Chapter 38A Chapter 38A -- RelativityRelativityA PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 2007

Objectives: Objectives: After completing this After completing this module, you should be able to:module, you should be able to:

•• State and discuss EinsteinState and discuss Einstein’’s two s two postulates regarding postulates regarding special relativityspecial relativity..

•• Demonstrate your understanding of Demonstrate your understanding of time time dilationdilation and apply it to physical and apply it to physical problems.problems.

•• Demonstrate and apply equations for Demonstrate and apply equations for relativistic lengthrelativistic length, , momentummomentum, , massmass, , and and energyenergy..

Special RelativitySpecial RelativityEinsteinEinstein’’s s Special Theory of RelativitySpecial Theory of Relativity, published , published in 1905, was based on two postulates:in 1905, was based on two postulates:

I. The laws of physics are the same for all frames of reference moving at a constant velocity with respect to each other.

I. The laws of physics are the same for all I. The laws of physics are the same for all frames of reference moving at a constant frames of reference moving at a constant velocity with respect to each other.velocity with respect to each other.

II. The free space velocity of light c is constant for all observers, independent of their state of motion. (c = 3 x 108 m/s)

II. The free space velocity of light II. The free space velocity of light cc is is constant for all observers, independent of constant for all observers, independent of their state of motion. (their state of motion. (cc = 3 x 10= 3 x 1088 m/sm/s))

Rest and MotionRest and MotionWhat do we mean when we say that an object What do we mean when we say that an object is at is at restrest . . . or . . . or in motionin motion? Is anything at rest?? Is anything at rest?

We sometimes say that We sometimes say that man, computer, phone, man, computer, phone, and desk are and desk are at restat rest..

What we really mean is that all are moving with What we really mean is that all are moving with the the same velocitysame velocity. We can only detect motion in . We can only detect motion in reference to something else.reference to something else.

We forget that the Earth We forget that the Earth is also in motion.is also in motion.

No Preferred Frame of ReferenceNo Preferred Frame of ReferenceWhat is the velocity of What is the velocity of the bicyclist?the bicyclist?

We cannot say without We cannot say without a a frame of referenceframe of reference..

Assume bike moves at Assume bike moves at 25 m/s,W relative to Earth25 m/s,W relative to Earth and that platform moves and that platform moves 10 m/s, E relative to Earth10 m/s, E relative to Earth..

What is the velocity of the bike What is the velocity of the bike relative to platform?relative to platform?

EarthEarth

25 m/s10 m/s

EastEast

WestWest

Assume that the platform is the reference, then Assume that the platform is the reference, then look at relative motion of Earth and bike.look at relative motion of Earth and bike.

Reference for Motion (Cont.)Reference for Motion (Cont.)

EarthEarth

25 m/s10 m/s

EastEast

WestWest

Earth as Reference

0 m/s0 m/s

To find the velocity of the bike To find the velocity of the bike relative to relative to platformplatform, we must imagine that we are sitting , we must imagine that we are sitting on the platform at rest (on the platform at rest (0 m/s0 m/s) relative to it.) relative to it.

We would see the Earth moving westward at 10 m/s and the bike moving west at 35 m/s. We would see the Earth moving westward at We would see the Earth moving westward at 10 m/s10 m/s and the bike moving west at and the bike moving west at 35 m/s35 m/s..

10 m/s10 m/s

35 m/s

0 m/sEastEast

WestWest

Platform as Reference

10 m/s10 m/s

35 m/s

0 m/sEastEast

WestWest

Platform as Reference

EarthEarth

25 m/s10 m/s

EastEast

WestWest

Earth as Reference

00

Frame of Frame of ReferenceReference

Consider the velocities Consider the velocities for three different for three different frames of reference.frames of reference.

0 m/s35 m/s

EastEastWestWest

Bicycle as Reference

25 m/s25 m/s

Constant Velocity of LightConstant Velocity of LightPlatform v = 30 m/s to right relative to ground.

10 m/s10 m/s

c c

Velocities observed inside car

The light from two flashlights and the two The light from two flashlights and the two balls travel in balls travel in opposite directionsopposite directions. The . The observed velocities of the ball differ, but the observed velocities of the ball differ, but the speed of light is independent of directionspeed of light is independent of direction..

