuland itcannotbemeasured; ifyouchange itbyaddingthegradientofascalar,thephysicsdoesn’tchange.

Some invariances have obvious physical meaning. It’s not hard to imagine tworeference frames in the same problem and translating between them: two physicalreference frames, yours and mine. Gauge invariance is different. It’s not abouttransformations of coordinates. It’s a redundancy of the description. Gauge invariancemeanstherearemanydescriptions,allofwhichareequivalenttoeachother.What’snewis that it involves a function of position. For example,whenwe rotate coordinates,wedon’t rotate differently at different locations. That sort of rotationwould not define aninvariance in ordinary physics.Wewould rotate once and only once, by some specificangle, inawaythatdoesnot involveafunctionofposition.Ontheotherhand,agaugetransformation involves a whole function—an arbitrary function of position. Gaugeinvariance isa featureofeveryknownfundamental theoryofphysics.Electrodynamics,thestandardmodel,Yang-Millstheory,gravity,allhavetheirgaugeinvariances.

You may have formed the impression that gauge invariance is an interestingmathematical property with no practical significance. That would be amistake. Gaugeinvariance allows us to write different but equivalent mathematical descriptions of aphysical problem. Sometimes we can simplify a problem by adding something to thevectorpotential.Forexample,wecanchooseanSthatsetsanyoneofthecomponentsofAμequaltozero.Typically,wewouldchooseaspecificfunctionSinordertoillustrateorclarify one aspect of a theory. That may come at the expense of obscuring a differentproperty of the theory. By looking at the theory from the point of view of all possiblechoicesofS,wecanseeallofitsproperties.

1They’realsoruledoutbyalargebodyoftheoreticalandexperimentalresults.

2General relativity (notcovered in thisbook) requires invarianceunderarbitrarycoordinate transformations.Lorentztransformations are a special case.Even so, the invariance principle is similar; instead of requiring to be a scalar,generalrelativityrequiresittobeascalardensity.

3Hint:TheanswerisNo!

4Thispropertyofsecondpartialderivativesistrueforthefunctionsthatinterestus.Theredoexistfunctionsthatdonothavethisproperty.

Lecture8

Maxwell’sEquations

Maxwell has a private table atHermann’sHideaway.He’s sitting there alone, having aratherintenseconversationwithhimself.

Art:LookslikeourfriendMaxwellishavinganidentitycrisis.

Lenny:Not exactly a crisis.He’s using two different identities on purpose, to build hisbeautifultheoryofelectromagnetism.

Art:Howfarcanhegetwiththem?Identitiesarenice,but…

Lenny:Abouthalfway.That’swheretherealactionbegins.

Asmostreadersknow,theTheoreticalMinimumbooksarebasedonmyseriesoflecturesof thesamenameatStanfordUniversity.It’s in thenatureof lectureseries that theyarenotalwaysperfectlyorderly.Myownlecturesoftenbeginwithareviewofthepreviouslecture or with a fill-in of some material that I didn’t get to. Such was the case withLecture8onMaxwell’sequations.TheactuallecturebegannotwithMaxwell’sequationsbutwiththetransformationpropertiesoftheelectricandmagneticfields—asubjectthatIonlyhintedat,attheendofLecture7.

Recently I had occasion to look at Einstein’s first paper on relativity. 1 I stronglyencourageyou to study thispaperonyourown.The firstparagraph inparticular is justmarvelous.Itexplainshislogicandmotivationextremelyclearly.I’dliketodiscussitfora moment because it’s deeply connected with the transformation properties of theelectromagneticfieldtensor.

8.1Einstein’sExampleEinstein’sexampleinvolvesamagnetandanelectricconductorsuchasawire.Anelectriccurrentisnothingbutacollectionofmovingchargedparticles.WecanreduceEinstein’ssetupto theproblemofachargedparticle,containedinawire, in thefieldofamagnet.Theessentialthingisthatthewireismovingintherestframeofthemagnet.

Fig.8.1shows thebasicsetup in the“laboratory”framewhere themagnet isat rest.Thewire isorientedalongtheyaxisandismovingalongthexaxis.Anelectron in thewire is dragged alongwith thewire. There’s also a stationarymagnet (not shown) thatcreatesauniformmagneticfieldwithonlyonecomponent,Bz,thatpointsoutofthepage.Therearenootherelectricormagneticfields.

Figure8.1:EinsteinExample—MovingChargeViewpoint.Laboratoryandmagnetareatrest.Chargeemovestotherightatspeedv.Theconstantmagneticfieldhasonlyonecomponent,Bz.Thereisnoelectric

field.

Whathappenstotheelectron?ItfeelsaLorentzforceequaltothechargeetimesthecrossproductof its velocityvwith themagnetic fieldBz. Thatmeans the forcewill beperpendiculartobothvandBz.Following theright-handrule,andremembering that theelectroncarriesanegativecharge,theLorentzforcewillbedirectedupward.Theresultisthatacurrentwillflowinthewire.TheelectronsarepushedupwardbytheLorentzforce,butbecauseofBenjaminFranklin’sunfortunateconvention(thechargeofanelectron isnegative)thecurrentflowsdownward.

Figure8.2:EinsteinExample—MovingMagnetViewpoint.Chargeeisatrest.Magnetandlaboratorymovetotheleftatspeedv.Theconstantelectricfield

hasonlyonecomponent,Ey.TheforceontheelectronisoppositetothedirectionofEybecausetheelectronhasa

negativecharge.

That’s how the laboratory observer describes the situation; the effective current isproducedbythemotionofthechargeinamagneticfield.ThesmalldiagramatthelowerrightofthefigureillustratestheLorentzforce

thatactsontheelectron.

Now let’s look at the same physical situation in the frame of the electron. In this“primed”frame,theelectronisatrest,andthemagnetmovestotheleftwithvelocityvasshowninFig.8.2.Sincetheelectronisatrest,theforceonitmustbeduetoanelectricfield.That’spuzzlingbecause theoriginal problem involvedonly amagnetic field.Theonly possible conclusion is that a movingmagnet must create an electric field. Boileddowntoitsessentials,thatwasEinstein’sargument:Thefieldofamovingmagnetmustincludeanelectriccomponent.

Whathappenswhenyoumoveamagnetpastawire?Amovingmagneticfieldcreateswhat Einstein and his contemporaries called an electromotive force (EMF), which

effectivelymeansanelectricfield.Takingamagnetwhosefieldpointsoutofthepageandmoving it to the left creates adownward-oriented electric field,Ey. It exerts anupwardforceeEy on a stationary electron. 2 In otherwords, the electric fieldEy in the primedframe has the same effect as the Lorentz force in the unprimed frame. By this simplethought experiment Einstein derived the fact that magnetic fields must transform intoelectricfieldsunderaLorentztransformation.Howdoesthispredictioncomparewiththetransformation properties of and when they are viewed as components of anantisymmetrictensorFμν?

8.1.1TransformingtheFieldTensorThetensorFμνhascomponents

Wewanttofigureouthowthecomponentstransformwhenwemovefromoneframeofreferencetoanother.Giventheprecedingfieldtensorinoneframeofreference,whatdoesitlookliketoanobservermovinginthepositivedirectionalongthexaxis?Whatarethenewcomponentsofthefieldtensorinthemovingframe?

Towork that out,we have to remember the rules for transforming tensorswith twoindices.Let’sgobacktoasimplertensor:4-vectors,andinparticularthe4-vectorXμ.4-vectors are tensorswith only one index.We know that 4-vectors transform by Lorentztransformation:

y′=y

z′=z

Aswe saw inLecture 6 (Eq. 6.3, for example),we canorganize these equations into amatrixexpressionusingtheEinsteinsummationconvention.WeintroducedthematrixLμνthat captures thedetailsof a simpleLorentz transformationalong thex axis.Using thisnotation,wewrotethetransformationas

(X′)μ=LμνXν,

whereνisasummationindex.ThisisjustshorthandfortheprecedingsetoffourLorentztransformationequations.InEq.6.4,wesawwhatthisequationlookslikeinmatrixform.Here’saslightlyrevisedversionofthatequation,with(t,x,y,z)replacedby(X0,X1,X2,X3):

Howdowetransformatensorwithtwoindices?Forthisexample,I’llusetheupper-indexversionofthefieldtensor,

I’vechosentheupper-indexversion(Eq.6.42)forconvenience,becausewealreadyhaverulestotransformthingswithupperindices;theformoftheLorentztransformationwe’vebeen using all along is set up to operate on things with upper indices. How do youtransformathingwith twoupper indices?It’salmost thesameas transformingasingle-indexthing.Wherethesingle-indextransformationruleis

withsummationindexσ,thetwo-indextransformationruleis

with two summation indices σ and τ. In other words, we just perform the single-indextransformationtwice.EachindexoftheoriginalobjectFστbecomesasummationindexinthetransformationequation.Eachindex(σandτ)transformsinexactlythesamewaythatitwouldifitweretheonlyindexbeingtransformed.

Ifyouhadanobjectwithmoreindices—youcouldhaveanynumberofindices—therulewouldbe thesame:Youtransformeveryindexin thesameway.That’s therulefortransformingtensors.Let’stryitoutwithFμνandseeifwecandiscoverwhetherthereisanelectricfieldalongtheyaxis.Inotherwords,let’strytocomputetheycomponentofelectricfield(E′)yintheprimedreferenceframe.

(E′)y=(F′)0y

WhatdoweknowabouttheoriginalunprimedfieldtensorFμνasitappliestoEinstein’sexample?We know that it represents a puremagnetic field along theunprimed z axis.Therefore,Fστhasonlyonenonzerocomponent,Fxy,whichcorrespondstoBz.3FollowingthepatternofEq.8.2,wecanwrite

Noticethat theindicesmatchupinthesamewayastheindicesinEq.8.2,eventhoughwe’reonlylookingatonesinglecomponentontheleftside.Thesummationsontherightcollapsetoasingleterm.ByreferringtothematrixinEq.8.1,wecanidentifythespecificelementsofL tosubstitute intothetransformationequation8.3.Specifically,wecanseethat

and

ApplyingthesesubstitutionsinEq.8.3resultsin

or

ButFxyisthesameasBzintheoriginalreferenceframe,andwecannowwrite

Thus,asEinsteinclaimed,apuremagneticfieldwhenviewedfromamovingframewillhaveanelectricfieldcomponent.

Exercise8.1:Consideranelectricchargeatrest,withnoadditionalelectricormagneticfields present. In terms of the rest frame components, (Ex, Ey, Ez), what is the xcomponentoftheelectricfieldforanobservermovinginthenegativexdirectionwithvelocityv?Whataretheyandzcomponents?Whatarethecorrespondingcomponentsofthemagneticfield?

Exercise8.2:Artissittinginthestationasthetrainpassesby.IntermsofLenny’sfieldcomponents, what is the x component ofE observed by Art?What are the y and zcomponents?WhatarethecomponentsofthemagneticfieldseenbyArt?

8.1.2SummaryofEinstein’sExampleWestartedoutbysettingupalaboratoryframewithanelectricfieldofzero,amagneticfieldpointingonlyin thepositivezdirection,andanelectronmovingwithvelocityv inthepositivex direction.We then asked for the electric andmagnetic fieldvalues in theframeoftheelectron.Wediscoveredtwothings:

1.There’snomagneticforceontheelectroninthenewframebecausetheelectronisatrest.

2.Thereis,however,aycomponentofelectricfieldinthenewreferenceframethatexertsaforceontheelectron.

Theforceonthe(moving)electronthatisduetoamagneticfieldinthelaboratoryframeis due to an electric field in the moving frame (the electron’s rest frame). That’s theessenceofEinstein’sexample.Now,ontoMaxwell’sequations.

Exercise 8.3: For Einstein’s example, work out all components of the electric andmagneticfieldsintheelectron’srestframe.

8.2IntroductiontoMaxwell’sEquationsRecallPauli’s(fictional)quote:

Fieldstellchargeshowtomove;chargestellfieldshowtovary.

InLecture6wespentagooddealoftimeonthefirsthalfofthequote,describingthewayfields tell chargeshow tomove.Now it’s thecharges’ turn to tell fieldshow tovary. Iffields control particles through the Lorentz force law, charges control fields throughMaxwell’sequations.

Myphilosophyofteachingelectrodynamicsissomewhatunorthodox,soletmeexplainit. Most courses take a historical perspective, beginning with a set of laws that werediscoveredinthelateeighteenthandearlynineteenthcenturies.You’veprobablyheardofthem—Coulomb’s law, Ampère’s law, and Faraday’s law. These laws are sometimestaughtwithoutcalculus,agravemistakeinmyopinion;physicsisalwaysharderwithoutthe mathematics. Then, through a somewhat torturous procedure, these laws aremanipulatedintoMaxwell’sequations.Myownwayofteaching,eventoundergraduates,istotaketheplunge—fromthestart—andbeginwithMaxwell’sequations.Wetakeeachoftheequations,analyzeitsmeaning,andinthatwayderiveCoulomb’s,Ampèere’sandFaraday’s laws. It’s the “cold shower” way of doing it, but within a week or two thestudentsunderstandwhatwouldtakemonthsdoingitthehistoricalway.

How many Maxwell’s equations are there altogether? Actually there are eight,although vector notation boils them down to four: two 3-vector equations (with threecomponentsapiece)andtwoscalarequations.

Four of the equations are identities that follow from definitions of the electric andmagneticfieldsintermsofthevectorpotential.Specifically,theyfollowdirectlyfromthedefinitions

togetherwithsomevectoridentities.There’snoneedtoinvokeanactionprinciple.Inthatsense,thissubsetofMaxwell’sequationscomes“forfree.”

8.2.1VectorIdentitiesAnidentityisamathematicalfactthatfollowsfromadefinition.Manyaretrivial,buttheinteresting ones are often not obvious. Over the years, people have stockpiled a largenumberofidentitiesinvolvingvectoroperators.Fornow,weneedonlytwo.YoucanfindasummaryofbasicvectoroperatorsinAppendixB.

TwoIdentitiesNowforourtwoidentities.Thefirstonesaysthatthedivergenceofacurlisalwayszero.Insymbols,

for anyvector field .You can prove this by a technique known as “brute force.” Justexpandtheentireleftsideincomponentform,usingthedefinitionsofdivergenceandcurl(AppendixB).You’ll find thatmanyof the terms cancel.For example, one termmightinvolve the derivative of somethingwith respect to x and thenwith respect to y,whileanotherterminvolvesthesamederivativesintheoppositeorder,andwithaminussign.Termsofthatkindcancelouttozero.

Thesecondidentitystatesthatthecurlofanygradientiszero.Insymbols,

whereSisascalarand Sisavector.Remember,thesetwoidentitiesarenotstatementsaboutsomespecialfieldlikethevectorpotential.They’retrueforanyfieldsSand ,aslongasthey’redifferentiable.ThesetwoidentitiesareenoughtoprovehalfofMaxwell’sequations.

8.2.2MagneticFieldWesawinEq.6.30thatthemagneticfield isthecurlofthevectorpotential ,

Therefore,thedivergenceofamagneticfieldisthedivergenceofacurl,

Ourfirstvectoridentitytellsusthattherightsidemustbezero.Inotherwords,

This isoneofMaxwell’sequations.Itsaysthat therecannotbemagneticcharges.Ifwesawaconfigurationwithmagneticfieldvectorsdivergingfromasinglepoint—amagneticmonopole—thenthisequationwouldbewrong.

8.2.3ElectricFieldWecanderiveanotherMaxwellequation—avectorequationinvolvingtheelectricfield—fromvectoridentities.Let’sgobacktothedefinitionoftheelectricfield,

NoticethatthesecondtermisjustthenthcomponentofthegradientofA0.Fromathree-dimensional point of view (just space), the time component of , that is, A0, can bethoughtofasascalar.4ThevectorwhosecomponentsarederivativesofA0canbethoughtofasthegradientofA0.Sotheelectricfieldhastwoterms;oneisthetimederivativeofthespacecomponentof ,and theother is thegradientof the timecomponent.WecanrewriteEq.8.6asavectorequation,

Nowlet’sconsiderthecurlof ,

Thesecondtermontherightsideisthecurlofagradient,andoursecondvectoridentitytellsusthatitmustbezero.Wecanrewritethefirsttermas

Whyisthat?Becausewe’reallowedtointerchangederivatives.Whenwetakeaderivativetomakeacurl,wecaninterchangethespacederivativewiththetimederivative,andpullthe time derivative on the outside. So the curl of the time derivative of is the timederivativeofthecurlof .Asaresult,Eq.8.8simplifiesto

Butwealreadyknow(Eq.6.30)thatthecurlof isthemagneticfield .SowecanwriteasecondMaxwellequation,

or

Thisvectorequationrepresentsthreeequations,oneforeachcomponentofspace.Eqs.8.5and8.10aretheso-calledhomogeneousMaxwellequations.

Although the homogeneous Maxwell equations were derived as identities, theyneverthelesshaveimportantcontent.Equation8.5,

expressestheimportantfactthattherearenomagneticchargesinnature,ifindeeditisafact. In elementary terms it states that, unlike electric fields, the lines ofmagnetic fluxnever end. Does that mean that magnetic monopoles—magnetic analogs of electricallycharged particles—are impossible? Iwon’t answer that here but at the end of the bookwe’llexaminetheissueinsomedetail.

WhataboutEq.8.10?Doesithaveconsequences?Certainlyitdoes.Equation8.10isone mathematical formulation of Faraday’s law that governs the workings of allelectromechanicaldevicessuchasmotorsandgenerators.Attheendofthislecturewe’llcomebacktoFaraday’slawamongothers.

8.2.4TwoMoreMaxwellEquationsThere are two more Maxwell equations 5 that are not mathematical identities, whichmeans that they cannot be derived just from the definitions of and . It will be oureventualgoaltoderivethemfromanactionprinciple,butoriginallytheywerediscoveredasempiricallawsthatsummarizedtheresultsofexperimentsoncharges,electriccurrents,and magnets. The two additional equations are similar to Eqs. 8.5 and 8.10, with theelectricandmagneticfieldsswitchingroles.Thefirstisasingleequation,

ThisissimilartoEq.8.5exceptthattherightsideisnotzero.Thequantityρisthedensityofelectriccharge,thatis,thechargeperunitvolumeateachpointofspace.Theequationrepresents the fact that electric charges are surrounded by electric fields that they carrywiththemwherevertheygo.ItiscloselyconnectedtoCoulomb’slaw,aswewillsee.

Thesecondequation—reallythreeequations—issimilartoEq.8.10:

Table8.1:MaxwellEquationsandTheirDerivationsfromtheVectorPotential.

Again,asidefromtheinterchangeofelectricandmagneticfieldsandachangeofsign,themostimportantdifferencebetweenEq.8.10andEq.8.13isthattherightsideofEq.8.13isnotzero.Thequantity isthecurrentdensity,whichrepresentstheflowofcharge—forexample,theflowofelectronsthroughawire.

ThecontentofEq.8.13isthatflowingcurrentsofchargearealsosurroundedbyfields.Thisequationisalsorelatedtoalaw,Ampèere’slaw,thatdeterminesthemagneticfieldproducedbyacurrentinawire.

Table 8.1 shows all four Maxwell equations.We derived the first two by applyingvector identities to the vector potential. The second two equations also come from thevectorpotential,buttheycontaindynamicalinformationandweneedanactionprincipletoderivethem.We’lldothatinthenextlecture.

Thesecondgroupofequationslooksalotlikethefirstgroup,butwiththerolesofand (almost) interchanged. I say almost because there are some small but importantdifferences in their structure. To begin with, the signs are a little different. Also, theequationsinthesecondgroupareinhomogeneous;theyinvolvequantitiesontherightsidecalledchargedensityρandcurrentdensity thatdonotappear in thefirstgroup.We’llhavemore to say aboutρ and in the next section.Charge densityρ is the amount ofcharge per unit volume, just as you would expect; in three dimensions, it’s a scalar.Currentdensityisa3-vector.We’llseeinamomentthatin4-vectorlanguagethingsarealittlemorecomplicated.We’lldiscoverthatρtogetherwiththethreecomponentsof arethecomponentsofa4-vector.6

8.2.5ChargeDensityandCurrentDensity

Whatarechargedensityandcurrentdensity?I’vetriedtoillustratetheminFigs.8.3and8.4,butthere’saproblem:Idon’tknowhowtodrawfourdimensionsofspacetimeonatwo-dimensionalpage.Keepthatinmindasyouinterpretthesefigures.Theyrequiresomeabstractthinking.

Charge density is exactly what it sounds like. You look at a small region of three-dimensional space and divide the total amount of charge in that region by the region’svolume.Chargedensityisthelimitingvalueofthisquotientasthevolumebecomesverysmall.

Let’sdescribe this idea from theperspectiveof four-dimensional spacetime.Fig.8.3triestodepictallfourdirectionsofspacetime.Asusual,theverticalaxisrepresentstime,andthexaxispointstotheright.Thespatialaxispointingoutofthepageislabeledy,zand represents both the y and z directions. This is the “abstract thinking” part of thediagram; it’s far fromperfect, but youget the idea.Let’s consider a little cell in space,showninthefigureasahorizontalsquare.Therearetwoimportantthingstonoticeaboutit.First,itsorientationisperpendiculartothetimeaxis.Second,it’snotreallyasquareatall.The“edges”paralleltothey,zaxisarereallyareas,andthesquareisreallyathree-dimensionalvolumeelement,inotherwords,acube.

Figure8.3:ChargeandCurrentDensityinSpacetime.They,zaxisrepresentstwo

directionsinspace.Curvedarrowsareworldlinesofchargedparticles.Thehorizontal

windowillustrateschargedensity.Theverticalwindowillustratesthexcomponentofcurrent

density.

Figure8.4:CurrentDensityinSpace.Thetimeaxisisnotshown.Thecurvedarrowsarenotworldlines,butshowtrajectoriesinspaceonly.

The curved arrows represent world lines of charged particles.We can imagine thatspaceandtimearefilledwiththeseworldlines,flowinglikeafluid.Considertheworldlinesthatpassthroughourlittlecube.Rememberthatourcube,beingperpendiculartothetime axis, represents a single instant of time. This means that the particles that “passthrough”thecubearesimplytheparticlesthatareinthecubeatthatinstantoftime.Wecount up all their charges to get the total charge passing through that little volume ofspace. 7 The charge density ρ is the limiting value of the total charge in the volumeelementdividedbyitsvolume,

Whataboutcurrentdensity?Onewaytothinkaboutcurrentdensityistoturnourlittlesquare onto its edge.The vertically oriented “square” is shown to the right in Fig. 8.3.This diagram calls for evenmore abstract thinking. The square still represents a three-dimensional cube in spacetime.But this cube is not purely spatial; becauseone edge isparalleltothetaxis,oneofitsthreedimensionsistime.Theedgeparalleltothey,zaxisconfirms that theother twodimensionsarespatial.Thissquare isperpendicular to thexaxis.Asbefore,theedgesparalleltothey,zaxisreallystandfortwo-dimensionalsquaresinthey,zplane.Let’sre-drawthislittlesquareinFig.8.4asapurespacediagramwithonlythex,y,andzaxes.Inthisnewdiagramthesquarereallyisasquarebecausethere’snotimeaxis.It’sjustalittlewindowperpendiculartothexaxis.Howmuchchargepassesthrough thatwindowperunit time?Thewindow isanarea, sowe’re reallyaskinghowmuchchargeflowsthroughthewindowperunitareaperunittime.8That’swhatwemeanbycurrentdensity.Becauseourwindowisperpendicular tothexaxis, itdefinescurrentdensity in the x direction, and we have similar components for the y and z directions.

