c is the speed of light, isn’t it?

c is the speed of light, isn’t it?George F. R. Ellisa)

Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7700,Capetown, South Africa

Jean-Philippe Uzanb)

Institut d’Astrophysique de Paris, GReCO, FRE 2435-CNRS, 98bis boulevard Arago, 75014 Paris, Franceand Laboratoire de Physique The´orique, CNRS-UMR 8627, Universite´ Paris Sud, Baˆtiment 210,F-91405 Orsay Ce´dex, France

~Received 24 February 2004; accepted 24 September 2004!

Theories for a varying speed of light have been proposed as an alternative way of solving severalstandard cosmological problems. Recent observational hints that the fine structure constant mayhave varied over cosmological scales have given impetus to these theories. However, the speed oflight is hidden in many physics equations and plays different roles in them. We discuss these rolesto shed light on proposals for varying speed of light theories. We also emphasize the requirementsfor attaining consistency of the resulting equations, when what was previously a constant is madea dynamical variable. ©2005 American Association of Physics Teachers.

@DOI: 10.1119/1.1819929#

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I. INTRODUCTION

Recent measurements of distant quasar absorption sphave lead to the claim that the fine structure constant mhave been smaller in the past,1,2 and have restarted a debaon the nature of various constants of nature. Many new~andsharper! experimental constraints on the variation of theconstants on different time scales have been determineda large variety of physical systems.3,4 The constancy of theconstants of physics has its origin in Einstein’s equivaleprinciple which states, beside the universality of free fathat the result of any nongravitational experiment shouldindependent of position in space and time and of the veloof the laboratory in which it is performed, two propertiereferred to as local position and local Lorentz invariancedemonstration that some constants have varied duringhistory of the universe would be a sign of the violationlocal position invariance and of the existence of a new fothat would likely be composition dependent.3,5 Motivationsfor such a variation have been proposed, mostly in the frawork of higher dimensional theories such as string theo6

Among the various proposed theories, the varying speelight models claims to be an alternative way~compared toinflation! of solving the standard cosmological problems,7–10

that is, the horizon and flatness problems.11 A recent study ofblack holes12 suggested that a variation of the speed of ligcan be differentiated from a variation of the elementacharge.13

It is well known that only the variation of dimensionlesquantities can be determined,3,14 mainly because measuringphysical quantity reduces to a comparison with a physsystem that is chosen as a reference. The values of thedard units depend on historical definitions, and the numervalues of the constants of nature depend on these choWhat is independent of the definition of the units are dimsionless ratios that, for example, characterize the relativeand strength of two objects or forces.15 Although only thevariation of dimensionless constants is meaningful, weimplement a theory in which the dimensionless constavary by assuming that a dimensional constant is varywhile specifying clearly what other quantities are kept fixe

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This assumption amounts to choosing a preferred systemunits. An example is Dirac theory16 in which the gravita-tional constant varies as the inverse of the cosmic timethe atomic units used by Dirac, the electron mass,me , isfixed as well as Planck’s constant and the speed of lightthat Dirac’s hypothesis corresponds to a model in whichdimensionless ratioGme

2/\c is varying. In this theory,atomic clocks, based for example on atomic transitioslowly drift with respect to gravitational clocks based, fexample, on planetary orbits.17

The role and status of the fundamental constants of phics have been widely debated. We shall define these cstants as the set of physical parameters that are not dmined by the theory we are considering. They afundamental in the sense put forward by Weinberg:18 ‘‘Wecannot calculate@them# with precision in terms of more fundamental constants, not just because the computation iscomplicated@...#, but because we do not know anything mofundamental’’~page 249 of Ref. 18!. These constants can bdivided ~nonuniquely! into two groups: dimensionless ratio~which are the fundamental parameters and are pure nbers! and fundamental units,14 which have dimensions. Howmany such fundamental units are needed is still debatedadd to this debate,14 we recall a central property of the fundamental units: each of them has acted as a ‘‘concsynthesizer,’’19,20 that is, each has unified concepts that wepreviously disconnected. For example, Planck’s constan\and the relationE5\v can be interpreted not only as a linbetween two classical concepts~energy and waves!, butrather as creating a new concept of broader scope, of wenergy and waves are just two facets. The speed of lightplayed such a synthesizing role by leading to the concepspace-time, as well as~with Newton’s gravitational constant!creating the link, through the Einstein equations, betwespacetime geometry and matter.19,20These considerations, awell as the fact that only three independent units~mass, dis-tance and time! are required to describe all the physicquantities,3,14 suggest that only three such quantities aneeded. For instance, from the gravitational constant andPlanck constant, one can construct units of length, mass

