Relativity Complete

RELATIVITY

Inertial Frame of Reference

A frame of reference where Newton’s Laws of Motion hold true.

FRAMES OF REFERENCE

Non-inertial Frame of Reference

A frame of reference that is at rest or moving with constant velocity is an inertial frame of reference.

A frame of reference where Newton’s Laws of Motion seem to be violated.

A frame of reference that is accelerating is a non- inertial frame of reference.

1st POSTULATE: The Principle of Relativity

The laws of physics are the same in all inertial frames of reference.

SPECIAL THEORY OF RELATIVITY (STR)

2nd POSTULATE: The Constancy of the Speed of Light

The speed of light in vacuum is the same in all frames of reference and is independent of the motion of the source,

1. Events that are simultaneous for one observer may not be simultaneous for another observer.

IMPLICATIONS OF THE STR

2. When observers moving relative to each other measure time, they may not get the same result.

3. When observers moving relative to each other measure length, they may not get the same result.

4. Newton’s 2nd Law, and equations for conservation of momentum and energy may have to be revised.

1st POSTULATE: The Principle of Relativity

The laws of physics are the same in all inertial frames of reference.

(a) Current is produced in the stationary coil by a moving nearby magnet.

(b) Current is produced in the coil by moving it through a stationary magnet.


The speed of light in vacuum is the same in all frames of reference and is independent of the motion of the source.


The speed of light in vacuum is the same in all frames of reference and is independent of the motion of the source.


It is impossible for an inertial observer to travel at the speed of light in vacuum.

cv SS /'

• Assume that the spaceship is travelling at c and then emits a light beam.

• According to Newton/Galileo: The observer on earth (S) would observe that the spaceship and the light move together (and are always at the same point in space).

• According to Einstein: The observer on the spaceship (S’) would observe that the light beam travels with a speed c relative to him (they cannot be at the same point in space).

The Galilean Coordinate Transformation

Consider two inertial frames S and S’ with S’ moving with constant velocity u (in the +x direction) relative to S and that their origins O and O’ coincide at t = t’ = 0.

Consider a particle P launched in S’.

S and S’ are separated at a later time by a distance equal to ut .


The position of the particle relative to S’ is x’ and its position relative to S is x, so that

x = x’ + utThe velocity of the particle relative to S’ is vx’ and its

velocity relative to S is vx, so that

vx = vx’ + u

vx =dx

dt

vx’=dx’

dt


Suppose the particle is the light beam launched from S’.

c = c’ + u

The velocity of the light beam relative to S’ is c’ and its velocity relative to S is c, so that

This is impossible according to Einstein’s 2nd Postulate. What’s wrong?

vx =dx

dtvx’=

dx’

dt

The Galilean relativity (incorrectly) assumes that both frames of reference use the same time scale, and that the velocities are defined as

Relativity of Simultaneity

vx =dx

dtvx’=

dx’

dt’

According to Einstein, the correct definition of velocities should be

The reference frame S’ uses a different time scale.

• In any given frame of reference, an event is an occurrence that has a definite position and time.

Definition of Simultaneous Events

• In any given frame of reference, two events are said to be simultaneous if they occur at the same time.

Two events that are simultaneous in a given frame of reference are NOT simultaneous in another frame moving relative to the first even if both frames are inertial frames.

Relativity of Time Intervals

Consider an observer (Mavis) on a reference frame S’ moving with a velocity u (where u < c) relative to another reference frame S.

She measures the time interval to between 2 events.

Event 1: A flash of light leaves a source at O’.

Event 2: The flash returns to O’ after having been reflected from mirror a distance d away.

to =2d

c

The time interval to between 2 events as measured by Mavis is

Relativity of Time Intervals

Consider an observer (Stanley) on reference frame S observing the same 2 events.

He measures a different time interval t between 2 events.

t =2l

c

22

2

tu

dl

where,

How are to and t related?

22

2

22

tu

dcc

lt but

2otcd

As measured by Stanley in S,

22

22

2

tutc

ct o

22

22

22

4 tutc

ct o

2

2222

c

tutt o

2

2

222

otc

tut

2

2

222

otc

tut

2

2

22 1 otc

ut

2

2

1cu

tt o

TIME DILATION

Since 22 /1 cu is always less than 1, Stanley

measures a longer time interval t than to for Mavis.