Currentdensity isa3-vector,

ThequantitiesΔAx,ΔAy,andΔAzrepresentelementsofareathatareperpendiculartothex,y,andz axes respectively. Inotherwords (seeFig.8.6), theΔA’s that appear inEqs.8.14are

ΔAx=ΔyΔzΔAy=ΔzΔxΔAz=ΔzΔ,y

andthereforewecanwriteEqs.8.14intheform

Youcanthinkofρitselfasaflowofchargeinthetimedirection,thexcomponentof asa flow in the x direction, and so forth.You can think of a spatial volume as a kind ofwindowperpendiculartothetaxis.We’llsoondiscoverthatthequantities(ρ,jx,jy,jz)areactually the contravariant components of a 4-vector. They enter into the other twoMaxwell equations, the equations thatwe’ll derive from an action principle in the nextlecture.

8.2.6ConservationofChargeWhatdoeschargeconservationactuallymean?Oneofthethingsitmeansisthatthetotalamountofchargeneverchanges;ifthetotalamountofchargeisQ,then,

Butthat’snotall.SupposeQcouldsuddenlydisappearfromourlaboratoryonEarthandinstantlyreappearonthemoon(Fig.8.5).AphysicistonEarthwouldconcludethatQisnotconserved.Conservationofchargemeanssomethingstronger than“the totalamountofQdoesnotchange.”ItreallymeansthatwheneverQdecreases(orincreases)withinthewallsof the laboratory, it increases (ordecreases)by the sameamount rightoutside thewalls.Evenbetter,achangeinQisaccompaniedbyafluxofQpassingthroughthewalls.

Whenwe say conservation, whatwe reallymean is local conservation. This importantideaappliestootherconservedquantitiesaswell,notjustcharge.

Tocapture the ideaof localconservationmathematically,weneed todefinesymbolsfor flux, or flow across a boundary.The terms flux, flow, andcurrent of a quantity aresynonymousandrefertotheamountflowingthroughasmallelementofareaperunitareaperunittime.ChangesintheamountofQinaregionofspacemustbeaccountedforbyafluxofQthroughtheboundariesoftheregion.

Fig.8.6showsathree-dimensionalvolumeelementthatcontainsachargeQ.Thewallsoftheboxarelittlewindows.Anychangeintheamountofchargeinsidetheboxmustbeaccompaniedbycurrentthroughitsboundaries.Forconvenience,let’sassumethevolumeof the box has a value of 1 in someunitswhere a small volume is just a unit volume.Similarly,wecantaketheedgesofthebox,(Δx,Δy,Δz),tohaveunitlength.

Figure8.5:LocalConservationofCharge.Theprocessinthispicturecannothappen.Chargecan’tdisappearinoneplaceandinstantlyreappearinsomefar-removedplace.

What is the time derivative of the charge inside the box?Because the box has unitvolume,thechargeinsidetheboxisthesameasthechargedensityρ.Therefore,thetimederivativeofthechargeisjust ,thetimederivativeofρ.Localchargeconservationsaysthat mustequaltheamountofchargeflowingintotheboxthroughitsboundaries.Forexample,suppose thechargeQ increases.Thatmeans theremustbesomenet inflowofcharge.

Thenetinflowofchargemustbethesumofthechargescominginthrougheachofthesixwindows(thesixfacesofthecube).Let’sconsiderthechargecominginthroughtheright-facing window, labeled −jx+ in the diagram. The x+ subscript means the xcomponentofcurrent thatenters theright-facingor+windowof thebox.Thisnotationseemsfussy,butinamomentyou’llseewhyweneedit.Thechargecomingthroughthatwindowisproportionaltojx,thexcomponentofthecurrentdensity.

Figure8.6:LocalChargeConservation.Forconvenience,weassumeΔx=Δy=Δz=1insome

smallunits.ThevolumeelementisΔV=ΔxΔyΔz=1.

The current density jx is defined in a way that corresponds to the flow from lowervalues of x to greater values of x. That’s why the charge coming into the right-facingwindowperunittimeisactuallythenegativeofjx,andtheleft-pointingarrowislabeled−jx+. Likewise, we use the subscript x−” for the leftmost or minus face. Similarconventionsapplytotheyandzdirections.Toavoidclutter,wedidnotdrawarrowsfor−jz+andjz−.

Let’s lookcloselyatbothof thecurrents (jx−and−jx+) flowing in thex direction inFig.8.6.Howmuchchargeflowsintotheboxfromtheright?Eq.8.15makesclearthattheamountofchargeenteringtheboxfromtherightduringtimeΔtis

ΔQright=−(jx−)ΔyΔzΔt

Similarly,theamountenteringfromtheleftis

ΔQleft=(jx+)ΔyΔzΔt

Thesetwochargesarenotingeneralequal.That’sbecausethetwocurrentsarenotequal;theyoccurattwolocationsthatareslightlydisplacedalongthexaxis.Thesumofthesetwo charges is the total increase of charge in the boxdue to currents in the positive or

negativexdirection.Insymbols,

ΔQtotal=−(jx−)ΔyΔzΔt+(jx+)ΔyΔzΔt

or

ΔQtotal=−(jx−−jx−)ΔyΔzΔt(8.17)

The term inparentheses is thechange incurrentovera small intervalΔxand iscloselyrelatedtothederivativeofjxwithrespecttox.Infact,it’s

SubstitutingthatbackintoEq.8.17resultsin

WeeasilyrecognizeΔxΔyΔzasthevolumeofthebox.Dividingbothsidesbythevolumegives

The left side of Eq. 8.19 is a change in charge per unit volume. In otherwords, it’s achangeinthechargedensityρ,andwecanwrite

DividingbyΔt,andtakingthelimitassmallquantitiesapproachzero,

NowEq.8.21isnotreallycorrect;itonlyaccountedforthechargecomingintooroutofthebox through the twowallsorperpendicular to thex axis.Toput it anotherway, thesubscript “total” in this equation is misleading because it only accounts for currentdensitiesinthatonedirection.Togetthewholepicture,wewouldrunthesameanalysisfortheyandzdirectionsand thenaddup thecontributionsfromcurrentdensities inallthreedirections.Theresultis

Therightsideisthedivergenceof ,andinvectornotationEq.8.22becomes

or

Thisequationisalocalneighborhood-by-neighborhoodexpressionoftheconservationofcharge.Itsaysthatthechargewithinanysmallregionwillonlychangeduetotheflowofcharge through thewalls of the region.Eq. 8.23 is so important that it has a name: it’scalledthecontinuityequation.Idon’tlikethisnamebecausethewordcontinuitygivestheimpressionthatthechargedistributionneedstobecontinuous,anditdoesnot.Ipreferto

callitlocalconservation.Itdescribesastrongformofconservationthatdoesnotallowacharge to disappear at one place and reappear at another place unless there’s a flow ofcharge(acurrent)inbetween.

WecouldaddthisnewequationtoMaxwell’sequations,butinfactwedon’tneedto.Eq.8.23isactuallyaconsequenceofMaxwell’sequations.Theproofisnotdifficult;it’safunexercisethatI’llleavetoyou.

Exercise8.4:UsethesecondgroupofMaxwell’sequationsfromTable8.1alongwiththetwovectoridentitiesfromSection8.2.1toderivethecontinuityequation.

8.2.7Maxwell’sEquations:TensorFormBynow,we’veseenallfourofMaxwell’sequationsin3-vectorform(Table8.1).Wehavenotyetderivedthesecondgroupoftheseequationsfromanactionprinciple,butwe’lldothat inLecture 10.Beforewe do, I’d like to show you how towrite the first group ofequations (along with the continuity equation) in tensor notation with upstairs anddownstairs indices. Thiswillmake clear that the equations are invariant under Lorentztransformations. In this section,we’ll switch back to 4-vector notation that follows therulesofindexology.Forexample,I’llstartwriting(Jx,Jy,Jz)or(J1,J2,J3) for thespacecomponentsofcurrentdensity.

Let’sstartwithEq.8.23,thecontinuityequation.Wecanrewriteitas

Nowlet’sdefineρtobethetimecomponentofJ:

ρ=J0.

Inotherwords, take(Jx,Jy,Jz)andput themtogetherwithρ to formacomplexof fournumbers.We’llcallthisnewobjectJμ:

Jμ=(ρ,Jx,Jy,Jz).

Usingthisnotation,wecanrewritethecontinuityequationas

Notice that taking the derivative with respect to Xμ does two things: First, it adds acovariantindex.Second,becausethenewcovariantindexisthesameasthecontravariantindex in Jμ, it triggers the summation convention. Written in this way, the continuityequationhastheappearanceofa4-dimensionalscalarequation.It’snothardtoprovethatthecomponentsofJμ reallydo transformas thecomponentsofa4-vector.Let’sdo thatnow.

To begin with, it’s a well-verified experimental fact that electric charge is Lorentzinvariant.IfwemeasureachargetobeQinoneframe,itwillbeQineveryframe.Nowlet’srevisitthesmallchargeρinsideourlittleunit-volumebox.Supposethisboxanditsenclosedchargearemovingtotherightwithvelocityvinthelaboratoryframe.9Whatare

thefourcomponentsofcurrentdensityintherestframeofthebox?Clearly,theymustbe

(J′)μ=(ρ,0,0,0),

because the charge is standing still in this frame. What are the components Jμ in thelaboratoryframe?Weclaimthey’re

orequivalently,

To see this, remember that charge density is determined by two factors: the amount ofcharge and the size of the region that contains the charge. We already know that theamount of charge is the same in both frames.What about the volume of the enclosingregion,namelythebox?Inthelaboratoryframe,themovingboxundergoesthestandardLorentzcontraction, .Becausetheboxcontractsonlyinthexdirection,itsvolumeisreducedbythesamefactor.Butdensityandvolumeareinverselyrelated,soadecreaseinvolumecausesareciprocal increaseindensity.ThatexplainswhythecomponentsontherightsideofEq.8.25arecorrect.Finally,noticethatthesecomponentsareactuallythecomponents of 4-velocity, which is a 4-vector. In other words, Jμ is the product of aninvariantscalarquantityρwitha4-vector.ThereforeJμitselfmustalsobea4-vector.

8.2.8TheBianchiIdentityLet’s take stockofwhatweknow: and formanantisymmetric tensorFμvwith twoindices.Thecurrentdensity3-vector togetherwithρ forma4-vector, that is, a tensorwithoneindex.WealsohavethefirstgroupofMaxwellequationsfromTable8.1,

These equations are a consequenceof thedefinitionsof and in terms of the vectorpotential.EachofthemhasatheformofpartialderivativesactingoncomponentsofFμv,witharesultofzero.Inotherwords,theyhavetheform

∂F=0.

I’musingthesymbol∂ loosely,tomean“somecombinationofpartialderivatives.”Eqs.8.26and8.27representfourequationsaltogether:onecorrespondingtoEq.8.26becauseit equates two scalars, and three corresponding to Eq. 8.27 because it equates two 3-vectors. How can we write these equations in Lorentz invariant form? The easiestapproachistojustwritedowntheanswerandthenexplainwhyitworks.Hereitis:

∂σFντ+∂νFτσ+∂τFσν=0.(8.28)

Eq.8.28iscalledtheBianchiidentity.10Theindicesσ,ν,andτcantakeonanyofthefourvalues(0,1,2,3)or(t,x,y,z).Nomatterwhichofthesevaluesweassigntoσ,ν,andτ,Eq.8.28givesaresultofzero.

Let’s convince ourselves that the Bianchi identity is equivalent to the twohomogeneousMaxwellequations.First,considerthecasewhereallthreeindicesσ,ν,andτrepresentspacecomponents.Inparticular,supposewechoosethemtobe

σ=y

ν=x

τ=z

Withalittlealgebra(recallthatFμνisantisymmetric),substitutingtheseindexvaluesintoEq.8.28gives

∂xFyz+∂yFzx+∂zFxy=0.

But thepure space components ofF, suchasFyx, correspond to the components of themagneticfield.Recall(Eq.6.41)that

Fyz=BxFzx=ByFxy=Bz.

Substitutingthosevalues,wecanwrite

∂xBx+∂yBy+∂zBZ=0.

Theleftsideisjustthedivergenceof ,sothisequationisequivalenttoEq.8.26.Itturnsoutthatifwechooseadifferentwayofassigningx,y,andztotheGreekindices,wegetthesameresult.Goaheadandtryit.

Whathappensifoneoftheindicesisatimecomponent?Let’strytheassignment

σ=y

ν=x

τ=t

resultingin

∂yFxt+∂xFty+∂tFyx=0.

Twoofthethreetermshavemixedspaceandtimecomponents—onelegintimeandoneleginspace.Thatmeansthey’recomponentsoftheelectricfield.Recall(Eq.6.41)that

Fxt=ExFty=−EyFyx=−Bz.

Thesubstitutionresultsin

∂yEx−∂xEy−∂tBz=0.

Ifweflipthesign(multiplyby−1),thisbecomes

∂xEy−∂yEx+∂tBz=0.

This is just the z component of Eq. 8.27, which says that the curl of plus the timederivativeof iszero.Ifwetriedothercombinationswithonetimecomponentandtwospacecomponents,wewoulddiscoverthexandycomponentsofEq.8.27.

Howmanywaysaretheretoassignvaluestotheindicesσ,ν,andτ?Thereare threeindices,andeachindexcanbeassignedoneofthefourvaluest,x,y,andz.Thatmeansthereare4×4×4=64differentwaystodoit.Howcanitbethatthesixty-fourequationsof theBianchi identity are equivalent to the fourhomogeneousMaxwell equations? It’ssimple;theBianchiidentityincludesmanyequationsthatareredundant.Forexample,anyindex assignmentwhere two indices are equalwill result in the trivial equation 0 = 0.Also,many of the index assignments are redundant; theywill yield the same equation.Every one of the nontrivial equations is equivalent either to Eq. 8.26 or to one of thecomponentsofEq.8.27.

There’sanotherwaytochecktheBianchiidentity.RememberthatFμνisdefinedtobe

Fμν=∂μAν−∂νAμorequivalently,

BymakingtheappropriatesubstitutionsintoEq.8.28,wecanwriteitas

If you expand these derivatives, you’ll discover that they cancel term by term, and theresultiszero.Eq.8.28iscompletelyLorentzinvariant.Youcancheckthatbylookingathowittransforms.

1OntheElectrodynamicsofMovingBodies,A.Einstein,June30,1905.

2Again,theforceonanelectronisoppositetotheelectricfielddirectionbecauseanelectroncarriesanegativecharge.

3TheantisymmetriccomponentFyx=−Fxyisalsononzero,butthisaddsnonewinformation.

4It’snotascalarfromthepointofviewoffour-dimensionalvectorspaces.

5Therearefourmoreifyouwritethemincomponentform.

6Infour-dimensionalspacetime,therefore,ρisnotascalar.Ittransformsasthetimecomponentofa4-vector.

7Passing through has a particular meaning in this context. The volume element exists at one instant of time. Theparticleswereinthepast,andthenthey’reinthefuture.Butattheinstantrepresentedbyourvolumeelement,theyareinthevolumeelement.

8That’sthepreciseanalogofturningthechargedensityrectangleonitsedgeinFig.8.3.

9Asusual,themovingframehasprimedcoordinates,andlaboratoryframeisunprimed.

10It’sactuallyaspecialcaseoftheBianchiidentity.

Lecture9

PhysicalConsequencesofMaxwell’sEquations1

Art:Lenny,isthatFaradaysittingovertherebythewindow?

Lenny:Yes,Ithinkso.Youcantellbecausehistableisnotclutteredwithfancyequations.Faraday’sresults—andthereareplentyofthem—comestraightfromthelaboratory.

Art:Well,Ilikeequations,evenwhenthey’rehard.ButIalsowonderifweneedthemasmuchaswethink.JustlookwhatFaradayaccomplisheswithouttheirhelp!

Justthen,FaradaynoticesMaxwellacrosstheroom.Theyexchangeafriendlywave.

9.1MathematicalInterludeThefundamentaltheoremofcalculusrelatesderivativesandintegrals.Letmeremindyouof what it says. Suppose we have a function F(x) and its derivative dF/dx. Thefundamentaltheoremcanbestatedsimply:

NoticetheformofEq.9.1:Ontheleftsidewehaveanintegralinwhichtheintegrandisaderivative of F(x). The integral is taken over an interval from a to b. The right sideinvolvesthevaluesofFattheboundariesoftheinterval.Eq.9.1isthesimplestcaseofafar more general type of relation, two famous examples being Gauss’s theorem andStokes’s theorem.These theoremsare important in electromagnetic theory, and so Iwillstateandexplainthemhere.Iwillnotprovethem,butyoucaneasilyfindproofsinmanyplaces,includingtheInternet.

9.1.1Gauss’sTheoremInstead of a one-dimensional interval a < x < b, let’s consider a region of three-dimensionalspace.Theinteriorofasphereoracubewouldbeexamples.Buttheregiondoesnothavetoberegularlyshaped.Anyblobofspacewilldo.Let’scalltheregionBforblob.

TheblobhasaboundaryorsurfacethatwewillcallSforsurface.Ateachpointonthesurfacewe can construct a unit vector pointing outward from the blob.All of this isshowninFigure9.1.Bycomparison to theone-dimensionalanalogy (Eq.9.1), theblobwouldreplacetheintervalbetweenaandb,andtheboundarysurfaceSwouldreplacethetwopointsaandb.

Figure9.1:IllustrationofGauss’sTheorem. isanoutwardpointingunitnormalvector.

InsteadofasimplefunctionFanditsderivativedF/dx,wewillconsideravectorfieldanditsdivergence .InanalogywithEq.9.1,Gauss’stheoremassertsarelation

betweentheintegralof overthethree-dimensionalblobofspaceB,andthevaluesofthevectorfieldon the two-dimensionalboundaryof theblob,S. Iwillwrite itand thenexplainit:

Let’sexaminethisformula.Ontheleftsidewehaveavolumeintegralovertheinteriorof

theblob.Theintegrandisthescalarfunctiondefinedbythedivergenceof .

Ontherightsidesisalsoanintegral,takenovertheoutersurfaceoftheblobS.WecanthinkofitasasumoverallthelittlesurfaceelementsthatmakeupS.Eachlittlesurfaceelementhasanoutward-pointingunitvector ,andtheintegrandofthesurfaceintegralisthedot product of with Anotherway to say this is that the integrand is the normal(perpendiculartoS)componentof .

Animportantspecialcaseisavectorfield thatissphericallysymmetric.Thismeanstwo things:First, itmeans that thefieldeverywherepointsalong theradialdirection. Inaddition,sphericalsymmetrymeansthatthemagnitudeof dependsonlyonthedistancefrom the origin and not the angular location. If we define a unit vector that pointsoutwardfromtheoriginateachpointofspace,thenasphericallysymmetricfieldhastheform

whereV(r) is some functionof distance from theorigin.Consider a sphereof radius r,centeredattheorigin:

x2+y2+z2=r2

The term sphere refers to this two-dimensional shell. The volume containedwithin theshell iscalledaball.Nowlet’sapplyGauss’stheorembyintegrating overtheball.TheleftsideofEq.9.2becomesthevolumeintegral

whereBnowmeanstheball.Therightsideisanintegralovertheboundaryoftheball—in other words, over the sphere. Because the field is spherically symmetric, V(r) isconstantonthesphericalboundary,andthecalculationiseasy.Theintegral

becomes

or

ButtheintegralofdSoverthesurfaceofasphereisjusttheareaofthatsphere,4πr2.ThenetresultisthatGauss’stheoremtakestheform

forasphericallysymmetricalfield .

9.1.2Stokes’sTheoremStokes’s theoremalso relates an integral over a region to an integral over its boundary.

This time theregion isnota three-dimensionalvolumebuta two-dimensionalsurfaceSboundedbyacurveC.Imagineaclosedcurveinspaceformedbyathinwire.Thetwo-dimensional surface is like a soap film attached to the wire. Fig. 9.2 depicts such aboundedsurfaceasashadedregionboundedbyacurve.

Itisalsoimportanttogivethesurfaceasenseoforientationbyequippingitwithaunitnormalvector, .Ateverypointon thesurfacewe imagineavector thatdistinguishesonesideofthesurfacefromtheother.

TheleftsideofStokes’stheoremisanintegralovertheshadedsurface.Thisintegralinvolves the curl of ; more precisely, the integrand is the component of in thedirection .Wewritetheintegralas

Let’sexamine this integral.We imaginedividing thesurfaceS into infinitesimal surfaceelementsdS.Ateverypoint,weconstructthecurlof andtakeitsdotproductwith theunitnormalvector atthatpoint.WemultiplytheresultbytheareaelementdSandaddthem all up, thus defining the surface integral . That’s the left side ofStokes’stheorem.

Figure9.2:Stokes’sTheoremandtheRight-HandRule.Thegray

surfaceisnotnecessarilyflat.Itcanballoonouttotheleftorrightlikeasoap

bubbleorarubbermembrane.

The right side involves an integral over the curved boundary line of S.We need todefineasenseoforientationalongthecurve,andthat’swheretheso-calledright-handrulecomesin.Thismathematicalruledoesnotdependonhumanphysiology.Nevertheless,theeasiestwaytoexplainitiswithyourrighthand.Pointyourthumbalongthevector .Yourfingerswillwrap around the bounding curveC in a particular direction. That directiondefinesanorientationforthecurveC.That’scalledtheright-handrule.

Wemay think of the curve as a collection of infinitesimal vectors , each pointingalong the curve in the direction specified by the right-hand rule. Stokes’s theorem then

relatesthelineintegral

aroundthecurveCtothesurfaceintegral

Inotherwords,Stokes’stheoremtellsusthat

9.1.3TheoremWithoutaNameLater we will need a theorem that to my knowledge has no name. There are lots oftheoremsthathavenoname—here’safamousone:

1+1=2.(9.5)

Therearesurelymanymoretheoremswithoutnamesthanwithnames,andwhatfollowsis one such.First,weneed a notation.LetF(x, y, z) be a scalar function of space.ThegradientofFisafield withthreecomponents

Now consider the divergence of . Call it , or more simply, ∇2F. Here it isexpressedinconcreteform,

The symbol ∇2 is called the Laplacian after the French mathematician Pierre-SimonLaplace.TheLaplacianstandsforthesumofsecondderivativeswithrespecttox,y,andz.