240p © 2005 American Association of Physics Teachers

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time known as the Planck length, mass and time. Inparticular system of units, the numerical values of these cstants reduce to 1.

The numerical values of the fundamental units are atrary and can be chosen to be unity in a natural systemunits, such as the Planck units, but their role as conceptthesizer remains. If these constants were to vary, thenalso would need to relax the synthesis they underpin. Itlows that we would need to develop a careful conceptframework to implement their possible variation. Our goato discuss in this context a class of theories in whichconstancy of the speed of light is relaxed. To help in undstanding the difficulty in building such a meaningful theowe examine the role of the speed of light in physical theand recall the constraints involved in turning a constant ia dynamical variable.

II. A SHORT HISTORY OF c

During antiquity, it was believed that our eyes were torigin of light and that its speed was infinite.19,20 We had towait for Galileo for this view to change. He was the firsttry to measure the speed of light experimentally, but hisperiment was unsuccessful.21 Ironically, by discovering thesatellites of Jupiter in 1610,22 he opened the door to the firsdetermination of the speed of light by the astronomer ORoemer in 1676.23 His measurement was then improvedJames Bradley in 1728, who utilized his discovery of taberration of light.24

During this period the status of the speed of light wasdifferent from that of the speed of sound: it was simplyproperty of light. Huygens proposed a description in termswaves, contrary to Newton’s corpuscular description. InNewtonian corpuscular approach to optics, the speed of lwas not a constant and had the same status as the speany other body. The wave description was supported expmentally by Foucault in 1850 who verified that the speedlight was smaller in a refractive medium than in vacuuThis behavior lead most physicists to believe that lightquires a medium, theether, in which to propagate. The speeof light was thus a property of the ether, in the same waythe speed of sound depends on the properties of the mein which it propagates. The speed of light was not thoughbe fundamental.

In 1855, Kirchhoff realized that («0m0)21/2 has the dimen-sion of a speed, where«0 andm0 are two constants enterinthe laws of electricity and magnetism. Weber and Kohlraumeasured this constant in 1856, using only electrostaticmagnetostatic experiments~see Ref. 25 for an historical presentation of these developments!. Within experimental accu-racy, it agreed with the speed of light. This agreementmained a coincidence until Maxwell formulated his theoryelectromagnetism in 1865 and concluded that ‘‘light iselectromagnetic disturbance propagated through the fieldcording to electromagnetic laws.’’26 At this stage, the statuof c increased tremendously because it became a charaistic of all electromagnetic phenomena. Note thatc is notonly related to a velocity of propagation, because it canmeasured by electrostatic and magnetostatic experiment

The next mutation ofc arose from the incompatibility oMaxwell’s equations with Galilean invariance: either wstick to Galilean invariance and Maxwell’s equations hoonly in a preferred frame, so that measurements of thelocity of light would allow one to determine this preferre

241 Am. J. Phys., Vol. 73, No. 3, March 2005

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frame, or we have to discard Galilean invariance. TMichelson–Morley experiments and Einstein’s interpretatimplied we should follow the second road: Galilean invaance was replaced by Lorentz invariance, andc triggered the~special! relativity revolution. The speed of light became thlink between space and time and hence entered most olaws of physics, mainly because it enters into the notioncausality. For example, it later became apparent thatc also isthe speed of propagation of gravitational waves and thaany massless particle.