40 m/s20 m/s

cc

Velocities observed from outside car

Velocity of Light (Cont.)Velocity of Light (Cont.)Platform moves 30 m/s to right relative to boy.

10 m/s10 m/s

c c 30 30 m/sm/s

Each observer sees c = 3 x 108 m/s

Outside observer sees very different velocities for balls.

The velocity of light is unaffected by relative motion and is exactly equal to:

c = 2.99792458 x 108 m/s

OT

OOEE

Simultaneous EventsSimultaneous EventsThe judgment of simultaneous events is also a The judgment of simultaneous events is also a matter of relativity. Consider observer matter of relativity. Consider observer OOTT sitting on sitting on top of moving train while observer top of moving train while observer OOEE is on ground.is on ground.

At t = 0, lightning At t = 0, lightning strikes train and strikes train and ground at ground at AA and and BB..

Observer Observer OOEE sees sees lightning events lightning events AAEE & & BBEE as as simultaneoussimultaneous..

Observer Observer OOTT says event says event BBTT occurs beforeoccurs before event event AATT due to motion of train. Each observer is correct!due to motion of train. Each observer is correct!

BBEE

BT

AAEE

AT

Not simultaneous

SimultaneousSimultaneous

A B

Time MeasurementsTime MeasurementsSince our measurement Since our measurement of time involves of time involves judgjudg-- mentsments about about simulsimul-- taneoustaneous events, we can events, we can see that time may also see that time may also be affected by relative be affected by relative motion of observers. motion of observers.

In fact, Einstein's theory shows that observers in relative motion will judge times differently - furthermore, each is correct.

In fact, Einstein's theory shows that observers in relative motion will judge times differently - furthermore, each is correct.

Relative TimeRelative TimeConsider cart moving with velocity Consider cart moving with velocity vv under a under a mirrored ceiling. A light pulse travels to ceiling and mirrored ceiling. A light pulse travels to ceiling and back in time back in time ttoo for rider and in time for rider and in time tt for watcher.for watcher.

Light path for rider

d0

2dct

toLight path for watcher

dxx

R

t

R2Rc

t

; 2 2

c t v tR x

Relative Time (Cont.)Relative Time (Cont.)

Substitution of:Substitution of:

0

2c td

22 20

2 2 2c tc v

t t

02 21

ttv c

Light path for rider

d0

2dct

to

dR

t

2c

t2v t

2 22

2 2c v d

t t

Time Dilation EquationTime Dilation Equation

Einstein’s Time dilation Equation:

02 21

ttv c

tt = = Relative timeRelative time (Time measured in frame (Time measured in frame moving relative to actual event).moving relative to actual event).

ttoo == Proper timeProper time (Time measured in the (Time measured in the same frame as the event itself).same frame as the event itself).

v v = = Relative velocity of two frames.Relative velocity of two frames.

c c = = Free space velocity of light (Free space velocity of light (cc = 3 x 10= 3 x 1088 m/sm/s).).

Proper TimeProper TimeThe key to applying the time dilation equation is The key to applying the time dilation equation is to distinguish clearly between to distinguish clearly between proper timeproper time ttoo and and relative timerelative time tt. Look at our example:. Look at our example:

Proper Time

d

to

Event Frame

Relative Time

t

Relative Frame

t > to

Example 1:Example 1: Ship Ship AA passes ship passes ship BB with a with a relative velocity of 0.8c (eighty percent of the relative velocity of 0.8c (eighty percent of the velocity of light). A woman aboard Ship velocity of light). A woman aboard Ship BB takes takes 4 s4 s to walk the length of her ship. What to walk the length of her ship. What time is recorded by the man in Ship time is recorded by the man in Ship AA??

v = 0.8c

A

B

Proper time to = 4 s

02 21

ttv c

Find relative time t

2 2

4 .00 s 4 .00 s1 - 0 .641- (0 .8 ) /

tc c

t = 6.67 s

The Twin ParadoxThe Twin ParadoxTwo twins are Two twins are on Earth. One on Earth. One leaves and leaves and travels for 10 travels for 10 years at 0.9c.years at 0.9c.

When traveler When traveler returns, he is 23 returns, he is 23 years older due years older due to time dilation!to time dilation!