Thenewtheoreminvolvesavectorfield . Itusesavectorversionof theLaplacian,described inAppendixB.Begin by constructing the curl, ,which isalso a vectorfield.Assuch,wecanalsotakeitscurl,

Whatkindofquantityisthisdoublecurl?Takingthecurlofanyvectorfieldgivesanothervector field, and so is also a vector field. I will now state the un-namedtheorem:2

Letmesayitinwords:Thecurlofthecurlof equalsthegradientofthedivergenceofminustheLaplacianof .Again,I’llrepeatitwithsomeparentheses:

Thecurlof(thecurlof )equalsthegradientof(thedivergenceof )minustheLaplacianof .

How can we prove Eq. 9.7? In the most boring way possible: write out all the termsexplicitlyandcomparebothsides.I’llleaveitasanexerciseforyoutoworkout.

Lateronin the lecture,wewilluse theun-namedtheorem.Fortunately,wewillonlyneed the special casewhere the divergence of is zero. In that case Eq. 9.7 takes thesimplerform,

9.2LawsofElectrodynamicsArt:Ouch,ouch!Gauss,Stokes,divergence,curl—Lenny,Lenny,myheadisexploding!

Lenny:Oh, that’s great, Art. It’ll give us a good example of Gauss’s theorem. Let’sdescribe the gray matter by a density ρ and a current . As your head explodes, theconservationofgraymatterwillbedescribedbyacontinuityequation,right?Art?Art?Areyouokay?

9.2.1TheConservationofElectricChargeAt the heart of electrodynamics is the law of local conservation of electric charge. InLecture 8, I explained that this means the density and current of charge satisfy thecontinuityequation(Eq.8.23),

Thecontinuityequationisalocalequationthatholdsateverypointofspaceandtime;itsays much more than “the total charge never changes.” It implies that if the chargechanges—let’ssay in the roomwhere I’mgiving this lecture—that itcanonlydosobypassingthroughthewallsoftheroom.Let’sexpandonthispoint.

Take the continuity equation and integrate it over a blob-like region of space, B,boundedbyasurfaceS.Theleftsidebecomes

whereQistheamountofchargeintheregionB.InotherwordstheleftsideistherateofchangeoftheamountofchargeinB.Therightsideis

HereiswherewegettouseGauss’stheorem.Itsays

Nowrecallthat istherateatwhichchargeiscrossingthesurfaceperunittimeperunitarea. When it’s integrated, it gives the total amount of charge crossing the boundarysurfaceperunittime.Theintegratedformofthecontinuityequationbecomes

anditsaysthatthechangeinthechargeinsideB isaccountedforbytheflowofchargethrough the surfaceS. If thecharge insideB is decreasing, that chargemustbe flowingthroughtheboundaryofB.

9.2.2FromMaxwell’sEquationstotheLawsofElectrodynamicsMaxwell’s equations were not derived by Maxwell from principles of relativity, leastaction,andgaugeinvariance.Theywerederivedfromtheresultsofexperiment.Maxwellwasfortunate;hedidn’thavetodotheexperimentshimself.Manypeople,theFoundingFathers, had contributed—Franklin, Coulomb, ∅rsted, Ampèere, Faraday, and others.Basic principles had been codified by Maxwell’s time, especially by Faraday. MostcoursesinelectricityandmagnetismtakethehistoricalpathfromtheFoundingFatherstoMaxwell.Butasinmanycases,thehistoricalrouteistheleastclear,logically.WhatIwilldo in this lecture is go backward, fromMaxwell to Coulomb, Faraday, Ampèere, and∅rsted,andthenfinishoffwithMaxwell’scrowningachievement.

AtthetimeofMaxwellseveralconstantsappearedinthetheoryofelectromagnetism—moreconstantsthanwerestrictlynecessary.Twoofthemwerecalledε0andμ0.Forthemostpart,onlytheproductε0μ0everappearsinequations.Infact,thatproductisnothingbuttheconstant1/c2,thesamec thatappearsinEinstein’sequationE=mc2.Ofcourse,whenthefoundersdidtheirwork,theseconstantswerenotknowntohaveanythingtodowith thespeedof light.Theywere justconstantsdeducedfromexperimentsoncharges,currents, and forces. So let’s forget that c is the speed of light and just take it to be aconstantinMaxwell’sdistillationofthelawslaiddownbyearlierphysicists.HerearetheequationsfromTable8.1,modifiedtoincludetheappropriatefactorsofc.3

9.2.3Coulomb’sLawCoulomb’slawistheelectromagneticanalogofNewton’sgravitationalforcelaw.Itstatesthatbetweenanytwoparticlesthereisaforceproportionaltotheproductoftheirelectricchargesandinverselyproportionaltothesquareofthedistancebetweenthem.Coulomb’slaw is the usual starting point for a course on electrodynamics, but herewe are,manypages intoVolume III, and we’ve hardlymentioned it. That’s because we are going toderiveit,notpostulateit.

Let’s imagine a point chargeQ at the origin, x = y = z = 0. The charge density isconcentratedatapointandwecandescribeitasathree-dimensionaldeltafunction,

WeusedthedeltafunctioninLecture5torepresentapointcharge.AsinLecture5,thesymbol δ3(x) is shorthand for δ(x)δ(y)δ(z). Because of the delta function’s specialproperties,ifweintegrateρoverspacewegetthetotalchargeQ.

Nowlet’slookatthethirdMaxwellequationfromEqs.9.10,

Whathappenstotheleftsideifweintegratethisequationoverasphereofradiusr?Fromthe symmetry of the situationwe expect the field of the point charge to be sphericallysymmetric, and therefore we can use Eq. 9.3. Plugging in , the left side of Eq. 9.3becomes

ButthethirdMaxwellequation, ,tellsuswecanrewritethisas

whichis,ofcourse,thechargeQ.TherightsideofEq.9.3is

4πr2E(r),

whereE(r)istheelectricfieldatdistancerfromthecharge.ThusEq.9.3becomes

Q=4πr2E(r)

Or, to put it anotherway, the radial electric field produced by a point chargeQ at theoriginisgivenby

NowconsiderasecondchargeqatadistancerfromQ.FromtheLorentzforcelaw(Eq.6.1),thechargeqexperiencesaforceq duetothefieldofQ.Themagnitudeoftheforceonqis

This is,ofcourse, theCoulomblawofforcesbetweentwocharges.Wehavederivedit,notassumedit.

9.2.4Faraday’sLawLet’sconsidertheworkdoneonachargedparticlewhenmovingitfrompointatopointbinanelectricfield.Theworkdoneinmovingaparticleaninfinitesimaldistance is

whereF is theforceon theparticle.Tomove theparticle froma tob, the forceFdoeswork

Iftheforceisduetoanelectricfield,thentheworkis

Ingeneral,theworkdependsnotonlyontheendpointsbutalsoonthepathalongwhichtheparticleistransported.Theworkdonecanbenonzeroevenifthepathisaclosedloop

inspace thatstartsandendsatpointa.Wecandescribe this situationbya line integralaroundaclosedloopinspace,

Canweactuallydoworkonaparticlebymovingitaroundaclosedpath—forexample,bymovingitthroughaclosedloopofconductingwire?Undercertaincircumstanceswecan.Theintegral iscalledtheelectromotiveforce(EMF)foracircuitmadefromawireloop.

Let’s explore thisEMFby usingStokes’s theorem (Eq. 9.4).Applied to the electricfield,itsays

ThisallowsustowritetheEMFforaclosedloopofwireintheform

wheretheintegralisoveranysurfacethathastheclosedcurveCasaboundary.

Now comes the pointwhereMaxwell’s equations have something to say.Recall thesecondMaxwellequationfromEqs.9.10:

Ifthemagneticfielddoesnotvarywithtime,thenthecurloftheelectricfieldiszero,andfromEq.9.17theEMFforaclosedpathisalsozero.Insituationswherethefieldsdonotvarywithtime,noworkisdonewhencarryingachargearoundaclosedpath.Thisisoftentheframeworkforelementarydiscussions.

Butmagnetic fields sometimes do varywith time; for example, justwave amagnetaroundinspace.ApplyingthesecondMaxwellequationtoEq.9.17,

or

Thus, theEMF is thenegativeof the time rateofchangeofacertainquantity .Thisquantity, the integralof themagnetic fieldover the surfaceboundedby theclosedwire,iscalledthemagneticfluxthroughthecircuitandisdenotedΦ.OurequationfortheEMFcanbesuccinctlywrittenas

TheEMFrepresentsaforcethatpushesachargedparticlearoundtheclosedloopofwire.Ifthewireisanelectricconductor,itwillcauseacurrenttoflowinthewire.

Figure9.3:Faraday’sLaw.

The remarkable fact that an EMF in a circuit can be generated by varying the fluxthroughthecircuitwasdiscoveredbyMichaelFaradayandiscalledFaraday’slaw.InFig.9.3,aclosedloopofwireisshownnexttoabarmagnet.Bymovingthemagnetclosertoandfartherfromtheloop,thefluxthroughtheloopcanbevariedandanEMFgeneratedintheloop.TheelectricfieldproducingtheEMFdrivesacurrentthroughthewire.Pullthemagnetawayfromthewireloopandacurrentflowsoneway.Pushthemagnetclosertotheloopandthecurrentflowstheotherway.That’showFaradaydiscoveredtheeffect.

9.2.5Ampèere’sLawHow about the other Maxwell equations? Can we recognize any other basicelectromagnetic laws by integrating them? Let’s try the fourthMaxwell equation fromEqs.9.10,

Fornow,we’llassumenothingvarieswithtime;thecurrent issteadyandallthefieldsarestatic.Inthatcase,theequationtakesthesimplerform

Onethingweseefromthisequationisthatthemagneticfieldwillbeverysmallunlessahugecurrentflowsthroughthewire.Inordinarylaboratoryunitsthefactor1/c2isaverysmall number. That fact played an important role in the interlude on crazy units thatfollowedLecture5.

Let’sconsideracurrentflowinginaverylong,thinwire—solongthatwemaythinkofitasinfinite.Wecanorientouraxessothatthewireliesonthexaxisandthecurrentflows to theright.Since thecurrent isconfined to the thinwire,wemayexpress itasadeltafunctionintheyandzdirections:

Nowimagineacircleofradiusrsurroundingthewire,asshowninFig.9.4.Thistimethecircle is an imaginary mathematical circle, not a physical wire. We are going to useStokes’s theoremagain, inaway that shouldbeclear from the figure.Whatwe’lldo is

integrate Eq. 9.19 over the disclike shaded region in Fig. 9.4. The left side gives theintegralof ,whichbyStokes’stheoremisthelineintegralofBoverthecircle.Therightsidejustgives1/c2timesthenumericalvalueofthecurrentthroughthewire,j.

Figure9.4:Ampère’sLaw.

Thus,

Itfollowsthatacurrentthroughawireproducesamagneticfieldthatcirculatesaroundthewire.BythatImeanthatthefieldpointsalongtheangulardirection,notalongthexaxisorinthedirectionawayfromthewire.

Because isparallel to ateverypointalong the loop, the integralof themagneticfieldisjustthemagnitudeofthefieldatdistancertimesthecircumferenceofthecircle,giving

SolvingforB(r),wefindthatthemagneticfieldproducedbyacurrent-carryingwireatadistancerfromthewirehasthevalue

Onthefaceofit,onemightthinkthefactor1/c2wouldbesooverwhelminglysmallthatnomagneticfieldwouldeverbedetectablefromanordinarycurrent.Indeed,intheusualmetricsystem,

WhatworkstocompensateforthistinynumberisthehugenumberofelectronsthatmovethroughthewirewhenanEMFiscreated.

Equation9.22 iscalled∅rsted’s lawafter theDanishphysicistHansChristian∅rsted.∅rsted noted that an electric current in awirewould cause themagnetized needle of acompass to align along a direction perpendicular to the wire and along the angulardirectionaroundthewire.WithinashorttimetheFrenchphysicistAndré-MarieAmpèeregeneralized ∅rsted’s law to more general flows of current. The ∅rsted-Ampèere lawstogetherwith Faraday’s lawwere the basis forMaxwell’s investigations that led to hisequations.

9.2.6Maxwell’sLaw

ByMaxwell’slawImeanthefact(discoveredbyJamesClerkMaxwellin1862)thatlightis composed of electromagnetic waves—wavelike undulations of electric andmagneticfields.Thediscoverywas amathematical one, not basedon a laboratory experiment. Itconsisted of showing that Maxwell’s equations have wavelike solutions that propagatewithacertainspeed,whichMaxwellcalculated.Bythattime,thespeedoflighthadbeenknown for almost two hundred years, and Maxwell observed that the speed of hiselectromagnetic waves agreed. What I will do in the rest of this lecture is show thatMaxwell’sequationsimplythatelectricandmagneticfieldssatisfythewaveequation.

WearegoingtoconsiderMaxwell’sequationsinregionsofspacewheretherearenosources—inotherwords,nocurrentsorchargedensities:

Ourgoal is to show that the electric andmagnetic fields satisfy the samekindofwaveequationthatwestudiedinLecture4.Let’sgrabthesimplifiedwaveequation(Eq.4.26)andtransposeallthespacederivativestotherightside:

TheLaplacianprovidesaconvenientshorthandfortheentirerightside;wecanrewritethewaveequationas

AsIexplainedinLecture4,thisequationdescribeswavesthatmovewiththevelocityc.

Now let’s put ourselves in the shoes ofMaxwell.Maxwell had his equations (Eqs.9.23),but thoseequationswerenotatallsimilar toEq.9.24.Infact,hehadnoobviousreasontoconnecttheconstantcwithanyvelocity,letalonethevelocityoflight.Isuspectthequestionheaskedwasthis:

Ihavesomecoupledequationsfor and .Istheresomewaytosimplifythingsbyeliminatingoneofthem(either or )toobtainequationsjustfortheother?

OfcourseIwasn’tthere,butit’seasytoimagineMaxwellaskingthisquestion.Let’sseeifwecanhelphim.TakethelastequationfromEqs.9.23anddifferentiatebothsideswithrespecttotime.Weget

Nowlet’susethesecondMaxwellequationtoreplace with ,whichindeedgives

anequationfortheelectricfield,

Thisisbeginningtolookmorelikeawaveequation,butwearenotquitethere.Thelaststep is touse theunnamed theorem(Eq.9.7).Fortunately, allweneed is the simplifiedform, Eq. 9.8. The reason is that the secondMaxwell equation, , is exactly theconditionthatallowsthesimplification.Theresultisjustwhatwewant:

Eachcomponentoftheelectricfieldsatisfiesthewaveequation.

Lenny: Iwasn’t thereArt,but I canwell imagineMaxwell’s excitement. I canhearhimsayingtohimself:

What are these waves? From the form of the equation, the wave velocity is that funnyconstantc thatappearsall overmy equations. If I’mnotmistaken, inmetric units it isabout3×108.YES!3×108meterspersecond!Thespeedoflight!

AndthusitcametopassthatJamesClerkMaxwelldiscoveredtheelectromagneticnatureoflight.

1Somehow,thisimportantlecturenevermadeittothevideositeforthecourse.ItrepresentsthemainpointofcontactbetweenLenny’sapproach to the subject and the traditionalapproach.Asa resultof this insertion,Lecture10 in thebookcorrespondstoLecture9in thevideoseries,andLecture11in thebookcorrespondstoLecture10in thevideoseries.

2Oneofourreviewerssuggestedcallingitthe“double-cross”theorem.

3IntheSIunitsadoptedbymanytextbooks,you’llseeρ/ 0and insteadofandρand .

Lecture10

MaxwellFromLagrange

Art:I’mworriedaboutMaxwell.He’sbeentalkingtohimselfforalongtime.

Lenny:Don’tworry,Art,itlookslikehe’sclosetoasolution.

[Lennygesturestowardthefrontdoor]Besides,helphasjustarrived.

TwogentlemenwalkintotheHideaway.Oneofthemiselderlyandnearlyblindbuthasnotroublefindinghisway.TheytakeseatsoneithersideofMaxwell.Maxwellrecognizesthemimmediately.

Maxwell:Euler!Lagrange!Iwashopingyou’dshowup!Yourtimingisperfect!

Inthislecture,we’lldotwothings.ThefirstistofollowuponLecture9andworkoutthedetails of electromagnetic plane waves. Then, in the second part of the lecture, we’llintroduce the actionprinciple for the electromagnetic field andderive the twoMaxwellequationsthatarenotidentities.Tohelpkeeptrackofwherewe’vebeenandwherewe’regoing,I’vesummarizedthefullsetofMaxwellequationsinTable10.1.

Table10.1:Maxwell equations in3-vector and4-vector form.The topgroup isderivedfromidentities;thebottomgroupisderivedfromtheactionprinciple.

10.1ElectromagneticWavesInLectures4and5wediscussedwavesandwaveequations,andinLecturewesawhowMaxwell derived the wave equation for the components of the electric and magneticfields.

Let’sbeginwiththeMaxwellequations(Eqs.9.23)intheabsenceofsources.I’llwritethemhereforconvenience:

Nowlet’sconsiderawavemovingalongthezaxiswithwavelength

wherekistheso-calledwavenumber.Wecanchooseanywavelengthwelike.Agenericplanewavehasthefunctionalform

FieldValue=Csin(kz−ωt),

whereCisanyconstant.Whenappliedtotheelectricfield,thecomponentshavetheform

where , and arenumericalconstants.Wecanthinkofthemasthecomponentsofafixedvectorthatdefinesthepolarizationdirectionofthewave.

We’veassumedthat thewavepropagatesalongthezaxis.Obviouslythis isnotverygeneral;thewavecouldpropagateinanydirection.Butwearealwaysfreetorealignouraxessothatthemotionofthewaveisalongz.

Another freedomwe have is to add a cosine dependence; but this simply shifts thewavealongthezaxis.Byshiftingtheoriginwecangetridofthecosine.

Now let’s use the equation . Because z is the only space coordinate that theelectricfieldcomponentsdependon,thisequationtakesanespeciallysimpleform,

IfwecombinethiswithEq.10.3,wefindthat

Inotherwords,thecomponentoftheelectricfieldalongthedirectionofpropagationmustbezero.Waveswiththispropertyarecalledtransversewaves.

Lastly,wecanalwaysalignthexandyaxessothatthepolarizationvector liesalongthexaxis.Thustheelectricfieldhastheform,

Next, let’s consider the magnetic field. We might try allowing the magnetic field topropagate in a different direction from the electric field, but that would violate theMaxwellequationsthatinvolveboththeelectricandmagneticfields.Themagneticfieldmustalsopropagatealongthezaxisandhavetheform

Bythesameargumentweusedfortheelectricfield,theequation impliesthatthezcomponentofthemagneticfieldiszero.Thereforethemagneticfieldmustalsolieinthex,yplane,butnotnecessarilyinthesamedirectionastheelectricfield.Infact,itmustbeperpendicular to the electric field and therefore must lie along the y axis. To see thispropertyweusetheMaxwellequation

Incomponentform,thisbecomes

Now,keepinginmindthatEzandEyarezero,andthatthefieldsonlyvarywithrespecttoz,weseethatonlytheycomponentofthemagneticfieldcanvarywithtime.Giventhatwe are discussing oscillatingwaves, it follows that only the y component of can benonzero.

Onemore fact follows fromEqs.10.9. Ifweplug in the forms and,wewillfindthat isconstrainedtobe

ThereisstillonemoreMaxwellequationwehaven’tused,namely

IfwetakethexcomponentofthislastMaxwellequationandusewhatwehavealreadylearned,wefindthatitreducesto

PluggingintheformsofExandBy,outcomesasimplerelationbetweenthefrequencyωandthewavenumberk:

ω=ck.

Thewaveformsin(kz−ωt)becomes

This is exactly the form of a wave propagating along the z axis with velocity c. Let’ssummarizethepropertiesofelectromagneticplanewaves:

•Theypropagatealongasingleaxisatthespeedoflight.

•Electromagneticwavesaretransverse,meaningthatthefieldslieintheplaneperpendiculartothepropagationaxis.

•Theelectricandmagneticfieldsareperpendiculartooneanother.

•Theratiooftheelectricandmagneticfieldsis

Inrelativisticunits(wherec=1),themagnitudesoftheelectricandmagneticfieldsareequal.

I’ll remind you of one property of light waves that you’re probably familiar with,

especially if you buy a better quality of sunglasses. Light is polarized. In fact, allelectromagneticwavesarepolarized.Thedirectionofpolarizationis thedirectionoftheelectricfield,soinourexamplewewouldsaythatthewaveispolarizedalongthexaxis.Thepropertiesofplaneelectromagneticwavescanbevisualizedinasinglepicture,showninFig.10.1.

10.2LagrangianFormulationofElectrodynamicsI’ve made this point so many times that I risk being repetitive. Once again: Thefundamental ideasofenergyconservation,momentumconservation,andtherelationshipbetweenconservationlawsandsymmetrylawsfollowonlyifyoustartwiththeprincipleof least action. You can write all the differential equations you like, and they may bemathematicallyconsistent.Allthesame,therewillbenoenergyconservation—therewillbenoenergytobeconserved—unlessthoseequationsarederivedfromaLagrangianandan action principle. Since we’re inclined to deeply believe in energy conservation, weshouldlookforaLagrangianformulationofMaxwell’sequations.

Let’s go through the fundamental principles and make our best guess about theLagrangian.Rememberthatthetwoequations

Figure10.1:SnapshotofanElectromagneticPlaneWavePropagatingtotheRightandOutofthePage(positivezdirection).The and fieldsareperpendiculartoeachother,andtothe

directionofpropagation.The fieldisshaded.

aremathematical identities that follow from the definitions of and (in termsof thevectorpotential).Thereisnomoreneedtoderivethemfromanactionprinciplethanthereisneedtoderive1+1=2.

Ourgoalnowistoderivethesecondset.In3-vectornotation,theseequationsare

We’veidentifiedthechargedensityρasthetimecomponentofthecurrent4-vectorJν,

ρ=J0.

Likewise,thethreecomponentsof correspondtothethreespacecomponentsofJv,

or

Alltheserelationshipsarecapturedincovariantformbythesingleequation

The time component of this equation is equivalent to Eq. 10.12, and the three spacecomponents conspire to give us Eq. 10.13. How do we derive these equations from aLagrangian?Beforewecaneventry,wemustguesswhatthecorrectLagrangianis.Thefundamentalprincipleswillhelpnarrowthesearch.