The last change occurred in 1983 and was driven byperimental needs. Up until then, the accuracy of the expmental determination ofc was limited by the accuracy anreproducibility of the realization of the meter. The BIPM~Bureau International des Poids et Mesures! redefined themeter in terms of a value ofc that was fixed by fiat. Thischoice of definition reflected Synge’s emphasis27 that thebest way to measure distance over large length scales iradar or equivalent methods dependent on the speed of lso that the best units for distance are light-seconds or ligyears. This realization was turned into technological reaby surveying instruments such as the Tellurometer28 andmore recent developments such as the GPS system wused for navigation.

We conclude from this summary that the speed of light hchanged from a simple property of light to the role offundamental constant29 that enters many laws of physics thare apparently disconnected from the notion of light itsel

III. ONE CONSTANT WITH MANY FACETS

As Sec. II shows,c is not only the speed of light, and wshall see that it is not even always the speed of light. It icomplex quantity that turns out to have different origins thlead to coincident values.

A. cEM : The electromagnetism constant

Maxwell equations in MKSA units are30

“"D5r, “"H50, “ÃH5J1] tD,

“ÃE52] tB, ~1!

where r is the density of free charge andJ is the currentdensity. The displacement,D, is related to the electric fieldE, and the polarization,P, by D5«oE1P and the magneticfield, H, is related to the magnetic induction~magnetic fluxdensity!, B, and the magnetization,M , by B5m0(H1M );«0 andm0 are respectively the permittivity and permeabiliof the vacuum.

We introduce the potential by the standard definitionE52“f2] tA andB5“ÃA ~which is unique up to a gaugtransformation!. In vacuum, it is easy to show that the wavequation for an electromagnetic wave is

~] t22cEM

2“ !~f,A!50, ~2!

where“ is the Laplacian, and

cEM2 [

1

«0m0. ~3!

Here cEM appears as the velocity of any electromagnewave, and thus of light in a vacuum.

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Calling cEM the speed of light is too restrictive becauseis characteristic of any electromagnetic phenomena. It shobe referred to as theelectromagnetism constant.

By settingx05cEMt, Maxwell’s equations can be recastthe form

]mFmn5 j n, ~4!

with Am5(f,A/cEM), j m5(r,J/cEM), and Fmn5]mAn

2]nAm , with m,n50,...,3. If we impose the Lorentz gaug(]mAm50), we recover the propagation equation forAm, Eq.~2!. Here the wave equation constant is embodied in thetation becausex0 is defined in terms ofcEM . Of course in arelativistic framework, we would obtain the same equatiobecause Maxwell electrodynamics is Lorentz invariant.

B. cST : The spacetime constant

The next role of the speed of light that we have mentionis as a synthesizer between space and time. This conenters the Lorentz transformation and the spacetime destion of special relativity.31

Following Ref. 32, we call the constant that appears inLorentz transformations the spacetime structure constcST, so that the Minkowski line element takes the form

ds252~dx0!21~dx1!21~dx2!21~dx3!2 ~5a!

52~cSTdt !21~dx1!21~dx2!21~dx3!2. ~5b!

It can be shown thatcST can be defined~completely indepen-dently of electromagnetism! as the universal invariant limispeed31 so that speeds are not additive, even though tmay be approximately so at low speeds. The conditionsthe composition of speed~denoted by%! satisfies are (i ) ithas an identity element,O ~that is,O% u5u% O5u for allu), (i i ) there is a universal element, in our casecST, suchthat cST% u5u% cST5u for all u, (i i i ) the associative ruleu% (v % w)5(u% v) % w, (iv) the differentials d(u% v)/duand d(u% v)/dv exist and are continuous inu andv, and (v)d(u% v)/du.0 and d(u% v)/dv.0 providedu,vÞ0. Theseconditions lead to the standard velocity combination rulesspecial relativity withcST as the limiting speed.31 ~Interest-ingly most of these constructions are based on axiomsare universal principles, and are not determined by the perties of any specific interaction of nature.! The limitingspeed is then given by the metric~5b! through the equationds250, corresponding to motion at the speedcST.

