Relativity Adapted from the AP Institute 2007 Objectives: After completing this module, you should be able to: State and discuss Einsteins two postulates.

RelativityAdapted from the AP Institute 2007

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

• State and discuss Einstein’s two State and discuss Einstein’s two postulates regarding postulates regarding special special relativityrelativity..

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

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

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

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

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

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

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

Rest and MotionWhat do we mean when we say that an What do we mean when we say that an object is at object is at restrest . . . or . . . or in motionin motion? Is ? Is anything at rest?anything at rest?We sometimes say We sometimes say that man, computer, that man, computer, phone, and desk are phone, and desk are at restat rest..

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

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

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

We cannot say We cannot say without a without a frame of frame of referencereference..Assume bike moves at Assume bike moves at 25 m/s,W relative to 25 m/s,W relative to EarthEarth and that platform moves and that platform moves 10 m/s, E 10 m/s, E relative to Earthrelative to Earth..What is the velocity of the bike What is the velocity of the bike relative to relative to platform?platform?

EarthEarth

25 m/s

10 m/s

EastEast

WestWest

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

Reference for Motion (Cont.)

EarthEarth

25 m/s

10 m/s

EastEast

WestWest

Earth as Reference

0 0 m/sm/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 , we must imagine that we are sitting on the platform at rest (sitting on the platform at rest (0 m/s0 m/s) ) relative to it.relative to it.We would see the Earth moving We would see the Earth moving westward at westward at 10 m/s10 m/s and the bike moving and the bike moving west at west at 35 m/s35 m/s..

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

10 10 m/sm/s

35 m/s

0 m/sEastEast

WestWest

Platform as Reference

10 10 m/sm/s

35 m/s

0 m/sEastEast

WestWest

Platform as Reference

EarthEarth

25 m/s

10 m/s

EastEast

WestWest

Earth as Reference

00

Frame of Reference

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

0 m/s

35 m/sEastEast

WestWest

Bicycle as Reference

25 25 m/sm/s

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

ground.

10 m/s

10 m/s

c c

Velocities observed inside

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

40 m/s

20 m/s

cc

Velocities observed from

outside car

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

boy.

10 m/s

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

At t = 0, lightning At t = 0, lightning strikes the train strikes the train and ground at and ground at AA and and BB..Observer Observer OOEE sees sees lightning events lightning events AAEE & B & BEE as as simultaneoussimultaneous..Observer Observer OOTT says event says event BBTT occurs beforeoccurs before

event event AATT due to motion of train. Each due to motion of train. Each observer is correct!observer is correct!

BBEE

BT

AAEE

AT

Not simultaneous

SimultaneousSimultaneous

A B

Time MeasurementsSince our Since our measurement of time measurement of time involves judg-ments involves judg-ments about simul-taneous about simul-taneous events, we can see events, we can see that time may also be that time may also be affected by relative 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 TimeConsider cart moving with velocity Consider cart moving with velocity vv under a under a mirrored ceiling. A light pulse travels to ceiling mirrored ceiling. A light pulse travels to ceiling and back in time and back in time ttoo for rider and in time for rider and in time tt for for watcher.watcher.

Light path for

rider

d0

2dc

t

to

Light path for

watcher

dxx

R

t

R2R

ct

; 2 2

c t v tR x

Relative Time (Cont.)

Substitution of:Substitution of:

0

2

c td

0

2 21

tt

v c

0

2 21

tt

v c

Light path for

rider

d0

2dc

t

to

dR

t

222

22d

tvtc

2tc

2tv

2

0

22

222

tctvtc 2

022222 tctvtc

Time Dilation Equation

Einstein’s Time dilation

Equation:

0

2 21

tt

v c

tt = = Relative timeRelative time (Time measured in (Time measured in frame moving relative to actual frame moving relative to actual event).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 TimeThe key to applying the time dilation The key to applying the time dilation equation is to distinguish clearly between equation is to distinguish clearly between proper timeproper time ttoo and and relative timerelative time tt. Look . Look at our example:at our example:

Proper Time

d

to

Event Fram

e

Relative Time

t

Relative Frame

t > to

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

the man in Ship A?

v = 0.8c

A

B

Proper time to = 4 s

0

2 21

tt

v c

0

2 21

tt

v c

Find relative time t

2 2

4.00 s 4.00 s

1- 0.641- (0.8 ) /t

c c t = 6.67

s

The Twin ParadoxTwo twins Two twins are on Earth. are on Earth. One leaves One leaves and travels and travels for 10 years for 10 years at 0.9c.at 0.9c.When traveler When traveler returns, The returns, The Earth twin is Earth twin is 23 years older 23 years older due to time due to time dilation!dilation!