We’vealreadydiscussed theneedforanactionprinciple.Wehavealsoseen that themathematicsoftheactionprincipleplaysoutslightlydifferentlyforfieldsthanitdoesforparticles.HerewefocusonfieldsbecausetheMaxwellequationsarefieldequations.Let’sseewhattheotherprinciplescanteachus.

10.2.1LocalityWhatever happens at a particular time and position can only be related to things thathappenatneighboringtimesandpositions.Howdoweguaranteethat?WemakesuretheLagrangiandensityinsidetheactionintegraldependsonlyonthefieldsthemselvesandontheirfirstderivativeswithrespecttothecoordinatesXμ.

Ingeneral,theLagrangiandensitydependsoneachfieldinthetheory;therecouldbeseveralofthem,butI’lljustusethesinglevariableϕtorepresentallthefieldsthatexist.1TheLagrangianalsodependsonthederivativesofϕwithrespecttospaceandtime.We’realready accustomed to the notation ∂μϕ.We’ll be using that, along with an evenmorecondensednotation,ϕ,μ,whichmeansthesamething:

This notation is standard. In the symbolϕ,μ the commameansderivative, andμmeanswithrespecttoXμ.ThelocalityrequirementsaysthattheLagrangiandensitydependsonlyonϕ,andϕ,μ.Inotherwords,theactionintegralmusthavetheform

Thesymbolϕ,μdoesn’tmeanoneparticularderivative.It’sagenericsymbolthatreferstoderivativeswithrespecttoallcomponents,timeandspace.Byrequiringtheactionintegral

totakethisform,weguaranteethattheresultingequationsofmotionwillbedifferentialequationsthatrelatethingslocally.

10.2.2LorentzInvarianceLorentzinvarianceisasimplerequirement:TheLagrangiandensityneedstobeascalar.Itmusthavethesamevalueineveryreferenceframe.Wecouldjustleaveitatthatandmoveontothenextprinciple.Instead,I’dliketoprovidesomecontextbyquicklyreviewingourpreviousresultsforscalarfields.TheMaxwellequationsarebasedonvectorfields,notonscalarfields,andit’sgoodtoseehowtheycompare.

Forascalarfieldϕ,theLagrangianmaycontainϕitselforanyfunctionofϕ.Let’scallthisfunctionU(ϕ).TheLagrangiancanalsocontainderivatives.2Ontheotherhand,thereare some things it cannot contain. For example, it can’t contain∂xϕ all by itself. Thatwouldbenonsensebecause∂xϕisnotascalar.It’sthexcomponentofavector.Whilewecan’t just throwinarbitrarycomponentsofavector,we’reallowedtocarefullypackagethemuptoformascalar.Thequantity

∂μϕ∂μϕ=ϕ,μϕ,μ

is a perfectly good scalar and could certainly appear in a Lagrangian. Because μ is asummationindex,wecanexpandthisas

∂μϕ∂μϕ=−(∂tϕ)2+(∂xϕ)2+(∂yϕ)2+(∂zϕ)2,

orin“comma”notation,

ϕ,μϕ,μ=−(∂tϕ)2+(∂xϕ)2+(∂yϕ)2+(∂zϕ)2.

InLecture 4,we used terms like the preceding ones towrite a simpleLagrangian (Eq.4.7).Inournewernotation,wecanwritethatLagrangianas

or

BasedonthisLagrangian,weusedtheEuler-Lagrangeequationstoderivetheequationsof motion. I’ve written only one field in this example, but there could be several. Ingeneral,wemightadduptheLagrangiansforseveralfields.Wemightevencombinetheminmorecomplicatedways.

Foreachindependentfield—inourexamplethere’sonlyone—webeginbytakingthepartialderivativeoftheLagrangianwithrespecttoϕ,μ,whichis

ThenwedifferentiatethattermwithrespecttoXμtoget

That’stheleftsideoftheEuler-Lagrangeequation.Therightsideisjustthederivativeofwithrespecttoϕ.Thecompleteequationis

ThisistheEuler-Lagrangeequationforfields.It’sthedirectcounterparttotheLagrangeequationsforparticlemotion.Asyourecall,theequationforparticlemotionis

Toderivetheequationofmotion,weevaluatetheLagrangian(Eq.10.15)usingtheEuler-Lagrangeequation(Eq.10.16).WedidthisinSection4.3.3andfoundthattheequationofmotionis

whichisasimplewaveequation.

Startingwiththevectorpotential,wewillfollowexactlythesameprocesstoarriveatMaxwell’sequations.It’smorecomplicatedbecausetherearemoreindicestokeeptrackof.Intheend,Ithinkyou’llagreethatit’sabeautifulpieceofwork.Beforewetakethatplunge, we have one more principle to review: gauge invariance. This is always arequirementinelectrodynamics.

10.2.3GaugeInvarianceTomakesure theLagrangian isgauge invariant,weconstruct it fromquantities thatarethemselvesgauge invariant.Theobviouschoicesare thecomponentsofFμν, theelectricandmagneticfields.Thesefieldsdon’tchangewhenyouaddthederivativeofascalartoAμ.Inotherwords,theydon’tchangeifyoumakethereplacement

Therefore, any Lagrangian we construct from the components of F will be gaugeinvariant.

Aswesawinthecaseofascalarfield,therearealsosomequantitieswecannotuse.Forexample,wecan’tincludeAμAμ.Thismayseemcounterintuitive.Afterall,AμAμisaperfectlygoodLorentzinvariantscalar.3However,it’snotgaugeinvariant.IfyouaddthegradientofascalartoA,thenAμAμwouldindeedchange,soit’snogood.Itcan’tbepartoftheLagrangian.

10.2.4TheLagrangianintheAbsenceofSourcesWe’llbuildupourLagrangianintwostages.First,we’llconsiderthecasewherethere’san electromagnetic field but no charges or currents. In otherwords, the casewhere thecurrent4-vectoriszero:

Jμ=0.

Lateron,we’llmodifytheLagrangiantocovernonzerocurrentvectorsaswell.

We can use the componentsFμν in anywaywe like without worrying about gaugeinvariance.However,Lorentzinvariancerequiresmorecare;wemustsomehowcombinethemtoformascalarbycontractingtheindices.

ConsidersomegeneraltensorTμνwithtwoindices.It’ssimpletoconstructascalarbyraisingoneindex,thencontracting.Inotherwords,wecouldformtheexpression

with μ as a summation index. That’s the general technique for making scalars out oftensors.WhathappensifwetrythatwithFμν?It’snothardtoseethattheresultis

becauseall thediagonalcomponents (componentswithequal indices)arezero. Inotherwords,

F00=0

F11=0

F22=0

F33=0.

The expression tells us to add these four terms together, and of course the result iszero.ThisisnotagoodchoicefortheLagrangian.Infact,itappearsthatanytermlinearinFμνwouldnotbeagoodchoice.Iflineartermsarenogood,wecantrysomethingthat’snonlinear.Thesimplestnonlineartermwouldbethequadraticterm

FμνFμν.

Thisexpressionisnontrivial.Itiscertainlynot identicallyequaltozero.Let’sfigureoutwhatitmeans.First,considerthemixedspace-and-timecomponents

F0nF0n.

These are the componentswhereμ is zero and therefore represents time. Latin indexnrepresents the space components of ν.What does the whole expression represent? It’snearly identical to the square of the electric field. As we’ve seen before, the mixedcomponentsofFμνaretheelectricfieldcomponents.ThereasonF0nF0nisnotexactlythesquareoftheelectricfieldisthatraisingonetimecomponentproducesachangeofsign.Becauseeachtermcontainsoneuppertimeindexandonelowertimeindex,theresultisminusthesquareof ,

F0nF0n=−E2.

Amoment’sthoughtshowsthatthistermappearstwiceinthesummationbecausewecaninterchangetherolesofμandν.Inotherwords,wemustalsoconsidertermsoftheform

Fn0Fn0=−E2.

Thesetermsalsoaddupto−E2becauseFμνisantisymmetric.Inthissecondform,thefirstindex represents the space components, while the second index represents time. Theoverallresultofallthissquaringandaddingis−2E2.Inotherwords,

Youmight think that the antisymmetry ofFμ ν would cause these two sets of terms tocancel.However,theydon’tbecauseeachcomponentissquared.

Nowweneedtoaccountforthespace-spacecomponentsofFμν.Thesearethetermswhereneitherindexiszero.Forexample,onesuchtermis

F12F12.

Raisingandloweringthespaceindicesdoesnothing;there’snochangeofsign.Intermsofelectricandmagneticfields,F12isthesameasB3,akaBz,thezcomponentofmagneticfield. Therefore, F12F12 is the same as (Bz)2. When we consider its antisymmetriccounterpartF21F21,wefindthattheterm(Bz)2entersthesumtwice.

WehavenowaccountedforallthetermsinthesumFμνFμνexceptthediagonals,wherethetwoindicesareequal toeachother.Butwealreadyknowthat thosecomponentsarezero, and so we’re done. Combining the space-space terms with the space-time termsresultsin

FμνFμν=−2E2+2B2,

or

By convention, this equation is written in a slightly different form. When I say byconvention,Imeanitwouldaffectnothingifweignoredtheconvention;theequationsofmotionwouldbethesame.OneconventionisthattheE2termispositiveandtheB2termisnegative.Thesecondconvention incorporatesa factorofonequarter.Asa result, theLagrangianisusuallywritten

or

Thefactorof hasnophysicalcontent.Itsonlypurposeis tomaintainconsistencywithlong-standinghabits.

10.3DerivingMaxwell’sEquationsEq. 10.18 is Lorentz invariant, local, and gauge invariant. It’s not only the simplestLagrangian we can write down, it’s also the correct one for electrodynamics. In thissection,we’ll see how it gives rise to theMaxwell equations. The derivation is a little

trickybutnot ashardasyoumight think.We’llproceed in small simple steps.To startwith,we’llignoretheJμtermsandworkthingsoutforemptyspace.Thenwe’llbringJμbackintothepicture.

Onceagain,herearetheEuler-Lagrangeequationsforfields:

Foreachfield,wewriteaseparateequationofthiskind.Whatarethesefields?Theyarethevectorpotentialcomponents

A0,A1,A2,A3or

At,Ax,Ay,Az.

These are four distinct and independent fields. What about their derivatives? To helpthingsalong,we’llextendourcommanotationtoinclude4-vectors.4Inthisnotation,thesymbolAμ,ν,or“Asubμcommaν,”standsforthederivativeofAμwithrespecttoXν.Inotherwords,wedefineAμ,νas

Thecommainthesubscriptoftheleftsideisessential.Ittellsyoutotaketheappropriatederivative.WithoutitthesymbolAμνwouldnotbeaderivativeatall. Itwouldjustbeacomponentofsometwo-indextensor.Whatdoesthefieldtensorlooklikeinthisnotation?Asweknow,thefieldtensorisdefinedas

It’seasytotranslatethisintocommanotation.It’sjust

Whyputsomucheffortintocondensingthenotationforpartialderivatives?Ithinkyou’llsee the value of this technique in the next few paragraphs. Of course it saves a lot ofwriting,butitdoesmuchmore;itmakesthesymmetriesinourequationspoprightoutofthepage.Evenmoreimportantly,itexposesthesimplicityofworkingthroughtheEuler-Lagrangeequations.

We’lltakethefourcomponentsofAμtobeseparateindependentfields.There’safieldequation for each one of them.To translate the Lagrangian (Eq. 10.17) into condensednotation,wesimplysubstituteforFμνfromEq.10.20:

That’s theLagrangian. Ifyou like,youcanwrite itout indetail in termsofderivatives.ThecomponentsAμandAν are the things that correspond toϕ inEq.10.19.There’s anEuler-LagrangeequationforeachcomponentofA.Tobeginwith,we’llchooseasinglecomponent,Ax,toworkon.Thefirstthingtodoiscomputethepartialderivativeof with

respecttoAx,μ,thatis,

Let’sbreakthiscalculationintosmallsteps.

Weneedtocalculatethederivativeof withrespecttospecificderivativesofspecificcomponentsofA.Toseewhatthatmeans,consideracase,thederivativeof withrespecttoAx,y.Eq.10.21tellsusthat isasummationoverthefourvaluesofμandν.WecouldwriteoutallsixteentermsofthisexpansionandthenlookfortermsthatcontainAx,y. Ifwedidthat,wewouldfindthatAx,yappearsinonlytwotermsthathavetheform

(Ax,y−Ay,x)(Ax,y−Ay,x).

Wetemporarilyignoredthefactorof .Becausebothxandyarespaceindices,loweringanyoftheindicesmakesnodifference,andwecanrewritethistermmoresimplyas

(Ax,y−Ay,x)(Ax,y−Ay,x).

Simplifyingfurtherandrestoringthenumericalfactor,thisbecomes

WhydidIwrite insteadof ?Becausetherearetwotermsofthisformintheexpansion;onethatoccurswhenμ=xandν=y,andanotherwhenμ=yandν=x.

Atthispoint,thereshouldbenomysteryabouthowtodifferentiate withrespecttoAx, y. Because none of the other terms contain Ax, y, they all go to zero when wedifferentiateandwecanignorethem.Inotherwords,wecannowwrite

This derivative is straightforward as long as we remember thatAx, y andAy, x are twodifferentobjects.We’reonlyinterestedinthedependenceonAx,y,andthereforeweregardAy,xasaconstantforthispartialderivative.Theresultis

andwecannowrecognizetherightsideasanelementofthefieldtensor,namely−Fyx.Theresultofallthisworkis

or,usingtheantisymmetryofF,

Thefarrightsideofthisequation,Fxy,comesforfreebecauseupperindicesareequivalenttolowerindicesforspacecomponents.

Ittookalongtimetoreachthispoint,buttheresultisquitesimple.Ifyougothrough

thesameexerciseforeveryothercomponent,you’lldiscoverthateachofthemfollowsthesamepattern.Thegeneralformulais

The next step, according to Eq. 10.19, is to differentiate Eq. 10.22with respect toXν.WhenwedothistobothsidesofEq.10.22,weget

or

Butwait a minute! The right side of Eq. 10.23 is nothing but the left side of the lastMaxwellequationfromTable10.1.

Coulditreallybethateasy?Let’swithholdjudgmentforamoment,untilweworkouttherightsideoftheEuler-Lagrangeequation.Thatmeanstakingthederivativeof withrespecttothefieldsthemselves.Inotherwords,takingthederivativeof withrespecttotheundifferentiatedcomponentsofA,suchasAx.ButtheundifferentiatedcomponentsofAdonotevenappearin ,andthereforetheresultiszero.That’sallthereistoit.TherightsideoftheEuler-Lagrangeequationiszero,andtheequationofmotionforemptyspace(nochargesorcurrents)matchestheMaxwellequationperfectly:

10.4LagrangianwithNonzeroCurrentDensityHowcanwemodify theLagrangianto includeJμ, thecurrentdensity?5Weneed toaddsomething to that includes all the components of Jμ. The current density Jμ has fourcomponents: The time component is ρ, and the space components are the threecomponentsof .Themthspacecomponentisthechargeperunitareaperunittimethatpassesthroughalittlewindoworientedalongthemaxis.Insymbols,wecanwrite

Jμ=ρ,jm.

Ifweconsideralittleboxlikecellinspace,thechargeinsidethecellisρtimesthevolumeofthecell.Inotherwords,it’sρtimesdxdydz.Therateofchangeofchargeinsidethecellisjustthetimederivativeofthatquantity.Theprincipleoflocalchargeconservationsaysthattheonlywaythischargecanchangeisforchargestopassthroughthewallsofthecell.Thatprinciplegaveusthecontinuityequation,

whichweworkedoutinLecture8.Thesymbol (thedivergenceof )isdefinedas

The first termon the right side (partialof jxwith respect tox) represents thedifferencebetweentheratesatwhichchargeflowsthroughthetwox-orientedwindowsofthecell.Theothertwotermscorrespondtotheratesofflowthroughtheyandz-orientedwindows.The sum of these three terms is the overall rate at which charge flows through all theboundariesofthebox.Inrelativisticnotation,thecontinuityequation(Eq.10.24)becomes

But how does this help us derive Maxwell’s equations? Here’s how: Given thecontinuityequation,wecanconstructagaugeinvariantscalarthatinvolvesbothJμandAμ.Thenewscalar isJμAμ,whichmaybe the simplestpossibleway tocombine these twoquantities. It doesn’t look gauge invariant, butwe’ll see that it is.We consider each ofthesequantitiestobeafunctionofposition.We’llusethesinglevariablextostandforallthreespacecomponents.

Now let’s think about the impact this new scalar would have if we add it to theLagrangian.Theactionwouldcontaintheadditionalterm

Theminussignisaconvention,ultimatelyduetoBenjaminFranklin.Jμ(x)Aμ(x)isascalarbecauseit’sthecontractionofa4-vectorwithanother4-vector.Itinvolvesboththecurrentdensityandthevectorpotential.Howcanwetellifit’sgaugeinvariant?Simple:Justdoagauge transformationandseewhathappens to theaction.Doingagauge transformationmeansaddingthegradientofsomescalartoAμ.Thegauge-transformedactionintegralis

Whatwereallycareabouthereisthechangeinactionduetotheextraterm.Thatchangeis

Thisdoesn’t lookmuchlikezero.If it’snotzero, thentheactionisnotgaugeinvariant.Butitiszero.Let’sseewhy.

Ithelpstorememberthatd4xisshorthandfordtdxdydz.Let’sexpandthesummationoverμ.I’llstopwritingtheleftsideatthispointbecausewedon’tneedit.Theexpandedintegralis

We’re going tomake one key assumption: that if you go far enough away, there is nocurrent.All currents in the problemare containedwithin a big laboratory so that everycomponentofJgoestozeroatlargedistances.Ifwecomeacrossasituationwherethereisanonzerocurrentatinfinity,wehavetotreatitasaspecialcase.Butforanyordinaryexperiment,wecanimaginethatthelaboratoryisisolatedandsealed,andthatnocurrentsexistoutsideit.

Let’slookatoneparticulartermintheexpandedintegral,theterm

whichwe’llnowwriteas

If you’ve read thepreviousbooks in this series, you alreadyknowwhere this is going.We’re about to use an important technique called integration by parts. Because we’reconsideringonlythexcomponentfornow,wecantreatthistermasanintegraloverxandignore thedydzdtportionofd4x.To integratebyparts,weswitch thederivative to theotherfactorandchangetheoverallsign.6Inotherwords,wecanrewritetheintegralas

WhathappensifwedothesamethingwiththenextterminEq.10.26?Thenexttermis

BecausethistermhasthesamemathematicalformastheJ1term,wegetasimilarresult.Infact,allfourtermsinEq.10.26followthispattern,andwecancapturethatideanicelyusingthesummationconvention.WecanreformulateEq.10.26as

or

Does thesum∂μJμ in this integral look familiar? It should. It’s just the left sideof thecontinuityequation(Eq.10.25).Ifthecontinuityequationiscorrectthenthisterm,andinfacttheentireintegral,mustbezero.Ifthecurrentsatisfiesthecontinuityequation—andonlyifitsatisfiesthecontinuityequation—thenaddingthepeculiar-lookingterm

totheLagrangianisgaugeinvariant.

How does this new term affect the equations of motion? Let’s quickly review ourderivationforemptyspace.WestartedwiththeEuler-Lagrangeequation,

WethenfilledinthedetailsofthisequationforeachcomponentofthefieldA.Fortheleftsidetheresultwas

The right side, of course,was zero. This result is based on the originalLagrangian foremptyspace,Eq.10.17.

Thenewterm(Eq.10.27)involvesAitselfbutdoesnotinvolveanyderivativesofA.Therefore,ithasnoimpactontheleftsideoftheEuler-Lagrangeequation.Whataboutthe

right side? When we differentiate Eq. 10.27 with respect to Aμ, we just get Jμ. Theequationofmotionbecomes

These are theMaxwell equationswe’vebeen looking for.They are, of course, the fourequationsthatconstitutethesecondsetofMaxwellequations,

To summarize: We’ve found that Maxwell’s equations do follow from an action orLagrangianformulation.Whatismore,theLagrangianisgaugeinvariant,butonlyifthecurrent4-vectorsatisfiesthecontinuityequation.Whatwouldhappenifthecurrentfailedtosatisfycontinuity?Theansweristhattheequationswouldsimplynotbeconsistent.WecanseetheinconsistencyfromEq.10.28bydifferentiatingbothsides,

Therightsideisjusttheexpressionthatappearsinthecontinuityequation.TheleftsideisautomaticallyzerobecauseFμνisantisymmetric.Ifthecontinuityequationisnotsatisfied,theequationsarecontradictory.

1Forelectromagnetism,thefieldsturnouttobecomponentsofthevectorpotential,butlet’snotgetaheadofourselves.

2It’snotgoingtoofartosaythatitmustcontainthem.Withoutderivatives,theLagrangianwouldbeuninteresting.

3Remember that theμ’s in this expression do not signify derivatives. They just represent the components ofA. Toindicatedifferentiation,wewoulduseacomma.

4Untilnowwe’veusedthecommanotationonlyforderivativesofascalar.

5Eq.10.17istheLagrangianintermsofFμν.Eq.10.21isthesameLagrangianintermsofthevectorpotential.

6Ingeneral, integrationbypartsinvolvesanadditionaltermcalledtheboundaryterm.TheassumptionthatJgoestozeroatgreatdistancesallowsustoignoretheboundaryterm.

Lecture11

FieldsandClassicalMechanics

“I’mannoyed,Lenny.Youalwaysgoonandonabouthowourequationshave tocomefromanactionprinciple.Okay,youconvincedme thataction isanelegant idea,butassomeonesaid,‘Eleganceisfortailors.’Frankly,Idon’tseewhyweneedit.”

“Don’tbegrumpy,Art.Therereallyisareason.IbetthatifIletyoumakeupyourownequations, energy won’t be conserved. A world without energy conservation would beprettyweird.Thesuncouldsuddenlygoout,orcarscouldspontaneouslystartmovingfornoreason.”

“Okay,Igetit:Lagrangiansleadtoconservationlaws.Irememberthatfromclassicalmechanics.Noether’stheorem,right?Butwe’vehardlymentionedenergyandmomentumconservation.Doelectromagneticfieldsreallyhavemomentum?”

“Yup.AndEmmycanproveit.”

11.1FieldEnergyandMomentumElectromagneticfieldsclearlyhaveenergy.Standinthesunforafewminutes;thewarmththatyoufeel is theenergyofsunlightbeingabsorbedbyyourskin. Itexcitesmolecularmotion and becomes heat. Energy is a conserved quantity, and it’s carried byelectromagneticwaves.