Finally we stress that the special relativity conversion ftor between energy and mass is given bycST

2 because it arisesfrom the study of the dynamics of a point particle~see, forexample, Ref. 31!. Thus the correct equation unifying the thconcepts of energy and mass isE5mcST

2 .We now consider the case of the curved spacetime of g

eral relativity and the corresponding measurement oftances, and consider for that purpose a spacetime withelement ds25gmndxmdxn. Observers can define their proptime by the relation

ds252cST2 dt2, ~6!

which is the most natural choice because it implies the liting speed given by ds250 is the same for all observerindependently of their position in space and time and of th


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motion; it also gives the standard time dilation effeClearly, we have that

2cST2 dt25g00~dx0!2 ~7!

for any observer at rest relative to the coordinate sys(xa). In the standard case wherecEM5cST, we can definethe spatial distance, d,, between two points with coordinatexi andxi1dxi as the radar distance given bycST/2 times theproper time measured by an observer located atxi for a sig-nal to go and return betweenxi andxi1dxi . ~The factor of1

2

is included because the light goes out and returns, so thpropagates twice the distance between the points.! It follows~see, for example, Ref. 33! that

d,25g i j dxidxj , ~8!

with g i j 5gi j 2g0ig0 j /g00 and i , j 51,2,3. Thus the determination of distance requires both the measurement of timethe use of a signal exchanged between the distant pointthe standard case, there is no ambiguity because we canlight that propagates at the universal speedcST5cEM . Ifthese velocities do not coincide, the determination of dtance would become more involved because light will folloa timelike ~and not a null! geodesic of the metric. Furthermore, in any situation where this standard matter-geomcoupling does not hold, we need to be told what geomemeaning, if any, the metric tensor has, and by what altertive method time and spatial distance is to be determine

C. Agreement ofcEM and cST

The historical path was from electrodynamics to the deonstration that the speed of light was constant to the Loretransformation and the group structure of spacetime. Thewas realized from the study of relativistic dynamics that aparticle with vanishing mass will propagate with the speedlight. But the speed of lightcEM agrees with the universaspeed,cST, only to within the experimental precision oMichelson–Morley type experiments~or, put differently, thephoton has zero mass only within some accuracy!, and thecausal cone need not coincide with the light cone. If we wto show experimentally that the photon has nonzero mthen the standard derivation of relativity from electromagntism would have to be abandoned.

It is clear that we can retain the basic principles of a metheory of gravity and obtain a variable speed of light,cEM ,by modifying Maxwell equations.34 Many ways can be fol-lowed and many terms can be added to the electromagnLagrangian. In the Proca theory, Maxwell’s equation is mofied to obtain a massive spin 1 state,35

]mFmn1m2An50. ~9!

It follows that the Lorentz gauge condition is automaticasatisfied if the photon is massive, but we have lost the frdom of gauge transformation. Teyssandier36 consideredterms that do not violate gauge invariance and couple tocurvature as

LEM5 14 ~11jR!FmnFmn1 1

2 hRmnFmrFrn

1 14 zRmnrsFmnFrs, ~10!

whereR is the Ricci scalar. From this Lagrangian, he cocluded that


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when the Ricci tensor takes the form 3Rmn5(4U2R)umun

1(U2R)gmn , whereum is a unit timelike vector field. Wealso emphasize that in quantum electrodynamics, the phhas an effective mass due to vacuum polarization37 usuallydescribed by adding the Euler–Heisenberg Lagrangian38 tothe standard electromagnetic Lagrangian.

We see that, on the one hand, if the universal speefixed by electromagnetism, it is difficult to understand wother interactions are locally Lorentz invariant. On the othhand, the invariance of the laws of physics under the Pocare group allows the existence of massless particles,does not imply that every gauge boson must be masslesis therefore important conceptually to carefully distinguithe speed of light,cEM , from the universal speed,cST, thatdictates the properties of spacetime.