Observers measure a clock to run slow if it moves relative to them.

PROPER TIME to

There is only one frame of reference where a clock is at rest.

• The time interval measured by a clock in the rest frame is called the proper time to.

• The time interval between two events that occur in the same point in space is proper time to.

• The time interval between two events that occur in different points in space is dilated time t.

• The time interval measured by a clock outside the rest frame is called the dilated time t.

The Twin Paradox Albert Einstein has a twin brother William, who left the earth aboard a spaceship that travels at a speed close to the speed of light (while Albert stays on earth). Albert observes that since William is moving, William’s clock is ticking slower. Albert concludes that William will be younger when he returns to earth.

William observes that Albert is moving (and he is the one at rest) and Albert’s clock is ticking slower. William concludes that Albert will be younger when he returns to earth. Who is correct?

The Twin Paradox

The twins are NOT identical in all respects.

Albert remains in an approximately inertial frame (the earth) at all times.

William must accelerate relative to earth in order to leave, turn around, and return to earth. William’s frame of reference is not inertial.

Albert’s observation is correct. When William returns, he is younger than Albert.

(a) The two events occurred at the same point in space as observed by the Martian, so the Martian measures the proper time.

(b) As observed by the pilot, the two events occurred at two different points in space.

ss

cu

tt o

435985.01

75

12

2

2

(a) The time interval that you measured is s

c

mxt 500.0

8.0

102.1 8

The time interval that the pilot measured is ssto 300.08.01)500.0( 2

(b) mxsc 7102.7)300.0)(8.0(

The proper time to is measured by the clock on the space probe.

t is the time that elapses as measured in the earth frame. The distance 42.2 ly is measured in the earth frame.

yc

lyt 6.42

9910.0

2.42

yyto 70.59910.01)6.42( 2

The biological age of the astronaut would be 19 y + 5.70 y = 24.70 y!

Relativity of Length

Mavis measures the length of the meterstick to be lo since it is in her rest frame S’. A light source is attached at one end of the meterstick and a mirror at the other end.

Consider an observer (Mavis) on a reference frame S’ moving with a velocity u (where u < c) relative to another reference frame S.

The time interval for the back and forth motion of the light (as measured by Mavis )is to.

to =2loc


As observed by Stanley on the platform (S), the meterstick is moving relative to him with a velocity u and he measures its length to be l.

On the way to the mirror, light travels a distance

d1 = l + ut1

On the way back to the source, light travels a distance

d2 = l - ut2


11 tuld

11 tcd

11 tctul uc

lt

1

On the way to the mirror:

On the way back to the source:

22 tuld

22 tcd

22 tctul uc

lt

2


21 ttt The total time of travel is

uc

l

uc

lt

2

2

1

2

cu

c

lt

Recall: 2

2

1c

utto

c

lt oo

2

2

2

12

c

ut

c

lo

2

2

2

2

1

2

1

2

cu

c

l

cu

c

l o

2

2

1

2

cu

c

lt o

Relativity of Length: Length Contraction

2

2

1c

ull o

Observers measure an object to be shorter (than when measured in its rest frame) if it moves relative to them.

2

2

2

2

11cu

l

cu

l o

The length of a body measured in a frame in which the body is at rest is called the proper length (lo).

The length of a body measured in any frame moving relative to the frame in which the body is at rest is called the contracted length (l).

Length Contraction

There is no contraction of lengths perpendicular to the direction of motion.

2

2

1c

ull o

2

2

1cu

llo

The length of the moving spacecraft as measured on Coruscant is l = 74.0 m.

mm

cc

llo 5.92

8.0

0.74

)6.0(1 2

2

2

222 1

c

ull o

2

2

2

2

1c

u

l

l

o

2

2

2

2

1ol

l

c

u

22

2

1 cl

lu

o

ccu 9524.01

3048.01 2

2

2

smxu /1086.2 8

(a) The scientist measures the distance 45km in his rest frame, so lo = 45.0km. The time as measured by the scientist is

sxc

kmt 41051.1

99540.0

45

(b) The distance as measured in the particle’s frame is contracted length, so

kmc

ckml 31.4

)99540.0(145

2

2

(c) The time as measured in the particle’s frame is

sxc

kmto

51044.199540.0

31.4

Also,

sxsxto524 1044.199540.011051.1

The side of the cube parallel to the direction of motion is contracted.