Electromagneticfieldsalsocarrymomentum,althoughit’snotaseasytodetect.Whensunlightisabsorbedbyyourskin,somemomentumistransferredanditexertsaforceorpressureonyou.Fortunatelytheforceisfeeble,andstandinginthesunlightwon’texertmuchofapush.Butit’sthere.Futurespacetravelmayutilizethepressureofsunlight(orevenstarlight)onalargesailinordertoaccelerateaspaceship(Fig.11.1).Whetherornotthis is practical, the effect is real. Light carriesmomentum,which,when it’s absorbed,exertsaforce.1

Energy and momentum are the key concepts that connect field theory to classicalmechanics.We’ll start by having a closer look atwhatwe actuallymean by thewordsenergyandmomentum.

In this lecture I am going to assume you are familiarwith the concepts of classicalmechanicsastheywerepresentedinVolumeIoftheTheoreticalMinimum.

11.2ThreeKindsofMomentumWehaveencounteredthreedifferentconceptsofmomentum.Thesearenotthreewaysofthinkingaboutthesamething,butthreewaystothinkaboutthreedifferentthings—thingsthat in general have different numerical values. The first and simplest concept ismechanicalmomentum.

Figure11.1:SolarSail.Lightwavescarrymomentum.Theyexertpressureonmaterial

objectsandcausethemtoaccelerate.

11.2.1MechanicalMomentumIn non-relativistic physicsmechanicalmomentum, called , is justmass times velocity.Moreprecisely,itisthetotalmassofasystemtimesthevelocityofthecenterofmass.It’sa vector quantitywith three space components,px,py, andpz. For a single particle, thecomponentsare

Foracollectionofparticleslabeledithecomponentsofmechanicalmomentumare

and similarly for the y and z components. Relativistic particles also have mechanicalmomentumgivenby

11.2.2CanonicalMomentumCanonical momentum is an abstract quantity that can apply to any kind of degree offreedom. For any coordinate that appears in the Lagrangian,2 there is a canonicalmomentum. Suppose theLagrangian depends on a set of abstract coordinatesqi. Thesecoordinatescouldbethespatialcoordinatesofaparticle,ortheangleofarotatingwheel.Theycouldevenbethefieldsrepresentingthedegreesoffreedomofafieldtheory.Eachcoordinatehasaconjugatecanonicalmomentumthatisoftendenotedbythesymbol i.iisdefinedasthederivativeoftheLagrangianwithrespectto :

Ifthecoordinateqihappenstobecalledx,andtheLagrangianhappenstobe

whereV(x)isapotentialenergyfunction,thenthecanonicalmomentumisthesameasthemechanicalmomentum.

Even if thecoordinate inquestion represents thepositionofaparticle, thecanonicalmomentummaynotbethemechanicalmomentum.Infactwe’vealreadyseenanexamplein Lecture 6. Sections 6.3.2 and 6.3.3 describe the motion of a charged particle in anelectromagneticfield.TheLagrangian(fromEq.6.18)is

andyoucansee thatvelocityappears inmore thanone term.Thecanonicalmomentum(fromEq.6.20)is

Wecancallthisquantity p.Thefirsttermontherightsideisthesameasthe(relativistic)mechanical momentum. But the second term has nothing to do with mechanicalmomentum. It involves the vector potential and it’s something new.Whenwe developclassicalmechanicsinHamiltonianform,wealwaysusecanonicalmomentum.

In many cases the coordinates describing a system have nothing to do with thepositionsofparticles.Afieldtheoryisdescribedbyasetoffieldsateachpointofspace.

Forexample,onesimplefieldtheoryhasLagrangiandensity

ThecanonicalmomentumconjugatetoΦ(x)inthistheoryis

This “field momentum” is only distantly related to the usual concept of mechanicalmomentum.

11.2.3NoetherMomentum3

Noethermomentumisrelatedtosymmetries.Letussupposethatasystemisdescribedbyasetofcoordinatesordegreesoffreedomqi.Nowsupposewechangetheconfigurationofthesystembyshiftingthecoordinatesatinybit.Wemightwritethisintheform

ThesmallshiftsΔimaydependonthecoordinates.Agoodgeneralwaytowritethisis

whereεisaninfinitesimalconstantandthefi(q)arefunctionsofthecoordinates.

The simplest example is a translation of a system in space. A system of particles(labeledn)maybeuniformlydisplacedalongthexaxisbyamountε.Weexpressthisbytheequations

δXn=ε

δYn=0.

δZn=0.

Eachparticleisshiftedalongthexaxisbyamountε,whileremainingatthesamevaluesofyandz. If thepotential energyof the systemonlydependson thedistancesbetweenparticles,thenthevalueoftheLagrangiandoesnotchangewhenthesystemisdisplacedinthisway.Inthatcasewesaythatthereistranslationsymmetry.

Anotherexamplewouldbearotationaroundtheoriginintwodimensions.AparticlewithcoordinatesX,YwouldbeshiftedtoX+δX,Y+δY,where

Youcancheckthatthiscorrespondstoarotationoftheparticlearoundtheoriginbyangleε.IftheLagrangiandoesnotchange,thenwesaythatthesystemhasrotationinvariance.

A transformationof coordinates that does not change the value of theLagrangian iscalledasymmetryoperation.4AccordingtoNoether’stheorem,iftheLagrangiandoesn’tchangeunderasymmetryoperation, thenthereisaconservedquantitythatI’llcallQ.5Thisquantityisthethirdconceptofmomentum.Itmayormaynotbeequaltomechanicalorcanonicalmomentum.Let’sremindourselveswhatitis.

Wecanexpressacoordinateshiftwiththeequation

whereδqi represents an infinitesimal change to the coordinateqi, and the function fi(q)dependsonallof theq’s,not justqi. IfEq.11.8 isasymmetry, thenNoether’s theoremtellsusthatthequantity

isconserved.Q is a sumover all the coordinates; there’s contribution fromeachof thecanonicalmomenta .

Let’s consider a simple example where q happens to be the x position of a singleparticle.Whenyoutranslatecoordinates,xchangesbyasmallamountthatisindependentofx. In that case δq (or δx) is just a constant; it’s the amount bywhich you shift. Thecorrespondingfistriviallyjust1.TheconservedquantityQcontainsoneterm,whichwecanwriteas

Sincef(q)=1,weseethatQisjustthecanonicalmomentumofthesystem.Forthecaseofasimplenonrelativisticparticleitisjusttheordinarymomentum.

11.3EnergyMomentumandenergyarecloserelatives;indeedtheyarethespaceandtimecomponentsof a 4-vector. It should not be surprising that the law of momentum conservation inrelativitytheorymeanstheconservationofallfourcomponents.

Let’srecalltheconceptofenergyinclassicalmechanics.Theenergyconceptbecomesimportantwhen theLagrangian is invariantundera translationof the timecoordinate t.Shiftingthetimecoordinateplaysthesameroleforenergyasshiftingthespacecoordinatedoes for momentum. Shifting the time coordinate means that “t” becomes “t plus aconstant.”InvarianceoftheLagrangianunderatimetranslationmeansthattheanswertoaquestionaboutanexperimentdoesn’tdependonwhentheexperimentstarts.

GivenaLagrangianL(qi, ) that’sa functionofqiand , there’saquantitycalled theHamiltonian,definedas

TheHamiltonian is theenergy,andforan isolatedsystemit’sconserved.Let’s return tooursimpleexamplewhereqisthexpositionofasingleparticle,andtheLagrangianis

WhatistheHamiltonianforthissystem?ThecanonicalmomentumisthederivativeoftheLagrangianwithrespecttothevelocity.Inthisexample,thevelocity appearsonlyinthefirsttermandthecanonicalmomentumpiis

Multiplyingpiby (whichbecomes inthisexample)theresultism 2.Next,wesubtracttheLagrangian,resultingin

or

Werecognizethisasthesumofkineticenergyandpotentialenergy.Isitalwaysthateasy?

ThesimpleLagrangianofEq.11.12hasonetermthatdependson 2,andanothertermthatdoesnotcontain atall.WheneveraLagrangianisnicelyseparatedinthisway,it’seasy to identify the kinetic energywith the 2 terms, and the potential energywith theother terms.When the Lagrangian takes this simple form—squares of velocitiesminusthingsthatdon’tdependonvelocities—youcanreadofftheHamiltonianquicklywithoutdoinganyextrawork;justflipthesignsofthetermsthatdon’tcontainvelocities.

If Lagrange’s equations apply to everything on the right side of Eq. 11.11, then theHamiltonianisconserved;itdoesnotchangewithtime.6WhetherornottheLagrangianhassuchasimpleform,thetotalenergyofasystemisdefinedtobeitsHamiltonian.

11.4FieldTheoryField theory is a special caseofordinaryclassicalmechanics.Their closeconnection isslightlyobscuredwhenwetrytowriteeverythinginrelativisticfashion.It’sbesttostartout bygivingupon the ideaofmaking all equations explicitly invariant underLorentztransformation. Instead, we’ll choose a specific reference frame with a specific timecoordinateandworkwithinthatframe.Lateron,we’lladdresstheissueofswitchingfromoneframetoanother.

Inclassicalmechanics,wehaveatimeaxisandacollectionofcoordinatescalledqi(t).Wealsohaveaprincipleofstationaryaction,withactiondefinedasthetimeintegralofaLagrangian,

TheLagrangianitselfdependsonallthecoordinatesandtheirtimederivatives.That’sallofit:atimeaxis,asetoftime-dependentcoordinates,anactionintegral,andtheprincipleofstationaryaction.

11.4.1LagrangianforFieldsIn field theory we also have a time axis, and coordinates or degrees of freedom thatdependontime.Butwhatarethosecoordinates?

Forasinglefield,wecanviewthefieldvariableΦasasetofcoordinates.Thatseemsodd,butrememberthatthefieldvariableΦdependsnotonlyontimebutalsoonposition.It’sthenatureof thisdependencyonpositionthatsets itapartfromtheqivariables that

characterizeaparticleinclassicalmechanics.

Let’simagine,hypothetically,thattheΦdependenceonpositionisdiscreteratherthancontinuous. Fig. 11.2 illustrates this idea schematically. Each vertical line represents asingledegreeoffreedom,suchasΦ1,Φ2,andsoforth.Werefer to themcollectivelyasΦi. This naming convention mimics the coordinate labels (such as qi) of classicalmechanics.Wedothistoemphasizetwoideas:

1.EachΦiisaseparateandindependentdegreeoffreedom.

2.Theindexiisjustalabelthatidentifiesaparticulardegreeoffreedom.

Inpractice,thefieldvariableΦisnotlabeledbyadiscreteindexibutbyacontinuousvariablex,andweusethenotationΦ(t,x).However,we’llcontinuetothinkofxasalabelforanindependentdegreeoffreedom,andnotasasystemcoordinate.ThefieldvariableΦ(t,x)representsanindependentdegreeoffreedomforeachvalueofx.

Wecanseethisevenmoreconcretelywithaphysicalmodel.Fig.11.3showsalineararray of masses connected by springs. The masses can only move in the horizontaldirection,and themotionofeachmass isaseparatedegreeof freedom,qi, labeledwithdiscreteindicesi.Aswepackmoreandmoretinymassesandspringsintothesamespace,thissystemstartstoresembleacontinuousmassdistribution.Inthelimit,wewouldlabelthedegreesoffreedombythecontinuousvariablexratherthanadiscretesetofindicesi.Thisschemeworksforclassicalfieldsbecausethey’recontinuous.

Figure11.2:ElementsofFieldTheory.IfwepretendthatΦisdiscrete,wecanthinkofΦiin

thesamewaywethinkofqiinclassicalmechanics.Thesubscriptsiarelabelsfor

independentdegreesoffreedom.

What about derivatives? Just as we expect, field Lagrangians depend on the timederivatives ofthefieldvariables.ButtheyalsodependonderivativesofΦwithrespecttospace.Inotherwords,theydependonquantitiessuchas

Figure11.3:FieldAnalogy.Thinkofasetofdiscretedegreesoffreedomsuchasthesemassesconnectedbysprings.WelabeleachdegreeoffreedomwithasubscriptΦi.Nowsupposethemassesbecomesmallerandmoredenselyspaced.Inthelimitthereisaninfinitenumberofsmallmassesspacedinfinitesimallyclosely.Inthislimittheybecomecontinuous(justlikeafield)ratherthandiscrete,anditmakesmoresensetolabelthedegreesoffreedomΦ(x)rather

thanΦi.

This is different from classical particle mechanics, where the only derivatives in theLagrangianaretimederivativessuchas .ALagrangianforfieldtheorytypicallydependsonthingslike

and

ButwhatisthederivativeofΦwithrespecttox?It’sdefinedas

for some small value of ε. In this sense, the space derivatives are functions of theΦ’sthemselves;mostimportantly,thespacederivativesdonotinvolve .ThedependenceoftheLagrangianonspacederivativesreflectsitsdependenceontheΦ’sthemselves.Inthiscase,itdependsontwonearbyΦ’Satthesametime.

11.4.2ActionforFieldsLet’sthinkaboutwhatwemeanbyaction.Anactioninclassicalmechanicsisalwaysanintegralovertime,

Butforfields, theLagrangianitself isanintegraloverspace.We’vealreadyseenthisinpreviousexamples.Ifweseparatethetimeportionoftheintegralfromthespaceportion,wecanwritetheactionas

I’musingthesymbolLfortheLagrangian,andthesymbol fortheLagrangiandensity.Theintegralof overspaceisthesameasL.Thepointofthisnotationistoavoidmixingupthetimederivativeswiththespacederivatives.

11.4.3HamiltonianforFieldsTheenergyofafieldispartoftheuniversalconservedenergythatisassociatedwithtimetranslation.Tounderstandfieldenergy,weneedtoconstructtheHamiltonian.Todothat,we need to identify the generalized coordinates (theq’s), their corresponding velocitiesandcanonicalmomenta(the ’sandthep’s),andtheLagrangian.We’vealreadyseenthatthe coordinates areΦ(t, x). The corresponding velocities are just (t, x). These are notvelocities in space, but quantities that tell us how fast the field changes at a particularpoint in space. The canonical momentum conjugate to Φ is the derivative of theLagrangianwithrespectto .Wecanwritethisas

where Φ(x) is thecanonicalmomentumconjugate toΦ.Bothsidesof theequationare

functionsofposition.Iftherearemanyfieldsinaproblem—suchasΦ1,Φ2,Φ3—therewillbeadifferent (x)associatedwitheachof them.With thesepreliminaries inplace,we’reabletowritetheHamiltonian.TheHamiltoniandefinedbyEq.11.11,

isasumoveri.Butwhatdoesirepresentinthisproblem?Itlabelsadegreeoffreedom.Because fields are continuous, their degrees of freedom are instead labeled by a realvariablex,andthesumoveribecomesanintegraloverx.Ifwereplacepiwith Φ(x),andwith (x),wehavethecorrespondence

ToturnthisintegralintoaHamiltonian,wehavetosubtractthetotalLagrangianL.ButweknowfromEq.11.13thattheLagrangianis

Thisisalsoanintegraloverspace.Therefore,wecanputbothtermsoftheHamiltonianinsidethesameintegral,andtheHamiltonianbecomes

Thisequationisinteresting;itexpressestheenergyasanintegraloverspace.Therefore,theintegranditselfisanenergydensity.Thisisacharacteristicfeatureofallfieldtheories;conserved quantities like energy, and also momentum, are integrals of densities overspace.

Let’s go back to the Lagrangian of Eq. 4.7, which we’ve used on a number ofoccasions.Here,weviewitasaLagrangiandensity.We’llconsiderasimplifiedversion,withonlyasingledimensionofspace,

Thefirsttermisthekineticenergy.7Whatis Φ?It’sthederivativeof withrespectto ,whichisjust .Thatis,

andtheHamiltonian,(theenergy),is

Replacing Φwith ,thisbecomes

Pluggingintheexpressionfor gives

or

InEq.11.14(theLagrangiandensity),wehaveakineticenergytermcontainingthetimederivative 2.Thesecondtwoterms,containingV(Φ)andthespacederivativeofΦ,playtheroleofpotentialenergy—thepartoftheenergythatdoesnotcontaintimederivatives.8 TheHamiltonian in this example consists of kinetic energyplus potential energy andrepresents the total energy of the system.By contrast, theLagrangian is kinetic energyminuspotentialenergy.

Let’ssettheV(Φ)termasideforamomentandconsiderthetermsinvolvingtimeandspacederivatives.Bothofthesetermsarepositive(orzero)becausethey’resquares.TheLagrangian has both positive and negative terms, and is not necessarily positive. Butenergyhasonlynon-negativeterms.Ofcoursethesetermscanbezero.ButtheonlywaythatcanhappenisforΦtobeconstant;ifΦisconstantthenitsderivativesmustbezero.Itshouldbenosurprisethatenergydoesnottypicallybecomenegative.

What aboutV(Φ)? This term can be positive or negative. But if it’s bounded frombelow, thenwe can easily arrange for it to be positive by adding a constant.Adding aconstant toV(Φ) doesn’t change anything about the equations of motion. However, ifV(Φ)isnotboundedfrombelow,thetheoryisnotstableandeverythinggoestohellinahandwagon;youdon’twantanypartofsuchatheory.SowecanassumethatV(Φ),andconsequentlythetotalenergy,iszeroorpositive.

11.4.4ConsequencesofFiniteEnergyIfx is justa labelandΦ(t,x) is an independentdegreeof freedomforeachvalueofx,whatpreventsΦfromvaryingwildlyfromonepointtoanother?IsthereanyrequirementforΦ(t,x)tovarysmoothly?Supposehypotheticallythattheoppositeistrue:namely,thatthevalueofΦmayjumpsharplybetweenneighboringpointsinFig.11.2.Inthatcase,thegradient (or space derivative) would get huge as the separation between points getssmaller.9

Thismeans theenergydensitywouldbecome infiniteas theseparationdecreased. Ifwe’re interested in configurations where the energy doesn’t blow up to infinity, thesederivativesmustbefinite.FinitederivativestellΦ(t,x)tobesmooth;theyputthebrakesonhowwildlyΦcanvaryfrompointtopoint.

11.4.5ElectromagneticFieldsviaGaugeInvarianceHowcanweapplytheseideasaboutenergyandmomentumtotheelectromagneticfield?WecouldcertainlyusetheLagrangianformulationthatwesawearlier.ThefieldsarethefourcomponentsofthevectorpotentialAμ.ThefieldtensorFμνiswrittenintermsofthespaceand timederivativesof thesecomponents.We thensquare thecomponentsofFμνandaddthemuptoget theLagrangian.Insteadofhavingjustonefield,wewouldhavefour.10

Butthereisahelpfulsimplificationbasedongaugeinvariance.Thissimplificationnotonlymakesourworkeasierbutillustratessomeimportantideasaboutgaugeinvariance.Rememberthatthevectorpotentialisnotunique;ifwe’reclever,wecanchangeitinwaysthatdonotaffectthephysics.Thisissimilartothefreedomtochooseacoordinatesystem,andwe can use that freedom to simplify our equations. In this case, our use of gaugeinvariancewillallowustoworkwithonlythreecomponentsofthevectorpotentialratherthanworryingaboutallfour.

Let’srecallwhatagaugetransformationis;it’satransformationthataddsthegradientofanarbitraryscalarStothevectorpotentialAμ.Itamountstothereplacement

WecanusethisfreedomtopickSinawaythatsimplifiesAμ.Inthiscasewe’llchooseSinawaythatmakes the timecomponentA0becomezero.Allwecareabout is the timederivative ofS, becauseA0 is the time component ofAμ. In other words, we’d like tochooseanSsuchthat

or

Isthatpossible?Theanswerisyes.Rememberthat isaderivativewithrespecttotimeatafixedpositioninspace.Ifyougotothatfixedpositioninspace,youcanalwayschooseSsothatitstimederivativeissomeprespecifiedfunction,−A0.Ifwedothis,thenthenewvectorpotential,

willhavezeroforitstimecomponent.Thisiscalledfixingthegauge.Differentchoicesofgauges have names—the Lorentz gauge, the radiation gauge, the Coulomb gauge. ThegaugewithA0=0goesundertheelegantname“theA0=0gauge.”Wecouldhavemadeotherchoices;noneofthemwouldaffectthephysics,buttheA0=0gaugeisespeciallyconvenientforourpurposes.Basedonourchoiceofgauge,wewrite

A0=0.

Thiscompletelyeliminatesthetimecomponentfromthevectorpotential.Allwehaveleftarethespacecomponents,

Am(x).

Withthisgaugeinplace,whataretheelectricandmagneticfields?Intermsofthevectorpotential,theelectricfieldisdefined(seeEq.8.7)as

ButwithA0 set equal to zero, the second term disappears andwe canwrite a simpler

equation,

The electric field is just the time derivative of the vector potential. What about themagneticfield?Itonlydependsonthespacecomponentsofthevectorpotential,andthesearenotaffectedbyourchoiceofgauge.Thusit’sstilltruethat

This simplifies thingsbecausewecannow ignore the timecomponentofAμaltogether.ThedegreesoffreedomintheA0=0gaugearejust thespacecomponentsofthevectorpotential.

Let’sconsider the formof theLagrangian in theA0=0gauge. In termsof the fieldtensor,recall(Eq.10.17)thattheLagrangianis

But (aswe saw inEq.10.18) that happens to equal1/2 times the squareof the electricfieldminusthesquareofthemagneticfield,

ThefirstterminEq.11.19is

ButfromEq.11.16weknowthatthisisthesameas

Withthissubstitution,Eq.11.19isstartingtoresembleEq.11.14,whichwas

Noticethat isnotjustasingleterm,butthreeterms;itcontainsthesquaresofthetimederivativesofAx,Ay,andAz.It isasumoftermsofexactlythesametypeasthetermofEq.11.19—one termforeachcomponentof thevectorpotential.Theexpandedformis

ThesecondtermofEq.11.19isthesquareofthecurlofA.ThefullLagrangiandensityis

or

The resemblance to Eq. 11.14 is even stronger now: squares of time derivativesminussquaresofspacederivatives.

What is the canonicalmomentum conjugate to a particular component of the vectorpotential?Bydefinition,it’s

whichisjust

Intermsofindividualcomponents,

But it’s also true that the time derivative ofA isminus the electric field. Sowe foundsomethinginteresting.Thecanonicalmomentahappentobeminusthecomponentsoftheelectricfield.Inotherwords,

Thus the physicalmeaning of canonicalmomentumconjugate to the vector potential is(minus)theelectricfield.

What’stheHamiltonian?Now thatwe know the Lagrangian, we canwrite down theHamiltonian.We could gothrough the formal construction of theHamiltonian, but in this casewe don’t need to.That’s because our Lagrangian has the form of a kinetic energy term that depends onsquaresoftimederivatives,minusapotentialenergytermthathasnotimederivativesatall.WhentheLagrangianhasthisform—kineticenergyminuspotentialenergy—weknowwhat the answer is: The Hamiltonian is kinetic energy plus potential energy. Thus theelectromagneticfieldenergyis

Onceagain,theLagrangianisnotnecessarilypositive.Inparticular,ifthere’samagneticfieldwith no electric field, theLagrangian is negative.But the energy, 1/2(E2 +B2), ispositive.Whatdoesthissayaboutanelectromagneticplanewavemovingalonganaxis?