Traveling twin ages more!

02 21

ttv c

Paradox:Paradox: Since motion is Since motion is relative, isnrelative, isn’’t it just as true that t it just as true that the man who stayed on earth the man who stayed on earth should be 23 years older?should be 23 years older?

The Twin Paradox ExplainedThe Twin Paradox Explained

The traveling twinThe traveling twin’’s s motion was not motion was not uniform. Acceleration uniform. Acceleration and forces were and forces were needed to go to and needed to go to and return from space.return from space.

The traveler The traveler ages more and ages more and not the one who not the one who stayed home.stayed home.

Traveling twin ages more!

This is NOT science fiction. This is NOT science fiction. Atomic clocks placed aboard Atomic clocks placed aboard aircraft sent around Earth and aircraft sent around Earth and back have verified the time back have verified the time dilation.dilation.

Length ContractionLength Contraction

0.9cLo

L

Since time is affected by Since time is affected by relative motion, length relative motion, length will also be different:will also be different:

2 20 1L L v c

LLoo is is properproper length length

L L is is relativerelative length length

Moving objects are foreshortened due to relativity.Moving objects are foreshortened due to relativity.Moving objects are foreshortened due to relativity.

Example 2:Example 2: A meter stick moves at 0.9c A meter stick moves at 0.9c relative to an observer. What is the relative relative to an observer. What is the relative length as seen by the observer?length as seen by the observer?

0.9c

1 mLo

L = ?

2 20 1L L v c

2 2(1 m) 1 (0.9 ) /L c c

(1 m) 1 0.81 0.436 mL

Length recorded by observer:Length recorded by observer: L = 43.6 cmL = 43.6 cm

If the ground observer held a meter stick, the If the ground observer held a meter stick, the same contraction would be seen from the ship.same contraction would be seen from the ship.

Foreshortening of ObjectsForeshortening of ObjectsNote that it is theNote that it is the lengthlength in the direction of in the direction of relative motion that contracts and not the relative motion that contracts and not the dimensions perpendicular to the motion.dimensions perpendicular to the motion.

0.9c

Wo

W<Wo1 m=1 m

If meter stick is 2 cm If meter stick is 2 cm wide, each will say the wide, each will say the other is only 0.87 cm other is only 0.87 cm wide, but they will wide, but they will agree on the length.agree on the length.

Assume Assume each each holds a holds a meter stick, in example.meter stick, in example.

Relativistic MomentumRelativistic MomentumThe basic conservation laws for momentum and The basic conservation laws for momentum and energy can not be violated due to relativity.energy can not be violated due to relativity.

NewtonNewton’’s equation for momentum s equation for momentum ((mvmv) must be ) must be changed as follows to account for relativitychanged as follows to account for relativity::

02 21

m vpv c

Relativistic momentum:

mmoo is the is the properproper massmass, often called the , often called the rest rest massmass. Note that for large values of . Note that for large values of vv, this , this equation reduces to Newtonequation reduces to Newton’’s equation.s equation.

Relativistic MassRelativistic MassIf momentum is to be conserved, the relativistic mass If momentum is to be conserved, the relativistic mass mm must be consistent with the following equation:must be consistent with the following equation:

02 21

mmv c

Relativistic mass:

Note that as an object is accelerated by a Note that as an object is accelerated by a resultant force, its mass increases, which resultant force, its mass increases, which requires even more force. This means that:requires even more force. This means that:

The speed of light is an ultimate speed!The speed of light is an ultimate speed!The speed of light is an ultimate speed!

Example 3: Example 3: The rest mass of an electron is The rest mass of an electron is 9.1 x 109.1 x 10--3131 kgkg. What is the relativistic mass if . What is the relativistic mass if its velocity is its velocity is 0.80.8cc ??

- 0.80.8cc

mmoo = 9.1 x 10= 9.1 x 10--3131 kgkg02 21

mmv c

-31 -31

2 2

9.1 x 10 kg 9.1 x 10 kg0.361 (0.6 )

mc c

m = 15.2 x 10-31 kgThe mass has The mass has

increased by 67% !increased by 67% !

Mass and EnergyMass and EnergyPrior to the theory of relativity, scientists Prior to the theory of relativity, scientists considered mass and energy as separate considered mass and energy as separate quantities, each of which must be conserved.quantities, each of which must be conserved.