D. cGW : The speed of gravitational waves in vacuum

As long as we are in a vacuum, the Einstein equatiderived from the action

S5E RA2g d4x ~12!

can be linearized around Minkowski spacetime. It canshown that the only degrees of freedom are massless statspin 2 that are gravitational waves. With the line elemen

ds252~cSTdt !21~h i j 1hi j !dxidxj , ~13!

andhi j hi j 50, ] ih

i j 50, the linearized Einstein vacuum fielequations reduce to the propagation equation

~] t22cST

2“ !hi j 50, ~14!

so that the speed of propagation of gravitational wavegiven by the universal speed of our spacetime,cGW5cST. Aslong as we assume that general relativity is valid, this cclusion cannot be avoided. If we were able to formulatetheory with light massive gravitons~see, for example, Ref39 for some attempts!, then the speed of propagation of graity might be different from the universal speed and woulead to a modification of Newton’s law of gravitation oastrophysical scales.40 We emphasize that there are dangous issues associated with the presence of extra-polarizstates of massive gravitons.41 Among other problems,41 thesemassive gravitons have a scalar polarization state whosepling to matter does not depend on the mass of the gravand that is, at the linear level, analogous to a Jordan–BraDicke coupling withv50. This coupling would modify thestandard relation of interaction between matter and light bfactor of 3

4 so that we would expect a 25% discrepancy wthe predictions of general relativity in the tests of gravitatiin the solar system, which are actually at a level of 0.01%42

From this discussion, it seems difficult to consider a spof gravity that differs fromcST. Nevertheless, it is good tokeep an open mind and not forget that these two speedsdiffer.

E. cE : The „Einstein… spacetime-matter constant

Let us now consider gravity coupled to matter. The Estein field equations,


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involve a coupling constanta/258pG/c4 between the Ein-stein tensorGmn and the stress-energy tensorTmn , wherechas the dimensions of speed; let us call itcE. To interpretcE, we need to consider the weak field limit of the fieequations~15! in which the spacetime metric is given by

ds252~11h00!~cSTdt !21h i j dxidxj . ~16!

The geodesic equation of a massive particle@um“mun50

with um[dxm/ds5(1,v/cST)] then reduces to

] tvj52cST

2 h i j ] ih00/2. ~17!

To obtain the Newtonian law, we need to identify the metperturbationh00 with the Newtonian gravitational potentiaf, giving

h0052f/cST2 . ~18!

The second step is to compare the Einstein equation to Pson’s equation. BecauseR00;] iG00

i ;2gi j ] i j g00, we findthat

2Df/cST2 5a~T002

12 g00T!/2. ~19!

For a fluid,T005rcST2 so that we need

a516pG/cST4 ~20!

to obtain the correct Newtonian limit for gravity. In the context of general relativity, it is thus clear thatcE5cST. It is,however, important to distinguish these concepts whencussing varying speed of light models: they may differsuch theories, depending on how the theory is formulate

F. Conclusion

If we wish to formulate a theory in which the speedlight is varying, the first step is to specify unambiguouswhich of the speeds that we have identified is varying, athen to propose a theoretical formulation, that is, a Lagraian, to achieve this goal. There is no reason why after reing the property of constancy of the speed of light, the dferent facets ofc described in this section will still coincideIt is important to clearly state which are the quantities thare kept fixed when one or another aspect ofc is assumed tovary ~see, for example, the discussion in Ref. 43!.

IV. DISCUSSION

Many models have been proposed to implement a varyspeed of light, starting with the proposal of Moffat44 in 1992who realized that it could be a solution to the same cosmlogical puzzles as inflation. Different versions have recenbeen widely popularized and various papers have claimthat these theories could resolve these cosmologproblems.7–9 We do not aim to review all these models~see,for example, Ref. 45 for a recent review and Ref. 46recent criticisms! and make only some brief general comments in the light of the two previous sections. As Jorda47

first pointed out, it is usually not consistent to allow a costant to vary in an equation that has been derived fromvariational principle with the hypothesis of this quantity bing constant. He stressed that one needs to go back toLagrangian and derive new equations, after having repla


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the constant by a dynamical field. This necessity is in pticular the case in scalar-tensor theories of gravity presein the previous section, and also applies to varying-c theo-ries.