The example we saw in Lecture 10 had an E component in one direction and a Bcomponent in theperpendiculardirection.TheB fieldhas the samemagnitudeas theEfieldandisinphasewithit,butpolarizedintheperpendiculardirection.Thistellsusthattheelectric andmagnetic field energies are the same.Anelectromagneticwavemovingdownthezaxishasbothelectricandmagneticenergy,andbothcontributionshappentobethesame.

MomentumDensityHowmuchmomentumdoesanelectromagneticwavecarry?Let’sgoback toNoether’sconceptofmomentumfromSection11.2.3.ThefirststepinusingNoether’stheoremistoidentifyasymmetry.Thesymmetryassociatedwithmomentumconservationistranslationsymmetryalongaspatialdirection.Forexample,wemay translateasystemalong thexaxis by a small distance ε (see Fig. 11.4). Each fieldΦ (x) gets replaced byΦ(x− ε).Accordingly,thechangeinΦ(x)is

which(ifεisinfinitesimal)becomes

Figure11.4:NoetherShift.TakeafieldcongurationΦ(x)andshiftittotheright(towardhigherx)byan

infinitesimalamountε.ThechangeinΦataspecific

pointis−εdΦ.

In the case of electromagnetism the fields are the space components of the vectorpotential.Theshiftsofthesefieldsunderatranslationbecome

Nextwerecall fromEq.11.9 that theconservedquantityassociatedwith thissymmetry

hastheform

Inthiscasethecanonicalmomentaareminustheelectricfields

andthefiarejusttheexpressionsmultiplyingεinEqs.11.23,

Thusthexcomponentofthemomentumcarriedbytheelectromagneticfieldisgivenby

Wecould,ofcourse,dothesamethingfortheyandzdirectionsinordertogetallthreecomponentsofmomentum.Theresultis

Evidently, like theenergy, themomentumofanelectromagneticfield isanintegraloverspace. 11 We might therefore identify the integrand of Eq. 11.24 as the density ofmomentum,

ThereisaninterestingcontrastbetweenthismomentumdensityandtheenergydensityinEq.11.22.Theenergydensityisexpresseddirectlyintermsoftheelectricandmagneticfields,butthemomentumdensitystillinvolvesthevectorpotential.Thisisdisturbing;theelectricandmagneticfieldsaregaugeinvariant,butthevectorpotentialisnot.Onemightthink that quantities like energy and momentum densities should be similar and onlydependonEandB. Infact there isasimplefix that turns themomentumdensity intoagaugeinvariantquantity.Thequantity

ispartoftheexpressionforthemagneticfield.Infact,wecanturnitintoamagneticfieldcomponentbyaddingtheterm

Ifwecouldsneakinthischange,we’dbeabletorewritetheintegralas

Canwegetawaywithit?HowwouldPnchangeifweinsertthisadditionalterm?Tofindout,let’slookatthetermwewanttoadd,

Let’sintegratethisbyparts.Aswe’veseenbefore,wheneverwetaketheintegralwhoseintegrandisaderivativetimessomethingelse,wecanshiftthederivativetotheotherterm

atthecostofaminussign:12

Butduetothesummationindexm,theterm isjustthedivergenceoftheelectricfield.We know fromMaxwell’s equations for empty space that is zero. Thatmeans theadditional integralchangesnothing,and therefore thatPn canbewritten in termsof theelectricandmagneticfields,

Alittlealgebrawillshowthattheintegrandisactuallythevector .Thatmeansis the momentum density. Dropping the subscripts and reverting to standard vectornotation,Eq.11.25nowbecomes

isavector,and itsdirection tellsyou thedirectionof themomentum.Themomentumdensity iscalledthePoyntingvector,oftendenotedby .

ThePoyntingvectortellsussomethingabouthowthewavepropagates.Let’sgobacktoFig.10.1andlookatthefirsttwohalf-cyclesofthepropagatingwave.Inthefirsthalf-cycle, pointsupward,and pointsoutof thepage.Following theright-handrule,wecan see that points to the right, along the z axis. That’s the direction in whichmomentum (and thewave itself) propagates.What about the next half-cycle?Since thedirectionofbothfieldvectorshasbeenreversed,thePoyntingvectorstill“poynts”totheright.We’llhavemoretosayaboutthisvectoraswego.

Noticehowimportantitisthat and areperpendiculartoeachother;themomentum(thePoyntingvector)isaconsequence.ThisalltracesbacktoEmmyNoether’swonderfultheorem about the connection between conserved momentum and spatial translationinvariance.It’sallcominghomeinonebigpiece—classicalmechanics,fieldtheory,andelectromagnetism.Itallcomesbacktotheprincipleofleastaction.

11.5EnergyandMomentuminFourDimensionsWe know that energy and momentum are conserved quantities. In four-dimensionalspacetime, we can express this idea as a principle of local conservation. The ideaswepresenthereareinspiredbyourpreviousworkwithchargeandcurrentdensitiesandwilldrawonthoseresults.

11.5.1LocallyConservedQuantitiesBack in Section 8.2.6, we explored the concept of local conservation of charge; if thecharge within a region increases or decreases, it must do so by moving across theboundaryoftheregion.Chargeconservationislocalinanysensiblerelativistictheoryandgivesrisetotheconceptsofchargedensityandcurrentdensity,ρand .Together, these

twoquantitiesmakea4-vector,

Charge density is the time component, and current density (or flux) is the spacecomponent.

The same concept applies to other conserved quantities. Ifwe thinkmore generally,beyondcharges,wecanimagineforeachconservationlawfourquantitiesrepresentingthedensity and flux ofany conserved quantity. In particular,we can apply this idea to theconservedquantitycalledenergy.

InLecture3,we learned that the energyof aparticle is the timecomponentof a4-vector.Thespacecomponentsofthat4-vectorareitsrelativisticmomenta.Symbolically,wecanwrite

Infieldtheory,thesequantitiesbecomedensities,andenergydensitycanbeviewedasthetimecomponentof a four-dimensional current.We’vealreadyderivedanexpression forenergydensity;Eq.11.22givestheenergydensityofanelectromagneticfield.Let’sgivethisquantitythetemporarynameT0:

WhatwewanttodoisfindacurrentofenergyanalogoustotheelectriccurrentJm.LikeJm,theenergycurrenthasthreecomponents.Let’scallthemTm.IfwedefineTmcorrectly,theyshouldsatisfythecontinuityequation

OurstrategyforfindingthecurrentofenergyTmissimple:DifferentiateT0withrespecttotime,andseeiftheresultisthedivergenceofsomething.TakingthetimederivativeofEq.11.26resultsin

or

where and arederivativeswithrespect to time.At firstglance, therightsideofEq.11.27 doesn’t look like a divergence of anything, because it involves time derivativesratherthanspacederivatives.ThetrickistousetheMaxwellequations

toreplace and .PluggingtheseintoEq.11.27givessomethingalittlemorepromising:

In this form, the right sidecontains spacederivativesand thereforehas somechanceofbeingadivergence.Infact,usingthevectoridentity

wefindthatEq.11.29becomes

Loandbehold,ifwedefinethecurrentofenergyas

wecanwritethecontinuityequationforenergyas

Inrelativisticnotation,thisbecomes

The vector representing the flow of energy should be familiar; it’s the Poyntingvector, named after John Henry Poynting, who discovered it in 1884, probably by thesame argument. We already met this vector in Section 11.4.5 under the heading“MomentumDensity.”ThePoyntingvectorhastwomeanings:Wecanthinkofitaseitheranenergyfloworamomentumdensity.

To summarize: Energy conservation is local. Just like charge, a change of energywithin some region of space is always accompanied by a flow of energy through theboundaries of the region. Energy cannot suddenly disappear from our laboratory andreappearonthemoon.

11.5.2Energy,Momentum,andLorentzSymmetryEnergyandmomentumarenotLorentzinvariant.Thatshouldbeeasytosee.Imagineanobjectofmassmatrestinyourownrestframe.Theobjecthasenergygivenbythewell-knownformula

E=mc2

orinrelativisticunits,

E=m

Becausetheobjectisatrest,ithasnomomentum.13Butnowlookatthesameobjectfromanotherframeofreferencewhereitismovingalongthexaxis.Itsenergyisincreased,andnowitdoeshavemomentum.

Iffieldenergyandmomentumarenotinvariant,howdotheychangeunderaLorentztransformation?Theansweristhattheyforma4-vector,justastheydoforparticles.Let’scallthecomponentsofthe4-vectorPμ.Thetimecomponentistheenergy,andthethreespacecomponentsareordinarymomentaalongthex,y,andzaxes.Allfourcomponentsareconserved:

This suggests that each component has a density and that the total value of eachcomponentisanintegralofthedensity.Inthecaseofenergywedenotedthedensityby

thesymbolT0.Butnowwe’regoingtochangethenotationbyaddingasecondindexandcallingtheenergydensityT00.

I’m sure you’ve already guessed that the double indexmeanswe’re building a newtensor. Each index has a specific meaning. The first index tells us which of the fourquantitiestheelementrefersto.14Theenergyisthetimecomponentofthe4-momentum,andthereforethefirstindexis0fortime.Tobeexplicit:Avalueof0forthefirstindexindicates that we’re talking about energy. A value of 1 indicates the x component ofmomentum.Valuesof2and3indicatetheyandzcomponentsofmomentumrespectively.

Thesecondindextellsuswhetherwe’retalkingaboutadensityoraflow.Avalueof0indicates a density.A value of 1 indicates a flow in the x direction.Values of 2 and 3indicateflowsintheyandzdirectionsrespectively.

Forexample,T00 is thedensityofenergy;weget the totalenergyby integratingT00overspace:

Nowlet’sconsiderthexcomponentofmomentum.Inthiscasethefirstindexisx(orthenumber1),indicatingthattheconservedquantityisthexcomponentofmomentum.Thesecondindexagaindifferentiatesbetweendensityandfloworcurrent.Thus,forexample,

ormoregenerally,

Whataboutthe flowofmomentum?Eachcomponenthasitsownflux.Forexample,wecanconsiderthefluxofx-momentumflowingintheydirection.15ThiswouldbedenotedTxy.SimilarlyTzxisthefluxofz-momentumflowinginthexdirection.

ThetrickinunderstandingTμνistoblotoutthesecondindexforamoment.Thefirstindextellsuswhichquantitywearetalkingabout,namelyP0,Px,Py,orPz.Then,onceweknowwhichquantity,weblotoutthefirstindexandlookatthesecond.Thattellsusifwearetalkingaboutadensityoracomponentofthecurrent.

WecannowwritedownthecontinuityequationforthecomponentofmomentumPmas

or

There are three such equations, one for each component ofmomentum (that is, one foreach value of m). But if we add to these three the fourth equation representing theconservationofenergy(byreplacingmwithμ)wecanwriteallfourequationsinaunified

relativisticform,

This all takes some getting used to, so itmight be a good time to stop and review thearguments.Whenyou’rereadytocontinue,we’llworkout theexpressionsformomentaandtheirfluxesintermsofthefieldsEandB.

11.5.3TheEnergy-MomentumTensorTherearelotsofwaystofigureoutwhatTμνisintermsoftheelectricandmagneticfields.Somearemoreintuitivethanothers.We’regoingtouseatypeofargumentthatmayseemsomewhatlessintuitiveandmoreformal,butit’scommoninmoderntheoreticalphysicsandisalsoverypowerful.WhatIhaveinmindisasymmetryorinvarianceargument.Aninvariance argument begins with listing the various symmetries of a system and thenaskinghowthequantityofinteresttransformsunderthosesymmetries.

ThemostimportantsymmetriesofelectrodynamicsaregaugeinvarianceandLorentzinvariance.Let’sbeginwithgaugeinvariance:HowdothecomponentsofTμν transformunderagaugetransformation?Theanswerissimple;theydon’t.Thedensitiesandfluxesofenergyandmomentumarephysicalquantities thatmustnotdependon thechoiceofgauge.Thismeansthattheyshoulddependonlyonthegauge-invariantobservablefieldsand andnothaveadditionaldependenceonthepotentialAμ.

Lorentz invariance ismore interesting.What kind of object isTμν, and how do thecomponentschange ingoing fromone frameof reference toanother? It isclearlynotascalarbecauseithascomponents labeledμandν. Itcannotbea4-vectorbecause ithastwoindicesandsixteencomponents.Theanswerisobvious.Tμν isa tensor—aranktwotensor,meaningthatithastwoindices.

NowwecangiveTμν itspropername:theenergy-momentum?tensor.16Because it issuchanimportantobject,notjustinelectrodynamicsbutinallfieldtheories,I’llrepeatit:Tμν is the energy-momentum tensor. Its components are the densities and currents ofenergyandmomentum.

WecanbuildTμνbycombiningthecomponentsofthefieldtensorFμν.Ingeneral,therearemanytensorswecouldbuildinthisway,butwealreadyknowexactlywhatT00is.It’stheenergydensity,

ThistellsusthatTμνisquadraticinthecomponentsofthefieldtensor;inotherwords,itisformedfromproductsoftwocomponentsofFμν.

The question then is, howmany differentways are there to form a tensor from theproductof twocopiesofFμν?Fortunatelytherearenot toomany.Infact, thereareonlytwo.AnytensorbuiltquadraticallyoutofFμνmustbeasumoftwotermsoftheform

whereaandbarenumericalconstantsthatwe’llfigureoutinamoment.

Let’spausetolookatEq.11.40.Thefirstthingtonoticeisthatwe’vemadeuseoftheEinsteinsummationconvention.Inthefirsttermtheindexσ issummedover,andinthesecondtermσandτaresummedover.

Thesecondthingtonoticeistheappearanceofthemetrictensorημν.Themetrictensorisdiagonalandhascomponentsη00=−1,η11=η22=η33=+1.Theonlyquestionishowtodeterminethenumericalconstantsaandb.

Here the trick is to realize thatwe already know one of the components:T00 is theenergydensity.AllweneedtodoisuseEq.11.40todetermineT00andthenplugitintoEq.11.39.Hereiswhatweget:

Comparingthetwosides,wefindthata=1andb=−1/4.Withthesevaluesofaandb,Eq.11.40becomes

Fromthisequationwecancomputeallthevariouscomponentsoftheenergy-momentumtensor.

AsidefromT00,whichisgivenbyEq.11.39,themostinterestingcomponentsareT0nandTn0, where n is a space index. T0n are the components of the flux (or current) ofenergy,andifweworkthemoutwefindthattheyare(asexpected)thecomponentsofthePoyntingvector.

Nowlet’slookatTn0.HerewecanuseaninterestingpropertyofEq.11.40.AlittlebitofinspectionshouldconvinceyouthatTμνissymmetric.Inotherwords,

Tμν=Tνμ.

ThusTn0isthesameasT0n;botharejustthePoyntingvector.ButTn0doesnothavethesamemeaningashT0n.T01 is thefluxofenergyin thexdirection,butT10 isanentirelydifferentthing;namely,itisthedensityofthexcomponentofmomentum.It’shelpfultovisualizeTμνasfollows:

where(Sx,Sy,Sz)arecomponentsofthePoyntingvector.Inthisform,it’seasytoseehowthe mixed space-time components (the top row and leftmost column) differ from thespace-spacecomponents(the3×3submatrixinthelowerright).Aswenotedearlier,σmnarethecomponentsofatensorcalledtheelectromagneticstresstensor,whichwehavenotdiscussedindetail.

Exercise11.1:ShowthatT0nisthePoyntingvector.

Exercise11.2:CalculateT11andT12intermsofthefieldcomponents(Ex,Ey,Ez)and(Bx,By,Bz).

Supposewerestoredthefactorsofthespeedoflightcthatwepreviouslysetto1.Theeasiestwaytodothatisbydimensionalanalysis.Wewouldfindthattheenergyfluxandmomentumdensitydifferbyafactorofc2.Thedimensionallycorrectidentificationisthatthemomentumdensityis ,withnofactorofc.Evidentlywehavemadeadiscovery,andit’sonethathasbeenconfirmedbyexperiment:

For an electromagnetic wave, the density of momentum is equal to the flux of energydividedbythespeedoflightsquared.BothareproportionaltothePoyntingvector.

Nowwecanseewhyontheonehandsunlightwarmsuswhenit’sabsorbed,butontheotherhanditexertssuchafeebleforce.Thereasonisthattheenergydensityisc2 timesthemomentumdensity,andc2isaverybignumber.Asanexercise,youcancalculatetheforceonasolarsailamillionsquaremetersinarea,atEarthdistancefromtheSun.Theresult is tiny,about8newtonsor roughly2pounds.On theotherhand, if the samesailabsorbed(ratherthanreflected)thesunlightthatfellonit, thepowerabsorbedwouldbeaboutamillionkilowatts.

Thedensityandcurrentofelectricchargeplayacentralroleinelectrodynamics.Theyappear in Maxwell’s equations as the sources of the electromagnetic fields. One maywonder if the energy-momentum tensor plays any similar role. In electrodynamics theanswer isno;Tμνdoesnotdirectlyappear in theequations for and . It’s only in thetheoryof gravity that energy andmomentum take their rightful role as sources, but notsources of electromagnetic fields. The energy-momentum tensor appears in the generaltheoryofrelativityasthesourceofthegravitationalfield.Butthat’sanothersubject.

11.6ByeforNowClassical field theory isoneof thegreat accomplishmentsofnineteenth- and twentieth-century physics. It ties together the broad fields of electromagnetism and classicalmechanics, using the action principle and special relativity as the glue. It provides aframeworkforstudyinganyfield—gravity,forexample—fromaclassicalperspective.It’sacrucialprerequisiteforthestudyofquantumfieldtheoryandforgeneralrelativity(thetopicofournextbook).Wehopewe’vemade thesubjectunderstandableandeven fun.We’rethrilledthatyoumadeitallthewaytotheend.

Somewiseguyoncesaid,

Outsideofadog,abookisaman’sbestfriend.Insideofadogit’stoodarktoread.

Ifyouhappentoreviewourbookfromeithertheoutsideortheinsideofadog—ortheboundaryofadog,forthatmatter—pleaseincludesomereferencetoGroucho,howeversubtle,inyourreview.Wewilldeemthattobesufficientproofthatyouactuallyreadthebook.

AsourfriendHermannmighthavesaid,“Timeisup!”Seeyouingeneralrelativity.

FarmerLenny,outstandinginhisϕield.

Apairo’ducksdancestoArt’sΦiddle.

1Seehttps://en.m.wikipedia.org/wiki/IKAROSforadescriptionofIKAROS,anexperimentalspacecraftbasedonthisconcept.

2FornowI’llusethesymbolLinsteadof fortheLagrangian.

3IusethetermNoetherMomentumbecauseI’mnotawareofastandardtermforthisquantity.

4Inthiscontext,we’retalkingaboutanactivetransformation,whichmeanswemovethewholelaboratory,includingallthefieldsandall thecharges, toadifferent location inspace.This isdifferent fromapassive transformation that justrelabelsthecoordinates.

5 I explain Noether’s theorem in the first volume of this series, Classical Mechanics. For more about Noether’scontributions,seehttps://en.m.wikipedia.org/wiki/Emmy_Noether.

6Forafullexplanation,seeVolumeIoftheTheoreticalMinimumseries.

7Ifwewentbacktothinkingaboutrelativity,wewouldrecognizethefirsttwotermsontherightsideasscalars,andcombinethemintoasingletermsuchas ∂μΦ∂

μΦ.That’sabeautifulrelativisticexpression,butwedon’twanttousethisformjustnow.Instead,we’llkeeptrackoftimederivativesandspacederivativesseparately.

8SometimesV(Φ)iscalledthefieldpotentialenergy,butit’smoreaccuratetothinkofthecombinationofbothtermsasthepotentialenergy.Potentialenergyshouldincludeanytermsthatdon’tcontaintimederivatives.

9Rememberwhatthederivativeis.It’sthechangeinΦbetweentwonearbypoints,dividedbythesmallseparation.ForagivenchangeinΦ,thesmallertheseparation,thebiggerthederivative.

10We’llworkmainlyinrelativisticunits(c=1).Onoccasion,we’llbrieflyrestorethespeedsoflightwhenasenseofrelativescaleisimportant.

11Because these integrals referenceonespacecomponentata time,we’vewrittendx insteadofd3x.Amoreprecise

notationfortheintegrandofEq.11.24mightbe .We’vedecidedonthesimplerform,wheredxisunderstood

toreferencetheappropriatespacecomponent.

12Weassumethatthefieldsgotozerobeyondacertaindistance,andthereforetherearenoboundaryterms.

13InthesetwoequationsEstandsforenergy,notelectricfield.

14 The roles of the first and second indices here may be reversed in relation to their roles described in the video.Becausethetensorissymmetric,thismakesnorealdifference.

15The term fluxmaybe confusing to some.Perhaps it’smore clear ifwe call it the changeof (thexcomponentofmomentum) in theydirection.Change inmomentumshouldbea familiar idea; in fact, it represents force.But sincewe’re talking aboutmomentum density, it’s better to think of these space-space components ofTμν as stresses. Thenegativesofthesespace-spacecomponentsforma3×3tensorintheirownright,knownasthestresstensor.

16Someauthorscallitthestress-energytensor.

Leonard Susskind is the Felix Bloch Professor in Theoretical Physics at StanfordUniversity. He is the author of Quantum Mechanics (with Art Friedman) and TheTheoreticalMinimum (with George Hrabovsky), among other books. He lives in PaloAlto,California.

ArtFriedmanisadataconsultantandtheauthorofQuantumMechanics (withLeonardSusskind).Alifelongstudentofphysics,helivesinMurphys,California.

Alsobytheauthors

QuantumMechanics:TheTheoreticalMinimum

byLeonardSusskindandArtFriedman

TheTheoreticalMinimum:WhatYouNeedtoKnowtoStartDoingPhysics

byLeonardSusskindandGeorgeHrabovsky

TheBlackHoleWar:MyBattlewithStephenHawkingtoMaketheWorldSafeforQuantumMechanics

byLeonardSusskind

TheCosmicLandscape:StringTheoryandtheIllusionofIntelligentDesign

byLeonardSusskind

AnIntroductiontoBlackHoles,InformationandtheStringTheoryRevolution:TheHolographicUniverse

byLeonardSusskindandJamesLindesay

AppendixA

MagneticMonopoles:LennyFoolsArt

“Hey,Art,letmeshowyousomething.Here,takealookatthis.”

“Holymoly,Lenny,Ithinkyoufoundamagneticmonopole.Butholdon!Didn’tyoutellmethatmonopolesareimpossible?Comeon,comeclean;you’vegotsomethingupyoursleeveandIthinkIknowwhatitis.Asolenoid,right?Haha,goodtrick.”