2

2

1'c

uaa

The volume of the cube as measured in S’ is

2

23

2

22 11'

c

ua

c

uaaV

2

2

1c

uab

2

2

140.1c

ubb

ccu 700.040.1

11 2

2

smxu /1010.2 8

2. A new planet Jretsim is 20 lightyears from Earth. Jasper boards his Spaceship Astign from Earth and zips to Jretsim at 80.0% the speed of light. After 10 years (on Earth), Kilroy boards his newer Spaceship Aguro and flies to Jretsim. If Kilroy and Jasper arrive at Jretsim at the same time, how fast is Spaceship Aguro?

ANSWER: 0.970c

The Lorentz Coordinate Transformation

Consider two inertial frames S and S’ with S’ moving with constant velocity u (in the +x direction) relative to S and that their origins O and O’ coincide at t = t’ = 0.

When an event at coordinates (x, y, z) happens in S at time t as observed in S, what are the coordinates (x’, y’, z’) and time t’ of the same event as observed in S’?

2

2

1'c

uxutx

2

2

1''c

uxutx

2

2

1

'

cu

utxx

2

2

2

21'

1c

uxut

cu

utx

As observed in S:

As observed in S’:

Equating the two:

The Lorentz Coordinate Transformation

2

2

1

'

cu

utxx

2

2

2

1

/'

cu

cuxtt

where: x’ is the position measured in S’.

x is the position measured in S.

t is the time measured in S.

t’ is the time measured in S’.

u is the velocity of S’ relative to S.

NOTE: If u 0, x = x’ and t = t’

y = y’ and z = z’

2

2

1

'

cu

udtdxdx

2

2

2

1

/'

cu

cudxdtdt

The Lorentz Velocity Transformation

Consider a particle that moves a distance dx in a time dt as measured in S.

In frame S’, the particle moves a distance dx’ in a time dt’.

2/'

'

cudxdt

udtdx

dt

dx

dtdx

cu

udtdx

dt

dx

21'

'

x

xx

vcuuv

v

21'

The Lorentz Velocity Transformation

x

xx

vcuuv

v

21'

x

xx

vcuuv

v'1

'

2

where: vx ’ is the velocity of the particle measured in S’.

Anything moving with velocity equal to c in frame S is also moving with a velocity c in S’.

vx is the velocity of the particle measured in S.

NOTE: If u and vx << c, vx’ = vx – u and vx = vx‘+ u If vx = c, vx’

= c

Anything moving with velocity less than c in frame S is also moving with a velocity less than c in S’.

(a)c

ccc

cc

vcuuv

v

x

xx 806.0

)400.0(600.0

1

600.0400.0

'1

'

22

(b)c

ccc

cc

vcuuv

v

x

xx 974.0

)900.0(600.0

1

600.0900.0

'1

'

22

u = 0.650c, vx’ =- 0.950c

Let S be the laboratory’s frame and S’ the 1st particle’s frame.

=+

+=

+

+=

)c950.0(c

c650.01

c650.0c950.0

'vc

u1

u'vv

2x2

xx - 0.784c

Let S be the starfighter’s frame and S’ the enemy ship’s frame.

cu 400.0 cvx 700.0'

(a)c

ccc

cc

vcuuv

v

x

xx 859.0

)700.0(400.0

1

400.0700.0

'1

'

22

(b)s

c

mx

v

xt

x

0.31859.0

1000.8 9

Relativistic Momentum

Suppose a particle is at rest in a reference frame. The mass of the particle measured in this frame is called its rest mass mo.

When the particle is moving with a velocity v, its relativistic momentum is

2

2

1cv

vmp o

If v << c, p = mov

If v c, p

2

2

1cv

mm o

Relativistic Mass

If v << c, m = mo

If v c, m

Momentum is not directly proportional to velocity!!!

Relativistic Kinetic Energy (K)

If v c, K

Total Energy (E)

2

2

2

2

1

cm

cv

cmK o

o

2

2

2

1cv

cmE o

2cmE oo

where moc2 is called the rest energy Eo.

If v << c, K = ½ mv2

2cmKE o

2

2

2

2

1

cmK

cv

cmo

o

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Relativity Complete