Now mass and energy must Now mass and energy must be considered as the same be considered as the same quantity. We may express quantity. We may express the the massmass of a baseball in of a baseball in joulesjoules or its or its energyenergy in in kilogramskilograms! The motion ! The motion addsadds to the massto the mass--energy.energy.

Total Relativistic EnergyTotal Relativistic EnergyThe general formula for the relativistic total The general formula for the relativistic total energy involves the rest mass energy involves the rest mass mmoo and the and the relativistic momentum relativistic momentum p = p = mvmv..

Total Energy, E 2 2 20( )E m c p c

For a particle with zero For a particle with zero momentum momentum p = 0:p = 0:

For an EM wave, mFor an EM wave, m00 = 0, = 0, and and E E simplifies to:simplifies to:

E = moc2

E = pc

Mass and Energy (Cont.)Mass and Energy (Cont.)The conversion factor between mass m and energy E is:

Eo = moc2

The zero subscript refers to The zero subscript refers to properproper or or restrest values.values.

A 1A 1--kg block on a table has an energy kg block on a table has an energy EEoo and mass and mass mmoo relative to table:relative to table:

1 kg

EEoo = (1 kg)(3 x 10= (1 kg)(3 x 1088 m/s)m/s)22 Eo = 9 x 1016 J

If the 1If the 1--kg block is in relative motion, its kinetic kg block is in relative motion, its kinetic energy adds to the energy adds to the total energytotal energy..

Total EnergyTotal EnergyAccording to Einstein's theory, the According to Einstein's theory, the total energytotal energy E E of a particle of is given by:of a particle of is given by:

Total Energy: E = mc2

Total energy includes rest energy and energy Total energy includes rest energy and energy of motion. If we are interested in just the of motion. If we are interested in just the energy of motion, we must subtract energy of motion, we must subtract mmoo cc22..

Kinetic Energy: K = (m – mo)c2

((mmoo cc22 + K)+ K)

Kinetic Energy: Kinetic Energy: K K = = mcmc2 2 –– mmoo cc22

Example 4:Example 4: What is the kinetic energy of a What is the kinetic energy of a proton (mproton (moo = 1.67 x 10= 1.67 x 10--2727 kg) traveling at 0.8c?kg) traveling at 0.8c?

+ 0.70.7cc

mmoo = 1.67 x 10= 1.67 x 10--2727 kgkg02 21

mmv c

-27 -27

2 2

1.67 x 10 kg 1.67 x 10 kg0.511 (0.7 )

mc c

; m = 2.34 x 10; m = 2.34 x 10--27 27 kgkg

K = (m K = (m –– mmoo )c)c22 = = (2.34 x 10(2.34 x 10--27 27 kg kg –– 1.67 x 101.67 x 10--17 17 kg)kg)cc22

Relativistic Kinetic Energy K = 6.02 x 10-11 JRelativistic Kinetic Energy K = 6.02 x 10-11 J

SummarySummaryEinsteinEinstein’’s s Special Theory of RelativitySpecial Theory of Relativity, published , published in 1905, was based on two postulates:in 1905, was based on two postulates:

I. The laws of physics are the same for all frames of reference moving at a constant velocity with respect to each other.

I. The laws of physics are the same for all I. The laws of physics are the same for all frames of reference moving at a constant frames of reference moving at a constant velocity with respect to each other.velocity with respect to each other.

II. The free space velocity of light c is constant for all observers, independent of their state of motion. (c = 3 x 108 m/s)

II. The free space velocity of light II. The free space velocity of light cc is is constant for all observers, independent of constant for all observers, independent of their state of motion. (their state of motion. (cc = 3 x 10= 3 x 1088 m/sm/s))

Summary (Cont.)Summary (Cont.)

02 21

mmv c

Relativistic mass:

02 21

ttv c

Relativistic time:

2 20 1L L v c Relativistic

length:

Summary(Cont.)Summary(Cont.)

Total energy: E = mc2

Kinetic energy: K = (m – mo)c2

02 21

m vpv c

Relativistic momentum:

CONCLUSION: Chapter 38ACONCLUSION: Chapter 38A RelativityRelativity