“Nope,Art,notatrick.It’samonopoleallright,andnostringsattached.”

FigureA.1:ElectricMonopole.It’sjustapositivecharge.

Whatwouldamagneticmonopolebe,ifthereweresuchthings?First,whatdoesthetermmonopolemean?Amonopoleissimplyanisolatedcharge.Thetermelectricmonopole,ifitwerecommonlyused,wouldsimplymeananelectricallychargedparticle,alongwithitselectric Coulomb field. The convention (see Fig. A.1) is that the electric field pointsoutward forapositivecharge like theproton, and inward foranegativecharge like theelectron.Wesaythatthechargeisthesourceoftheelectricfield.

Amagneticmonopolewouldbeexactlythesameasanelectricmonopole,exceptthatitwouldbesurroundedbyamagneticfield.Magneticmonopoleswouldalsocomeintwokinds,thoseforwhichthefieldpointsoutward,andthoseforwhichitpointsinward.

Asailorofold,whomighthavehadamagneticmonopole,wouldhaveclassifieditaseitheranorthorasouthmagneticmonopole,dependingonwhetheritwasattractedtotheEarth’snorthorsouthpole.Fromamathematicalpointofviewit’sbettertocallthetwotypesofmagneticmonopolespositiveandnegative,dependingonwhether themagneticfieldpointsoutor in. If suchobjectsexist innature,wewouldcall themsourcesof themagneticfield.

Electric and magnetic fields are mathematically similar, but there is one crucialdifference.Electricfieldsdohavesources:electriccharges.Butaccordingtothestandardtheoryofelectromagnetism,magneticfieldsdonot.ThereasoniscapturedmathematicallyinthetwoMaxwellequations

and

Thefirstequationsaysthatelectricchargeisthesourceofelectricfield.IfwesolveitforapointsourcewegetthegoodoldCoulombfieldforanelectricmonopole.

Thesecondequationtellsusthatthemagneticfieldhasnosources,andforthatreasonthere is no such thing as a magnetic monopole. Nevertheless, despite this compellingargument, magneticmonopoles not only are possible, but they are an almost universalfeatureofmoderntheoriesofelementaryparticles.Howcanthisbe?

Onepossibleansweristojustchangethesecondequationbyputtingamagneticsourceontherightside.Callingthedensityofmagneticchargeσ,wemightreplaceEq.A.2with

SolvingthisequationforapointmagneticsourcewouldresultinaamagneticCoulombfieldthat’stheexactanalogoftheelectricCoulombfield,

Theconstantμwouldbethemagneticchargeofthemagneticmonopole.Byanalogywithelectrostatic forces, twomagneticmonopoleswouldbeexpected toexperiencemagneticCoulomb forces between them; the only difference would be that the product of theelectricchargeswouldbereplacedwiththeproductofthemagneticcharges.

Butit’snotsosimple.Wereallydon’thavetheoptionoftinkeringwith∇·B=0.TheunderlyingframeworkforelectromagnetismisnottheMaxwellequations,butrathertheactionprinciple,theprincipleofgaugeinvariance,andthevectorpotential.Themagneticfieldisaderivedconceptdefinedbytheequation

AtthispointitmightbeagoodideatogobacktoLecture8andreviewtheargumentsthatled us to Eq.A.5.What does this equation have to dowithmagneticmonopoles? Theanswerisamathematicalidentity,onethatwehaveusedseveraltimes:Thedivergenceofacurlisalwayszero.

Inotherwords,anyvectorfieldsuchas that’sdefinedasthecurlofanotherfield—the

vectorpotential inthiscase—automaticallyhaszerodivergence.Thus appearstobeanunavoidableconsequenceoftheverydefinitionof .

Nonetheless,mosttheoreticalphysicistsarefirmlypersuadedthatmonopolescan,andprobablydo,exist.TheargumentgoesbacktoPaulDirac,whoin1931explainedhowonecould “fake” a monopole. In fact, the fake would be so convincing that it would beimpossibletodistinguishfromtherealthing.

Start with an ordinary bar magnet (Fig. A.2) or even better, a electromagnet orsolenoid.Asolenoid(Fig.A.3)isacylinderwrappedbywirewithcurrentgoingthroughit. The current creates amagnetic field, which is like the field of the barmagnet. Thesolenoid has the advantage thatwe can vary the strength of themagnet by varying thecurrentthroughthewire.

FigureA.2:ABarMagnetwithNorthandSouth

Poles.

Every magnet, including the solenoid, has a north and south pole that we can callpositiveandnegative.Magneticfieldcomesoutofthepositivepoleandreturnsthroughthenegativepole.Ifyouignorethemagnetinbetween,itlooksverymuchlikeapairofmagneticmonopoles,onepositiveandonenegative.Butof courseyoucan’t ignore thesolenoid.Themagneticfielddoesn’tendatthepoles;itpassesthroughthesolenoidsothatthe the linesof fluxdon’t terminate; theyformcontinuous loops.Thedivergenceof themagneticfieldiszeroeventhoughitlookslikethereisapairofsources.

Nowlet’sstretchoutthebarmagnetorsolenoid,makingitverylongandthin.Atthesametimelet’sremovethesouth(negative)poletosuchafardistancethatitmightaswellbeinfinitelyfar.

Theremainingnorthpolelookslikeanisolatedpositivemagneticcharge(Fig.A.4).Ifone had several such simulated monopoles they would interact much like actualmonopoles, exerting magnetic Coulomb forces on each other. But of course the longsolenoid is unavoidably attached to the fake monopole, and the magnetic flux passesthroughit.

FigureA.3:ASolenoidorElectromagnet.The

magneticfieldisgeneratedbyacurrentthroughthewirewoundaroundthe

cylindricalcore.

Wecouldgofurtherandimaginethesolenoidtobeflexible(callitaDiracstring),asillustrated inFig.A.5.As longas the fluxgoes through thestringandcomesoutat theotherpole,Maxwellequation wouldbesatisfied.

Finally, we couldmake the flexible string so thin that it’s invisible (Fig. A.6). Youmight think that thesolenoidwouldbeeasilydetectable, so that themonopolecouldbeeasily unmasked as a fake. But suppose that it were so thin that any charged particlemoving in itsvicinitywouldhaveanegligiblechanceofhitting it andexperiencing themagneticfieldinsidethestring.Ifitwerethinenough,thestringwouldpassbetweentheatomsofanymaterial,leavingitunaffected.

It’snotpracticaltomakesolenoidsthataresothinthattheypassthroughallmatter,butit’sthethoughtexperimentthat’simportant.Itshowsthatmagneticmonopoles,oratleastconvincingsimulationsof them,mustbemathematicallypossible.Moreover,byvaryingthecurrentinthesolenoid,itispossibletomakethemonopolesattheendsofthesolenoidhaveanymagneticcharge.

FigureA.4:ElongatedSolenoid.

Ifphysicswereclassical—notquantummechanical—thisargumentwouldbecorrect:monopoles of anymagnetic chargewould be possible.ButDirac realized that quantummechanics introduces a subtle new element. Quantum mechanically, an infinitely thinsolenoid would not generally be undetectable. It would affect the motion of chargedparticlesinasubtleway,eveniftheynevergetclosetothestring.Inordertoexplainwhythisisso,wedoneedtouseabitofquantummechanics,butI’llkeepitsimple.

Let’simagineourthinsolenoidisextremelylong,stretchingacrossallspace,andthatwearesomewherenearthestringbutfarawayfromeitherend.Wewanttodeterminetheeffectonanatomwhosenucleusliesnearthesolenoidandwhoseelectronsorbitaroundthesolenoid.Classically,therewouldbenoeffectbecausetheelectronsdon’tpassthroughthemagneticfield.

Insteadofarealatom,whichisabitcomplicated,wecanuseasimplifiedmodel—acircular ring of radius r that an electron slides along (Fig. A.7). If the solenoid passesthroughthecenterofthering,theelectronwillorbitaroundthesolenoid(iftheelectronhas any angular momentum). To begin with, suppose that there is no current in thesolenoidso thatmagnetic field threading thestring iszero.1Suppose thevelocityof theelectronontheringisv.ItsmomentumpandangularmomentumLare

FigureA.5:DiracString:AThinandFlexibleSolenoid.

FigureA.6:LimitingCaseofaDiracString.The

stringcouldbesothinastobeunobservable.

and

FigureA.7:RingSurroundingaThinStringlikeSolenoid.Achargedelectronslidesalongthering.Byvaryingthecurrentinthesolenoidwecanvarythemagneticfield

threadingthestring.

Finally,theenergy oftheelectronis

whichwemayexpressasafunctionoftheangularmomentumL,

Nowfortheintroductionofquantummechanics.Allweneedisonebasicfact,discoveredbyNielsBohr in1913.ItwasBohrwhofirst realizedthatangularmomentumcomesindiscrete quanta. This is true for an atom, and just as true for an electronmoving on acircularring.Bohr’squantizationconditionwasthattheorbitalangularmomentumoftheelectronmustbeanintegermultipleofPlanck’sconstantħ.CallingL theorbitalangularmomentum,Bohrwrote

wherencanbeanyinteger,positive,negative,orzero,butnothinginbetween.Itfollowsthattheenergylevelsoftheelectronmovingontheringarediscreteandhavethevalues

Thusfar,wehaveassumedthatthereisnomagneticfieldthreadingthestring,butourrealinterestistheeffectontheelectronofthestringwithmagneticfluxϕrunningthroughit.Solet’srampupthecurrentandcreateamagneticfluxthroughthestring.Themagneticfluxstartsatzero;overatimeintervalΔtitgrowstoitsfinalvalueϕ.

Onemightthinkthatturningonthemagneticfieldhasnoeffectontheelectronsincethe electron is notwhere themagnetic field is located.But that’swrong.The reason isFaraday’s law: A changing magnetic field creates an electric field. It’s really just theMaxwellequation

This equation says that the increasing magnetic flux induces an electric field thatsurrounds the string and exerts a force on the electron. The force exerts a torque andaccelerates the electron’s angularmotion, therebychanging its angularmomentum (Fig.A.8).

Ifthefluxthroughthestringisϕ(t),thenEq.9.18tellsusthattheEMFis

BecauseEMFrepresentstheenergyneededtopushaunitchargeoncearoundtheentireloop,theelectricfieldattheringhastheoppositesignandis“spreadout”alongthelengthofthering.Inotherwords,

FigureA.8:Anotherviewofthesolenoid,ring,and

charge.

andthetorque(forcetimesr)is

ExerciseA.1:DeriveEq.A.13, based onEq. 9.18.Hint: The derivation follows thesamelogicasthederivationofEq.9.22inSection9.2.5.

A torque will change angular momentum in the same way that a force changesmomentum.Infact,thechangeinangularmomentumduetoatorqueTappliedforatimeΔtis

or,usingEq.A.14,

Thefinalstepincalculatinghowmuchtheangularmomentumchanges,istorealizethattheproduct isjustthefinalfluxϕ2.Thusattheendoftheprocessoframpinguptheflux,theelectron’sangularmomentumhaschangedbytheamount

Bythistime,thefluxthroughthesolenoidisnolongerchangingandthereforethereisnolongeranelectricfield.Buttwothingshavechanged:First,thereisnowamagneticfluxϕ

throughthesolenoid.Second,theangularmomentumoftheelectronhasshiftedby .Inotherwords,thenewvalueofLis

which is not necessarily an integermultiple ofħ. This in turn shifts the possible set ofenergylevelsoftheelectron,whethertheelectronlivesonaringorinanatom.Achangeinthepossibleenergiesofanatomwouldbeeasytoobserveinthespectrallinesemitted

bytheatom.But thishappensonlyforelectronsthatorbit thestring.Asaconsequence,onecouldlocatesuchastringbymovinganatomaroundandmeasuringitsenergylevels.ThiswouldputthekiboshonLenny’stricktofoolArtwithhisfakemonopole.

But there is anexception.Goingback toEq.A.18, suppose that the shiftof angularmomentum just happened to be equal to an integermultiplen′ of Planck’s constant. Inotherwords,suppose

Inthatcasethepossiblevaluesoftheangularmomentum(andtheenergylevels)wouldbeno different than they were when the flux was zero, namely some integer multiple ofPlanck’sconstant.Artwouldnotbeabletotellthatthestringwaspresentbyitseffectonatomsoranythingelse.Thisonlyhappensforcertainquantizedvaluesoftheflux,

Nowlet’sreturntooneendofthestring,saythepositiveend.ThemagneticfluxthreadingthestringwillspreadoutandmimicthefieldofamonopolejustasinFig.A.6.Thechargeofthemonopoleμisjusttheamountoffluxthatspillsoutoftheendofthestring;inotherwords it is ϕ. If it is quantized as in Eq. A.20, then the string will be invisible, evenquantummechanically.

Puttingitalltogether,LennycanindeedfoolArtwithhis“fake”magneticmonopole,butonlyifthechargeofthemonopoleisrelatedtotheelectronchargeqby

Thepoint,ofcourse, isnot just thatsomeonecanbefooledbyafakemonopole;forallintentsandpurposes,realmonopolescanexist,butonlyiftheirchargessatisfyEq.A.21.The argument may seem contrived, but modern quantum field theory has convincedphysiciststhatit’scorrect.

But this does leave the question ofwhy nomagneticmonopole has ever been seen.Whyaretheynotasabundantaselectrons?Theanswerprovidedbyquantumfieldtheoryisthatmonopolesareveryheavy,muchtoomassivetobeproducedinparticlecollisions,eventhosetakingplaceinthemostpowerfulparticleaccelerators.Ifcurrentestimatesarecorrect, no acceleratorwe are ever likely to buildwould create collisionswith enoughenergy to produce a magnetic monopole. So do they have any effect on observablephysics?

Diracnoticedsomethingelse:Theeffectofasinglemonopoleintheuniverse,orevenjustthepossibilityofamonopole,hasaprofoundimplication.Supposethatthereexistedanelectricallychargedparticleinnaturewhosechargewasnotanintegermultipleoftheelectron charge. Let’s call the electric charge of the new particleQ. A monopole thatsatisfiesEq.A.21,withq being the electron charge,might not satisfy the equation ifqwerereplacedbyQ.Inthatcase,ArtcouldexposeLenny’sfraudulentmonopolewithanexperimentusingthenewparticle.Forthisreason,Diracarguedthattheexistenceofevenasinglemagneticmonopoleintheuniverse,oreventhepossibilityofsuchamonopole,requiresthatallparticleshaveelectricchargesthatareintegermultiplesofasinglebasic

unitofcharge,theelectroncharge.Ifaparticlewithcharge timestheelectroncharge,or any other irrational multiple, were to be discovered, it wouldmean that monopolescouldnotexist.

Isittruethateverychargeinnatureisanintegermultipleoftheelectroncharge?Asfarasweknow,itis.Thereareelectricallyneutralparticlesliketheneutronandneutrino,for which the integer is zero; protons and positrons whose charges are −1 times theelectron charge; and many others. So far, every particle ever discovered—includingcomposites likenuclei, atomic ions, exotics like theHiggsboson, andallothers—hasachargethat’sanintegermultipleoftheelectroncharge.3

1Thisiswherethevariablefieldofthesolenoidishelpful.

2Technically,it’sthechangeinfluxovertimeperiodΔt,butbyassumptionthefluxstartsoutatzero.Thereforethesetwoquantitiesarethesame.

3 Youmay think that quarks provide a counterexample to this argument, but they don’t. It’s true that quarks carrychargesof or oftheelectroncharge.However,theyalwaysoccurincombinationsthatyieldanintegermultipleoftheelectroncharge.

AppendixB

Reviewof3-VectorOperators

Thisappendixsummarizes thecommonvectoroperators:gradient,divergence,curl,andthe Laplacian.We assume you have seen them before.We’ll restrict the discussion toCartesian coordinates in three dimensions. Because all of these operators use thesymbol,we’lldescribethatfirst.

B.1The OperatorInCartesiancoordinates,thesymbol (pronounced“del”)isdefinedas

where , ,and areunitvectorsinthex,y,andzdirectionsrespectively.Inmathematical

expressions,wemanipulatetheoperatorcomponents(suchas )algebraicallyasthoughtheywerenumericquantities.

B.2GradientThegradientofascalarquantityS,written S,isdefinedas

Inour“condensed”notation,itscomponentsare

wherethederivativesymbolsareshorthandfor

Thegradientisavectorthatpointsinthedirectionofmaximumchangeforthescalar.Itsmagnitudeistherateofchangeinthatdirection.

B.3DivergenceThedivergenceof ,writtenas · ,isascalarquantitygivenby

or

The divergence of a field at a specific location indicates the tendency of that field tospreadout from that point.Apositivedivergencemeans the field spreadsout from thislocation.Anegativedivergencemeans theopposite—the field tends toconverge towardthatlocation.

B.4CurlThecurlindicatesthetendencyofavectorfieldtorotateorcirculate.Ifthecurliszeroatsomelocation,thefieldatthatlocationisirrotational.Thecurlof ,written × ,isitselfavectorfield.Itisdefinedtobe

Itsx,y,andzcomponentsare

or

Forconvenience,we’llrewritetheprecedingequationsusingnumericalindices:

Thecurloperatorhas thesamealgebraicformas thevectorcrossproduct,whichwesummarizehereforeasyreference.Thecomponentsof are

Usingindexnotation,thisbecomes

B.5LaplacianTheLaplacianisthedivergenceofthegradient.Itoperatesonatwice-differentiablescalarfunctionS,andresultsinascalar.Insymbols,it’sdefinedas

ReferringbacktoEq.B.1,thisbecomes

Ifweapply∇2toascalarfunctionS,weget

Thevalueof∇2SataparticularpointtellsyouhowthevalueofSatthatpointcomparestotheaveragevalueofSatnearbysurroundingpoints.If∇2S>0atpointp,thenthevalueofSatpointpislessthantheaveragevalueofSatnearbysurroundingpoints.

We usuallywrite the∇2 operatorwithout an arrow on top because it operates on ascalar and produces another scalar. There is also a vector version of the Laplacianoperator,writtenwithanarrow,whosecomponentsare

Index

acceleration,169;4-acceleration,246;harmonicoscillatorsand,188;lightand,355(fig.);masstimes,248;ofparticles,172–173;relativistic,248–249

action:inclassicalmechanics,366;defining,118–119,366;field,117,167,366;nonrelativistic,95–96;ofparticlemotion,123–124;ofparticles,163;principleofleast,93–95,118;relativistic,96–99;stationary,122

actionintegral,159,164–165;gauge-transformed,348;Lorentzforcelawand,230–231

actionprinciple,256–257;electromagnetismand,398;offieldtheory,122–124;Maxwell’sequationsand,285(table)

activetransformations,359n

AlphaCentauri,58

ampere,201

Ampère,André-Marie,279

Ampère’slaw,279,318–321,320(fig.)

angularmomentum,403,405–406

antisymmetrictensors,225–227,228,250–251,276n

antisymmetry,340

approximationformula,159

arbitraryscalarfunction,264

atmospherictemperature,117

atoms,406;sizeof,194–196

Avogadro’snumber,194–195

barmagnet,399(fig.)

Bianchiidentity:checking,302;Maxwell’sequationsand,299–302;specialcasesof,300n

binomialtheorem,89

Bohr,Niels,403

boson,Higgs,409

boundarypoints,122;ofspacetimeregions,122,123(fig.)

boundaryterms,349n,380n

calculus,xv,139,182;fundamentaltheoremof,304

canonicalmomentum,235;conjugate,356;defining,356–357;vectorpotentialand,374–375

Cartesianframe,3

chargedensity,286;defining,286;Maxwell’sequationsand,286–291;rectangle,289n;inspacetime,287(fig.)

charges,231;electrical,298;fieldsand,208–209;Seealsoconservationofcharge

classicalfieldtheory,115,148–149,392

classicalmechanics,xix,101–102;actionsin,366

classicalphysics,xix

clocks,17–19;moving,41–42

columnvector,216n

commanotation,334

complexfunctions,186n

components:contravariant,140,183–184,213;covariant,136–145,184–185;diagonal,227n,339;electricfields,339;of4-vectors,127–128,144;of4-velocity,84–85;of4-velocityintermsofaninvariant,87–88;ofLaplacian,415;magneticfields,379;ofmomentum,110;space,244,346;space-and-time,339;space-space,340,386n;time,237,346;undifferentiated,346;vectorpotential,342;ofvelocity,85

condensednotation,412

conjugatecanonicalmomentum,356

conservationofcharge:currentdensityin,294;electric,311–312;local,292(fig.),293(fig.);Maxwell’sequationand,291–296;neighborhood-by-neighborhoodexpressionof,296

conservationofenergy,387

conservedquantities,382

constants,21;coupling,159;ofnature,194;Newton’sgravitational,194;Planck’s,194,403,407

continuityequation,296;forenergy,384;4-vectorsand,351;gaugeinvarianceand,347;rewriting,297;satisfying,382

contravariantcomponents,140,183;negatives,184;notationof,213

contravariantindices,178,211

convention,341

conventionalunits,Lorentztransformationof,32–34

coordinatespace,94

coordinatesystems:distancein,49;lightraysand,17;synchronizationof,16–17

coordinates:rotationof,53;inspacetime,25;transformationof,26

cosine,131

coulomb,199;defining,201–202

Coulombfield:electric,397;magnetic,397

Coulombgauge,372

Coulomb’slaw,199,203,279,284,314–315

couplingconstant,159

covariantcomponents,136–145;Lorentztransformationand,137–138;notationof,223;transforming,184–185

covariant4-vectors,142

covariantindices,178,213,297–298

covarianttransformations,of4-vector,219

covariantvectors,182

curl,282–283;divergenceof,280–281,398;double,310;3-vectoroperators,413–414

currentdensity:inconservationofcharge,294;defining,289;inelectrodynamics,391;Lagrangianwith,346–351;Maxwell’sequationsand,286–291;nonzero,346–351;inrestframe,298;inspace,288(fig.);inspacetime,287(fig.)

degreesoffreedom,367;fieldtheoryand,116,365(fig.);representationof,363

derivatives,338n,370;calculationof,343–344;finite,370–371;Lagrangianformulationand,335n,364–365;notation,213–214;partial,299,343,345

diagonalcomponents,227n,339

differentials,137,143

differentiation,168–169,338n

dimensionlessfractions,90

dimensions,14–16

Dirac,Paul,xvi,398,401,408

Diracdeltafunction,165,172

Diracstring,400,402(fig.)

distance:incoordinatesystems,49;invariantsand,53;spacetime,50n

divergence:ofcurl,280–281;3-vectoroperators,412–413

dotproduct,181

doublecurl,310

double-crosstheorem,310n

downstairsindices,173–176;Einsteinsummationconventionand,177

dummyindex,177–178

Einstein,Albert,xvi,xviii,5,31;firstpaperonrelativityof,270–278;historicalcontextof,60–61;Maxwell’sequationsand,60;Minkowskiand,54–55;onrelativisticmomentum,101;onspacetime,23;onsynchronization,13

Einsteinsummationconvention,137,176–181,389;downstairsindicesand,177;scalarfieldand,180;upstairsindicesand,177

electricCoulombfield,397

electriccurrent,200–201

electricfields:components,339;underLorentztransformation,273–274;magneticfieldsand,273–274,277,404;Maxwell’sequationsand,282–284,328–329;waveequationsand,323

electricforces,207–208

electricmonopole,396,396(fig.)

electricalcharge,Lorentzinvarianceof,298

electrodynamics:currentdensityin,391;gaugeinvarianceand,388;Lagrangianformulationof,331–333;lawsof,311–324;Lorentzinvarianceand,388;Maxwell’sequationsand,312–313

electromagneticfields,228–229,317;gaugeinvarianceand,371–375;Lorentzforcelawand,247–248;momentumof,378–389;vectorpotentialof,242

electromagneticplanewave,332(fig.)

electromagneticstresstensor,390

electromagneticunits,198–203

electromagneticwaves,327–331

electromagnetism,xvi,xviii,31,313,334n;actionprincipleand,398;vectorpotential,377–378

electromotiveforce(EMF),273

electrons:primedframesof,273;velocityof,271–272

electrostaticpotential,237

EMF.Seeelectromotiveforce

energy,354;continuityequationfor,384;defining,108,383–384;density,368;equation,191;field,354–355;fieldpotential,369n;finite,370–371;infourdimensions,381–391;kinetic,107,119,369,376;Lorentzsymmetryand,384–387;inLorentztransformation,106;ofmasslessparticles,109,111;momentumand,360–362;ofobjects,107;ofphotons,112;potential,369;relativistic,104–106;rest,108

energy-momentumtensor,387–391

etherdrag,xvii–xviii

Euclideangeometry,48;righttrianglesin,49–50

Euclideanplane,49(fig.)