To illustrate this important point, we consider thexamples7–9 in which a varying speed of light is supposesolve the standard cosmological problems. This concluswas based on the assumption that the Friedmann equaremain valid even whencÞ0, so that

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r13H~r1P/c2!53Kc2

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c. ~23!

It is obvious that Eqs.~21!–~23! are consistent. Howeverthese equations do not provide an equation for the evoluof c; rather, this evolution has to be postulated. The dynaics is incomplete@see the Appendix which gives the correequations~A8!–~A11! to use in that case and in particulEq. ~A11! which gives the dynamics of the ‘‘constant’’#.

We see from this example that when a constant becomdynamical field, we need to derive new equations of evotion. In particular a new equation describing the propagatof the new degree of freedom of the theory will have toobtained. The best way to construct such a theory is to ba Lagrangian with a corresponding new degree of freedincludedab initio and derive field equations by means ofstandard Lagrangian variation. In particular, such a formution allows us~1! to determine the true degrees of freedo~2! to check if there exists negative energies or acaupropagation of some modes in the theory,~3! to give a com-plete and self-consistent description of the dynamics by pviding the dynamical equation for the new variable, and~4!to give a clear specification of what we call the stress-enetensor~For instance, the example of the Appendix resultedthe appearance of bothTmn andTmn , whose physical meaning is now ambiguous, and it is very important not to deca priori what we call energy, for example.! In addition, wecan check if the equations are consistent, which will followthey are all derived from the same Lagrangian by the sdard universal method, which extremizes the specifiedgrangian under all small variations~in contrast to Refs.7–10, where nonstandard variation schemes were used!.

To conclude, we have emphasized that the nature ofspeed of light is complex and has many facets. These dient facets have to be distinguished if we wish to construtheory in which the speed of light is allowed to vary.particular, if the electromagnetic speed is assumed to varneeds to be shown how Maxwell’s equations are tochanged. If it is the causal speed that is to vary, we shouldshown how the space-time metric tensor structure and inpretation are altered. In either case, we need to be shhow the relation to the other aspects is to be treatedconsistent way. As we have emphasized, letting a consvary implies replacing it by a dynamical field consistent


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Simply letting it vary in equations derived under the assumtion that it is a constant leads to incorrect results, asexample of scalar-tensor theories of gravity illustrates.need to consider a Lagrangian that allows us to determinedegrees of freedom of the theory and to check if the vational principle is well defined.

The possibility that the fundamental constants may vduring the evolution of the universe offers an exceptiowindow onto higher dimensional theories, and is probalinked with the nature of the dark energy that makesuniverse accelerate today. Thus the topic is certainly of minterest, but it has to be developed on solid theoretical fodations.

ACKNOWLEDGMENTS

We thank Gilles Esposito-Fare`se, Je´rome Martin, FilippoVernizzi, Gilles Cohen-Tannoudji, and Rene´ Cuillierier fordiscussions on the constants of nature. We also thank JMagueijo, John Barrow, John Moffat, and Michael Claytofor commenting on their work and Christian ArmendariPicon, David Coule, Vittorio Canuto, and Thomas Dent. Jthanks the University of Buenos Aires, and particularly Dego Harari, for hospitality during the latest stages of twork. GE thanks the NRF and University of Cape Town fsupport.

APPENDIX: MAKING A CONSTANT DYNAMICAL:THE EXAMPLE OF SCALAR-TENSORTHEORIES OF GRAVITY

As emphasized in Sec. I, a complete proposal forvarying-c theory must include a specification of the dynamics underlying the proposed variation. It is not acceptablesimply change the constantc to a functionc(xi) while leav-ing the dynamical equations unchanged. The formulationtheories with varying constants implies replacing quantitthat were previously constant by dynamical fields. To be ctain of the correctness of the equations in this case, allrelations in the theory have to be rederived. To illustrate tpoint, we recall one of the simplest such constructions, tis, scalar-tensor theories of gravity which allow for a dnamical gravitational constant. To do so we need to geneize Eq. ~12! and introduce a new dynamical scalar fieldcand write an action for both this field and gravity coupledmatter.