Traveling twin ages more!

0

2 21

tt

v c

0

2 21

tt

v c

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

The Twin Paradox ExplainedThe traveling The traveling twin’s motion was twin’s motion was not uniform. not uniform. Acceleration and Acceleration and forces were forces were needed to go to needed to go to and return from and return from space. Butasume space. Butasume he coast 99.99% he coast 99.99% of the timeof the timeThe traveler The traveler ages less than ages less than the one who the one who stayed home.stayed home.

Traveling twin ages

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

Length Contraction

0.9cLo

L

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

2 20 1L L v c

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 Moving objects are foreshortened due to relativity.relativity.

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

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

seen by the observer?

0.9c

1 mLo

L = ?

2 20 1L L v c

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 Length recorded by observer:observer:

L = 43.6 cmL = 43.6 cm

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

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

0.9c

Wo

W<Wo

1 m=1 m

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

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

Relativistic MomentumThe basic conservation laws for The basic conservation laws for momentum and energy can not be violated momentum and energy can not be violated due to relativity.due to relativity.Newton’s equation for momentum Newton’s equation for momentum ((mvmv) ) must be changed as follows to account for must be changed as follows to account for relativityrelativity::

0

2 21

m vp

v c

Relativistic

momentum:

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

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

0

2 21

mm

v c

Relativistic

mass:

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

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

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

Example 3: The rest mass of an electron is 9.1 x 10-31 kg. What is the relativistic mass if its velocity

is 0.8c ?

- 0.80.8cc

mmoo = 9.1 x 10= 9.1 x 10-31-31 kg kg0

2 21

mm

v c

-31 -31

2 2

9.1 x 10 kg 9.1 x 10 kg

0.361 (0.6 )m

c c

m = 15.2 x 10-31 kg

The mass has The mass has increased by increased by

67% !67% !

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

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

Total Energy, E

For a particle with For a particle with zero momentum zero momentum p p = 0:= 0:For an EM wave, mFor an EM wave, m00 = = 0, and 0, and E E simplifies to:simplifies to:

E = moc2

E = pc

2220 )( cpcmE

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 1-kg block on a table has an A 1-kg block on a table has an energy energy EEoo and mass and mass mmoo relative relative to table: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 1-kg block is in relative motion, its If the 1-kg block is in relative motion, its kinetic energy adds to the kinetic energy adds to the total energytotal energy..

Total EnergyAccording to Einstein's theory, the According to Einstein's theory, the total total energyenergy 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 Total energy includes rest energy and energy of motion. If we are interested in energy of motion. If we are interested in just the energy of motion, we must just the energy of motion, we must subtract subtract mmoocc22..

Kinetic Energy: K = (m – mo)c2

((mmoocc22 + K) + K)

Kinetic Energy: Kinetic Energy: K K = = mcmc2 2 – – mmoocc22

Example 4: What is the kinetic energy of a proton (mo = 1.67 x 10-27 kg) traveling at 0.8c?

+ 0.70.7cc

mmoo = 1.67 x 10= 1.67 x 10-27-27 kg kg0

2 21

mm

v c

-27 -27

2 2

1.67 x 10 kg 1.67 x 10 kg

0.511 (0.7 )m

c c

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

kgkg

K = (m – mK = (m – moo)c)c22 = = (2.34 x 10(2.34 x 10-27 -27 kg – 1.67 x 10kg – 1.67 x 10-17 -17

kg)kg)cc22

Relativistic Kinetic Energy K = 6.02 x 10-11 J

Relativistic Kinetic Energy K = 6.02 x 10-11 J

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

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

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

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

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

Summary (Cont.)

0

2 21

mm

v c

Relativistic

mass:

0

2 21

tt

v c

Relativistic

time:

2 20 1L L v c Relativistic

length:

Summary(Cont.)

Total energy: E = mc2

Kinetic energy: K = (m – mo)c2

0

2 21

m vp

v c

Relativistic momentu

m:

CONCLUSION:It’s not all relativ(ity)!