Euler-Lagrangeequation,120,156,230,243,336;infieldtheory,127–129;forfields,342;leftsideof,237;Lorentzforcelawand,230,234–243;formotionequations,185;formultidimensionalspacetime,125–126;particlemotionand,159;rightsideof,237,346;forscalarfield,126,152–154;SeealsoLagrangian

events,26;simultaneous,19;spacelike,59n

exactexpansion,89

exponentials,forwaveequations,185–186

fakemonopoles,407–408

Faraday,Michael,xvi,312,318

Faraday’slaw,279,284,315–318,318(fig.),404

field:action,117,167,366;analogy,365(fig.);chargesand,208–209;energy,momentumand,354–355;equations,333;Euler-Lagrangeequationfor,342;Hamiltonianfor,367–370;Higgs,152;Lagrangianfor,363–366;momentum,357;mystery,151–152;particlesand,149–151,158–168;relativistic,132;spacetimeand,116–117;vector,117,305;Seealsospecificfields

fieldpotential,128;energy,369n

fieldtensor,250–253,345;Lagrangianand,373;Lorentztransformationof,253,274–277

fieldtheory,xvi,362;actionprincipleof,122–124;classical,115,148–149,392;degreesoffreedomand,116,365(fig.);elementsof,364(fig.);Euler-Lagrangeequationin,127–129;Lagrangianin,128,147–149;localityand,258;quantum,392,408;representationof,117

fieldtransformations,144(table);properties,133–136,134(fig.)

finitederivatives,370–371

finiteenergy,370–371

FitzGerald,George,31

fixeddifferences,36n

fixinggauges,372

flow,291

flux,291,386n,404

forces:electric,207–208;gravitational,207;long-range,207;magnetic,207,209;short-range,207

4-acceleration,246

4-momentum,110,385

4-vectors,220–221;componentsof,127–128,144;continuityequationand,351;covariant,142;covarianttransformationsof,219;defining,74;differencesof,174;fields,134;introductionto,72;lightraysand,70–71;Lorentztransformationof,74,138–139,141,274–275;Maxwell’sequationsinformof,326(table);notation,72–75,211–212;relativisticenergyand,106;representationof,175;scalarfieldsfrom,144–146;scalarsand,282n;spacecomponentsof,244;transformationpropertiesof,74

4-velocity:componentsexpressedasinvariants,87–88;componentsof,84–85;constructing,75–76;defining,75,84–85;equationswith,246;Lorentzforcelawand,246;Lorentztransformationof,85;Newtonand,86–87;slowerlookat,84–85;summaryof,88;three-dimensionalvelocitycomparedwith,85

four-dimensionalspacetime,81n

fourthdimension,timeas,55

frameindependence,94

frames.Seespecifictypes

Franklin,Benjamin,272–273,348

freeindex,178

fundamentalprinciples,255

fundamentaltheoremofcalculus,304

futurelightcone,55,71(fig.)

Galileo,xvii,5,61

gaugeinvariance,255;continuityequationand,347;defining,259,264;electrodynamicsand,388;electromagneticfieldsvia,371–375;Lagrangianformulationand,337–338,350;motionand,265–266;perspectiveon,267–268;symmetryand,259–262;vectorpotentialand,371

gauge-transformedactionintegral,348

Gauss’stheorem,304–307,312;illustrationof,305(fig.)

generalLorentztransformation,36

generalrelativity,392

generalizedchainrule,139

geometricrelationships,210

gravitationalattraction,6

gravitationalforces,207

Greekindices,244–245

Hamiltonian,361–362;forfields,367–370;formalconstructionof,376

Hamiltonianmechanics,92

Hamiltonians,defining,104

harmonicoscillators:accelerationtermfor,188;Lagrangianof,187

HeisenbergUncertaintyPrinciple,197

Higgsboson,409

Higgsfield,152

higher-orderderivatives,146

hydrogen,195

hypotenuse,45,49–50

identitymatrices,176

indices:contracting,179,214;contravariant,211;covariant,213,297–298;downstairs,173–177;dummy,177–178;free,178;Greek,244–245;Latin,244–245;notation,73,414;raisingandlowering,224–225;summation,177–178,239n,241,380;upstairs,173–177

inertialframes,intwinparadox,45

inertialreferenceframes(IRF),5–7

infinitemomentum,104

integrand,258

interactionterms,167

intervals:lightlike,58,60,71(fig.);spacelike,82–83;spacetime,50n,57–58,73;timelike,84

invariance,262–265;Lorentz,128,132,259,260;spacetimetrajectoryand,263;oftensorequations,223;Seealsogaugeinvariance

invariants,48;defining,49,56;distanceand,53;4-velocitycomponentsexpressedas,87–88;manifestly,245;inMinkowskispace,50–51;timeand,53,55;velocitycomponentsas,87–88

IRF.Seeinertialreferenceframes

kineticenergy,107,369;expressionof,119;Lagrangianas,376

Klein-Gordonequation,188–190

Kroneckerdelta,166

labels,363

Lagrangian,98–99,104,156,261;inabsenceofsources,338–341;density,257,334,357,366,374;derivativesand,335n,364–365;fieldtensorand,373;infieldtheory,128,147–149;forfields,363–366;gaugeinvarianceand,337–338,350;ofharmonicoscillators,187;andkineticenergy,376;localityand,334–335;Lorentzforcelawand,232–234;Lorentzinvarianceand,146,335–337;ofnonrelativisticparticles,119;withnonzerocurrentdensity,346–351;notationof,343;particles,149–150;physicalmodels,363–364;relativistic,145–147;timeintegralof,362

Lagrangianformulationofelectrodynamics,331–333

Lagrangianmechanics,92

Laplace,Pierre-Simon,310

Laplacian,322;componentsof,415;3-vectoroperators,415

Latinindices,244–245

length,speedoflightand,15

lengthcontraction,37–40,41(fig.)

light:accelerationand,355(fig.);compositionof,321;polarizationof,331;propertiesof,331

light,speedof,6–7,35–36;lengthand,15;momentumof,104;ofrelativisticunits,22;timeand,15

lightcones,70–71

lightrays:coordinatesand,17;4-vectorsand,70–71;lightconesand,70–71;movementof,70–71;slopeof,9;velocityof,10–11

lightlikeseparation,58,60

LimoandBugparadox,46;simultaneityin,47–48;spacetimediagramof,47(fig.)

linearalgebra,xv

locality:defining,257–258;fieldtheoryand,258;Lagrangianformulationand,334–335;Maxwell’sequationsand,334;quantummechanicsand,258

locallyconservedquantities,381–384

lodestone,209

long-rangeforces,207

Lorentz,Hendrik,31;historicalcontextof,61–62

Lorentzboosting,101

Lorentzcontraction,31,298–299;formula,86

Lorentzforcelaw,205–206,210;actionintegraland,230–231;electromagneticfieldsand,247–248;Euler-Lagrangeequationand,230,234–243;4-velocityand,246;illustrationof,273;nonrelativisticversionof,229;relativisticversion,229;with3-vectors,229;vectorpotentialand,230–231,247–248

Lorentzgauge,372

Lorentzinvariance,128,132,260,299–300,338;defining,259;ofelectricalcharge,298;electrodynamicsand,388;Lagrangianformulationand,146,335–337;ofMaxwell’sequations,302

Lorentzinvariantequations,243–246

Lorentzsymmetry:energyand,384–387;momentumand,384–387

Lorentztransformation,101;combinationof,64;ofconventionalunits,32–34;covariantcomponentsand,137–138;energyin,106;offieldtensor,253,274–277;of4-vectors,74,138–139,141,274–275;of4-velocity,85;general,36–37,214–218;magneticfieldsunder,273–274;Maxwell’sequationsand,61;Minkowskispaceand,50–51;momentumin,106;notationand,215;forrelativemotion,52;spacelikeintervalsand,83;alongxaxis,215–217;alongyaxis,217

Lorentztransformations,26–29;discoveryof,31;equations,32–36;betweenrestandmovingframes,30–31

magneticCoulombfield,397

magneticfields,373;components,379;electricfieldsand,273–274,277,404;underLorentztransformation,273–274;Maxwell’sequationsand,281–282,328–329

magneticforces,207,249;discoveryof,209

magneticmonopole,396,407

magnets,moving,270,272(fig.)

magnitude,111

manifestlyinvariant,245

mass:defining,108;rest,108;timesacceleration,248

masslessparticles:energyof,109,111;velocityof,109

matrixmultiplication,218

Maxwell,JamesClerk,xvi–xvii,31,312–313,324

Maxwell’sequations,228–229,269,351,383,397;actionprincipleand,285(table);Bianchiidentityand,299–302;chargedensityand,286–291;consequenceof,296;conservationofchargeand,291–296;currentdensityand,286–291;derivationof,341–346,347;derivationsfromvectorpotential,285(table);Einsteinand,60;electricfields

and,282–284,328–329;electrodynamicsand,312–313;4-vectorform,326(table);homogeneous,283;inhomogeneous,286;introductionto,279–280;aslawofphysics,60–61;localityand,334;Lorentzinvarianceof,302;Lorentztransformationand,61;magneticfieldand,281–282,328–329;non-identity,284–286;numberof,280;second,317,323;symmetryof,61;tensorform,297–299;3-vectorform,326(table);vectoridentitiesand,280–281;vectorpotentialand,337

Maxwell’slaw,321–324

mechanicalmomentum,355–356

metersticks:lengthsof,40;inmovingframes,39;inrestframe,40;instationaryframe,40

metric,175

Michelson,Albert,xvii,xix

Michelson-Morleyexperiment,60

Minkowski,Hermann,25,48–53;Einsteinand,54–55

Minkowskilightcone,54,54(fig.);future,55,71(fig.);past,55

Minkowskispace,26,245;invariantsin,50–51;Lorentztransformationand,50–51

Minkowskispacetime,124

momentum:angular,403,405–406;canonical,235,356–357,374–375;componentsof,110;conservationof,101;density,376–381;ofelectromagneticfields,378–389;energyand,360–362;field,357;fieldenergyand,354–355;flowof,386;infourdimensions,381–391;4-momentum,110,385;function,109;infinite,104;Lorentzsymmetryand,384–387;inLorentztransformation,106;mechanical,355–356;Noether,358–360;ofphotons,112;relativistic,100–104;ofspeedoflight,104;typesof,354–360

monopole:defining,396;electric,396,396(fig.);fake,407–408;magnetic,396,407

Morley,Edward,xvii,xix

motion:equationof,156,168–171,185,265–266,350–351;Euler-Lagrangeequationsfor,185;gaugeinvarianceand,265–266;Newton’sequationof,130;unforced,130

movingframe,7,16,28;Lorentztransformationbetweenrestframeand,30–31

movingframes,metersticksin,39

multidimensionalspacetime,Euler-Lagrangeequationsfor,125–126

multivariablegeneralizations,139

mysteryfields,151–152

neutrino,408–409

neutron,408–409

newscalar,183–184

Newton,Isaac,xvii,5;4-velocityand,86–87;onspacetime,23

Newtonianapproximations,89–92

Newtonianexpectations,ofvelocity,68–69

Newtonianframes,7–11,8(fig.);coordinatesin,9

Newtonianmechanics,6,103–104

Newtonianphysics,velocityin,64

Newton’sequationofmotion,130

Newton’sgravitationalconstant,194

Newton’ssecondlaw,103–104,121

Newton’stransformationlaw,11

Newton’suniversallawofgravitation,196–197,207–208

Noether,Emmy,381

Noethermomentum,358–360

Noethershift,377(fig.)

Noether’stheorem,359n

nonrelativisticaction,95–96

nonrelativisticlimits,99–100

nonrelativisticmomentum,102

nonrelativisticparticles:Lagrangianof,119;redux,118–121;trajectory,119(fig.)

nonzerocurrentdensity,Lagrangianwith,346–351

notation:comma,343;condensed,412;ofcontravariantcomponents,223;conventions,216n;ofcovariantcomponents,223;derivatives,213–214;extending,210;of4-vectors,72–75,211–212;importanceof,173;index,73,414;ofLagrangian,343;Lorentztransformationand,215;scalars,212;3-vector,332–333

objects,energyof,107

observer:inrelativemotion,17–19;andreferenceframes,4;stationary,40;third,ofvelocity,66–70

origin:propertimefrom,73;spacetimeintervalfrom,73

Ørsted,HansChristian,321

Ørsted’slaw,321

paradoxes:LimoandBug,46–48,47(fig.);twin,43–46,44(fig.)

parallelwires,200(fig.)

partialderivatives,299,343,345

particlemechanics:defining,92;positionin,92;velocityin,92

particlemotion,123–124

particles,156–157;accelerationof,172–173;actionof,163;Euler-Lagrangeequationandmotionof,159;fieldsand,149–151,158–168;Lagrangian,149–150;massless,109,111;nonrelativistic,118–121,119(fig.);positionof,238;inscalarfield,162(fig.);slow,106–108;timelikeparticletrajectory,93(fig.)

pastlightcone,55

ϕ,117–118;asscalarfield,126;stationaryactionfor,124–127;valueof,121

photons,111;energyof,112;momentumof,112

physics,lawsof,6–7;constantsofnature,194;Maxwell’sequationsas,60–61

Planck,Max,xvi

Planckunits,196–198;trafficsignin,198(fig.)

Poincaré,Henri,31

Poisson’sequation,170–171

polarization,328;oflight,331

position,inparticlemechanics,92

position-dependentmass,153

positron,112

positroniumdecay,111–113

potentialenergy,369

Poynting,JohnHenry,384

Poyntingvector,380,390,391;defining,384

primedframes,ofelectrons,273

principleofleastaction,93–95,118

principleofrelativity,6–7

propertime,50n,127;defining,52;fromorigin,73;physicalmeaningof,55–57

Pythagoreantheorem,49;three-dimensional,53

quantumfieldtheory,392,408

quantummechanics,xvi,xix;localityand,258

radiationgauge,372

railroads,52

rank2tensors,220–222

realfunctions,186

redundancyofdescription,267

referenceframes(RF),3;inertial,5–7;observerand,4;physicalrelationshipbetween,29;symmetryof,29;SeealsoNewtonianframes;SRframes

relativemotion:Lorentztransformationfor,52;alongxaxis,52

relativisticacceleration,248–249

relativisticaction,96–99

relativisticenergy,104–106;4-vectorand,106

relativisticfields,132

relativisticLagrangian,building,145–147

relativisticlaw:foradditionofvelocity,68;ofmotion,79–80

relativisticmomentum,100–104;Einsteinon,101

relativisticunits:speedoflightof,22;SRframesusing,18(fig.)

relativisticvelocity,102–103

relativity,xviii,368n;Einstein’sfirstpaperon,270–278;general,392;principleof,6–7;signconventionsin,58n;specialtheoryof,xix–xx,xviii

restenergy,108

restframe,7;currentdensityin,298;Lorentztransformationbetweenmovingframeand,30–31;metersticksin,40

restmass,108

RF.Seereferenceframes

righttriangles:inEuclideangeometry,49–50;twinparadoxand,50

Right-HandRule,Stoke’stheoremand,308(fig.)

ring,405(fig.)

scalarfields,117,209;conventions,181–183;defining,133;derivativesof,182;Einsteinsummationconventionand,180;Euler-Lagrangeequationfor,126,152–154;from4-vectors,144–146;particlesin,162(fig.);ϕas,126

scalars,116,231,368n;forming,212;4-vectorsand,282n;notation,212

scale,unitsand,194–196

Schrödingerequation,188–189,189n

short-rangeforces,207

signconventions,inrelativity,58n

simultaneity,13;inLimoandBugparadox,47–48;surfaceof,22

simultaneousevents,19

sine,131

single-indextransformationrule,276

slowparticles,106–108

solarsail,355(fig.)

solenoid,398,405(fig.);elongated,401(fig.);flexible,400;stringlike,403(fig.)

sourcefunction,164

space:currentdensityin,288(fig.);dimensionsof,116,118;timeand,25

spacecomponents,346;of4-vectors,244

space-and-timecomponents,339

spacelikeevent,59n

spacelikeintervals,82;Lorentztransformationand,83

spacelikeseparation,58–59

space-spacecomponents,340,386n

spacetime:chargedensityin,287(fig.);coordinatesin,25;currentdensityin,287(fig.);Einsteinon,23;fieldsand,116–117;four-dimensional,81n;Minkowski,124;multidimensional,125;Newtonon,23;samenessof,26;separation,70–71

spacetimediagram,7,15,22;ofLimoandBugparadox,47(fig.)

spacetimedistance,50n;defining,57–58;describing,57–58

spacetimeinterval,50n,57–58;fromorigin,73

spacetimetrajectory,76(fig.);invarianceand,263

specialtheoryofrelativity,xix–xx,xviii

sphere,306

SRframes:synchronizationin,12–14;usingrelativisticunits,18(fig.)

standardvectornotation,380

stationaryaction,122;forϕ,124–127

stationaryframe,28;metersticksin,40

Stokes’stheorem,307–309,319–320;Right-HandRuleand,308(fig.)

submatrix,3x3,253

subscript,73n

summationconvention,350

summationindex,177–178,239n,241,380

superscript,73n

surface,304;ofsimultaneity,22

symmetrictensors,225–227,250–251

symmetry:gaugeinvarianceand,259–262;leftandright,27;ofMaxwell’sequations,61;operations,359;ofreferenceframes,29;spherical,305;ofvectorfield,305

synchronization:ofcoordinatesystems,16–17;Einsteinon,13;inSRframes,12–14

tensorequations,invarianceof,223

tensorformofMaxwell’sequations,297–299

tensors,210;antisymmetric,225–227,228,250–251,228,276n;electromagneticstress,390;energy-momentum,387–391;field,250–253,274–277,345,373;ofhigherrank,222–223;rank2,220–222;symmetric,225–227,250–251;two-indexed,250–251,275,342;untransformed,222

ThalesofMiletos,207

theoremwithoutaname,309–310

3x3submatrix,253

3-vectoroperators,411–415;curl,413–414;divergence,412–413;gradient,412–413;Laplacian,415

3-vectors:Lorentzforcelawwith,229;Maxwell’sequationsinformof,326(table);notation,332–333

three-dimensionalvelocity,4-velocitycomparedwith,85

time,5;dimensionsof,116;asfourthdimension,55;invariantsand,53,55;proper,50n,55–57,127;spaceand,25;speedoflightand,15

timecomponents,346;ofvectorpotential,237

timedependence,171–173

timederivative,238

timedilation,41–43,43(fig.);formula,86

timedistance,45

timeintegral,ofLagrangian,362

timetranslations,257

timelikeintervals,84

timelikeparticletrajectory,93(fig.)

timelikeseparation,58–59

timeliketrajectory,83(fig.)

torque,405

trafficsigns,198(fig.)

transformationequations,28

translationsymmetry,358

transversewaves,328

triangles,right,49–50

TTM,xvi

twinparadox,43–46,44(fig.);inertialframesin,45;righttrianglesand,50

2x2unitmatrix,217–218

two-indexedtensors,250–251,275,342

undifferentiatedcomponents,346

unforcedmotion,130

uniformmotion,36

units,14–16;conventional,32–34;electromagnetic,198–203;Planck,196–198,198(fig.);scaleand,194–196

universaltime,13

unprimedframes,35

unprimedzaxis,276

upstairsindices,173–176;Einsteinsummationconventionand,177

vectoralgebra,209

vectorfield:symmetryof,305;velocityas,117

vectoridentities:Maxwell’sequationsand,280–281;two,280–281

vectorpotential:canonicalmomentumand,374–375;components,342;ofelectromagneticfields,242;electromagnetism,377–378;gaugeinvarianceand,371;Lorentzforcelawand,230–231,247–248;Maxwell’sequationsand,337;Maxwell’sequationsandderivationsfrom,285(table);timecomponentof,237

vectors.See4-vector;3-vectors

velocity:adding,64–70;combining,65(fig.);componentsasinvariant,87–88;componentsof,85;ofelectrons,271–272;integraloffunctionof,98;oflightrays,10–11;ofmasslessparticles,109;negative,8,27;Newtonianexpectationsof,68–69;inNewtonianphysics,64;inparticlemechanics,92;positive,9;relativistic,102;relativisticlawforadditionof,68;thirdobserverof,66–70;three-dimensional,85;asvectorfield,117;Seealso4-velocity

velocity-dependentterms,241

waveequations,130–131,169;electricfieldsand,323;exponentialsfor,185–186

wavenumber,327

waves,130–131,187–190;electromagnetic,327–331;electromagneticplane,332(fig.);numbers,191;transverse,328

worldline,17,26,94n

xaxis:Lorentztransformationalong,215–217;relativemotionalong,52

x1axis,19–25;SRframeswith,23(fig.)

yaxis,Lorentztransformationalong,217

y1axis,SRframeswith,23(fig.)

Yang-Millstheory,256,267

zaxis,unprimed,276

zero-over-zeroconundrum,109

zerothequation,245–246,248

z-momentum,386