We follow the notation and the main results for such theries as discussed in Ref. 48~see also Ref. 49!. These theoriesare characterized by an action,S, of the form50

S51

16pG E d4x A2g@F~c!R2Z~c!gmn]mc]nc

22U~c!#1Sm@fm ;gmn#, ~A1!

where G is the bare gravitational constant, which diffefrom the measured one, andR is the Ricci scalar of themetricgmn . The actionS has been divided into an action fothe gravitational field~the first term involving the metric andc! and an action for matter,Sm@fm ;gmn#. The latter is afunctional of some matter fieldscm and of the metric, butdoes not involve the scalar fieldc. For this form forS, calledthe Jordan frame, the new dynamical degree of freedomc,couples to the curvature term,R, so that the effective gravi-tational constant depends on the value ofc and thus becomes


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dynamical, while the mass and size of the standard partiare left unchanged. The Jordan frame defines the lengthstime actually measured by laboratory rods and clocks. Suform also implies that the weak equivalence principle hobecause all the matter fields feel the same metric andthus fall identically. In models with varying constants, thprinciple might no longer hold~see the following!. The dy-namics ofc depend on the functionsF, Z, andU, but notethat Z can always be set equal to unity by a redefinitionthe fieldc, so that only two arbitrary functions remain.

The variation ofS gives

F~c!Gmn58pGTmn1Z~c!@]mc]nc

2 12 gmn~]ac]ac!#1¹m]nF~c!

2gmn¹a¹aF~c!2gmnU~c!, ~A2!

2Z~c!¹a¹ac52dF

dcR2

dZ

dc~]ac!212

dU

dc, ~A3!

¹mTnm50, ~A4!

where the matter stress-energy tensor is definedTmnA2g52dSm /dgmn . The equations~A2!–~A4! are writ-ten in the Jordan frame. In the action~A1!, matter is univer-sally coupled to the metric tensorgmn so that this metric isthe one that defines the lengths and times as measurelaboratory rods and clocks, with the associated speed of lc

ST~which we have here set equal to unity by the appropr

choice of units!. All experimental and observational dahave their standard interpretation in this frame.

There exists another interesting frame, the Einstein fradefined by performing the conformal transformation

gmn5F~c!gmn , ~A5a!

S dw

dc D 2

53

4 S d lnF

dc D 2

11

2F~c!, ~A5b!

A~w!5F21/2, ~A5c!

2V~w!5U/F2, ~A5d!

so that the action in Eq.~A1! takes the form

S51

16pG E d4x A2g@R22gmn]mw]nw24V~w!#

1Sm@fm ;A2~w!gmn#. ~A6!

With this form, the action looks like the action of generrelativity, but the matter fields are now explicitly coupledthe metric, so that for instanceTmn5A2(w)Tmn . The maindifference between the two frames arises from the factthe spin 2 degrees of freedom, called the graviton, areturbations ofgmn and thatw is a spin 0 scalar. The perturbation of gmn actually mixes the spin 2 and spin 0 excitationThe dimensionless functionF needs to be positive for thgraviton to carry positive energy.

Note a difference between these two formulations ofsame theory. In the Einstein frame, the matter fieldscoupled toA2(w)gmn , so that the length and mass of anparticle will depend on the value of the scalar field while tgravitational constant is kept constant. It follows that, whin the Jordan frame,G is varying and the massesm of theparticles are constant; in contrast, in the Einstein frame


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masses are varying butG is constant with the same theory. Ifact, what is meaningful is that the dimensionless paramGm2/\c depends on the scalar field and is dynamical. Tcontrast illustrates that talking about a dynamical dimensiful quantity occurs because one cannot distinguish a varyG or varying masses and that the same theory can descthe two situations after a mathematical transformation.

An easy application is to start from the field equatio~A2!–~A4! and apply them to the cosmological contextwhich the metric is assumed to take the form

ds252dt21a2~ t !g i j dxidxj , ~A7!

wherea(t) is the scale factor of the universe describingexpansion and is a function of the cosmic timet only; g i j isthe metric of constant time hypersurfaces of constant cuture ~for example, in the case of an Euclidean spag i j dxidxj5d i j dxidxj ). The stress-energy tensor of the mathas only two independent components,T0

052r and Tij

5Pd ij , wherer is the energy density andP the pressure.

Due to the symmetry of the metric in Eq.~A7!, all dynamicalfields depend only ont (r(t),P(t),c(t),...) sothat the cos-mological equations are given by

3FS H21K

a2D58pGr11

2Zc223HF1U, ~A8!

22FS H2K

a2D58pG~r1P!1Zc21F2HF, ~A9!

r13H~r13P!50, ~A10!

Z~ c13Hc !53F8S H12H21K

a2D2Z8c2

22U8,

~A11!

where F85dF/dc, K561,0 is the curvature index, a dorefers to a derivative with respect to the cosmic timet, andH[a/a. We would not have obtained these correct eqtions by simply letting the constantG vary in the standardcosmological equations@compare the correct equation~A8!–~A11! with the incorrects Eqs.~21!–~23! obtained byletting G vary in the original dynamical equations deriveassuming thatG was constant#.

In the preceding framework, the universality of free-fallnot violated and all constants but the gravitational constare constant. It can be further generalized by allowing diffent couplings inSm , for example,

Sm5E d4x A2gS B

4~c!F21...D , ~A12!

which describes a theory in which both the gravitational costant and fine structure constant are varying. In that casefield couples differently to different particles so that we epect a violation of the universality of free fall~see also Ref.51 for a very general scalar-tensor theory in which the spof electromagnetic waves is calculated!.

The point of including this derivation is that exactly thsame principles govern the derivation of consistent dynamfor any ‘‘varying-constant’’ theory and in particular fovarying-c theories. In particular Eqs.~A8!–~A11! are theequations governing cosmological dynamics in such theoif correctly derived from a variational principle. They shoube compared with Eqs.~21!–~23! obtained by letting the


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constant vary in equations derived assuming that it wasally constant. Note in particular that we have one more eqtion, Eq. ~A11!, describing the dynamics of the new degrof freedom,c, that is of the varying constant. The samconclusions hold for varying-c theories,7–10 because in thesetheories the usual variational principles involve only the costantc in combination with the gravitational constantG.

a!Electronic mail: [email protected]!Author to whom correspondence should be addressed. Electronic

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13M. J. Duff, ‘‘Comment on time-variation of fundamental constantshep-th/0208093; V. V. Flambaum, ‘‘Comment on ‘Black hole constravarying constants,’ ’’ astro-ph/0208384; S. Carlip and S. Vaidya, ‘‘Blaholes may not constrain varying constants,’’ Nature~London! 421, 498–499 ~2003!; hep-th/0209249. This works refute this claim.

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20A translation and presentation of these texts can be found in R. Lehoand J.-P. Uzan,Les Constantes Fondamentales~Belin, Paris, France,2005!.


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21G. Galileo, inDialogues Concerning Two New Sciences~Presse Universi-taire de France, Paris, 1995!, pp. 159–161.

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48G. Esposito-Fare`se and D. Polarski, ‘‘Scalar-tensor gravity in an accelating universe,’’ Phys. Rev. D63, 063504-1–20~2001!; gr-qc/0009034.

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or

50T. Damour and G. Esposito-Fare`se, ‘‘Tensor-multi-scalar theories of gravitation,’’ Class. Quantum Grav.9, 2093–2176~1992!.

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f material toh moiste sold f

Blowpipes. These blowpipes are designed to raise the temperature of a candle or burner flame to a high enough value to heat a small bead oincandescence. Air from the mouth is blown through the flame, and the resulting hot flame is directed at the sample. The design with the bulb to catcurefrom the breath is due to Axel Fredrik Cronstedt. In the 1928 Welch catalogue, the plain model sold for twenty-four cents, and the Cronstedt stylorthirty cents. They are in the Greenslade Collection.~Photograph and notes by Thomas B. Greenslade, Jr., Kenyon College!


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c is the speed of light, isn’t it?