March 7, 2011

Special RelativityReading: Chapter 4, Rybicki & Lightman

PRELIMINARIES

History> SR developed out of a need to reconcile classical, Newtonian mechanics, Maxwell’s Equations and the laws of EM

> SR does not deal with gravitation, only electromagnetic forces

Inertial Coordinate Systems

A key concept in SR is the definition of inertial reference frames:

A reference frame is inertial if every particle within it which isinitially at rest remains at rest, and every test particle in motion

continues that motion without change in speed or direction.

Then in SR All the laws of physics are the same in every inertial reference frame.

“Principle of Relativity”

Example: An inertial reference frame is a space ship in free fall near the Earth

• A free particle at rest in the vehicle remains at rest• A particle given a gentle push moves across the vehicle in a straight line with constant speed.

Why? Because the particle and the ship are both falling with the same acceleration towards the Earth

• The space ship is only an inertial reference system when it issmall enough that differences in the accelerations over its size can be ignored.

• If particles didn’t have the same acceleration regardless of size, shape, composition, etc. then an observer inside the space shipwould notice a relative acceleration among different particles. some of the particles at rest would not remain at rest, others would.

A sufficiently large space ship would not be an inertial reference frame,because the acceleration by gravity is not uniform across it.

from Spacetime Physics by Taylor & Wheeler

“The laws of physics are the same in every inertial reference frame”

• This means that if you derive a law of physics in one frame, you can apply it in another.

• Both the form of the laws of physics and the numerical values of the physical constants that the laws contain are the same in every frame.

• All inertial frames are equivalent in terms of every law of physics the laws of physics cannot provide a way to distinguish one inertial reference frame from another.

• Although the laws of physics are the same in every frame, measured quantities (like the time between events A and B, the distance measured between two points, even the force) will not be the same.

• Only the mathematical form of the laws of physics are the same.

Note: Special relativity can deal with accelerated motion, but only in inertial frames

Events

The concept of an “event” is crucial in SR.

An “event” is something that “happens” at a specific locationin space (x,y,z) and at a specific time, t

The key question in SR is: How do the coordinates x,y,z,t of a given event relate to the coordinates of the same event as measured by an observer in another inertial coordinate system.

One imagines a lattice-work grid to measure x,y,z and a bunch of“clocks” which are somehow synchronized by pulses of light, say,and then can record when a particular “event” happens at point x,y,z

These things seem intuitively obvious, but what is meant by a measurement at a point x,y,z,t is crucial.

“The Observer”

The word “observer”is a shorthand way ofspeaking of thewhole collection ofrecording clocks associated with oneinertial frame.

For simplicity, we will define our inertial coordinate systems fortwo observers in such a way that

the origins are chosen and the clocks synchronized in such a way that when the origins coincide, the clocks read zero.

The speed between the two systems is parallel to the x-axis.

v

xx'

yy'

z z'So we want to connect the coordinatesof some event in one frame (x,y,z,t) tothe coordinates of the same event inthe other frame, (x‘,y‘,z‘,t‘)

Before SR, the transformation was obvious:

The Galilean Transformation

tt

zz

yy

txx

v

However, the fact that physical constants are the same in all framesimplies that the SPEED OF LIGHT (c) is the SAME IN ALL FRAMES:

A pulse of light along the x-axis in the “unprimed” system travels

AA ctx in time At

In the “primed” system, it also travels BB ctx since light travelsat c in all frames

But:

AB

AAB

tt

txx

v

So the Galilean transformation predicts

BBAB cttxx v

Instead, one can show that the proper transformation is

the Lorentz Transformation.

)(

) v(

2v xtt

zz

yy

txx

c

where

€

≡ 1− v 2

c 2( )−1/ 2

or letting c

v

21

1

The reasons why the transformation takes this form are discussed in texts about SR, but note that since

we do satisfy the “pulse of light” argument, since

tt

22222

22222

tczyx

tczyx

transform correctly

The inverse of

)(

) v(

2v xtt

zz

yy

txx

c

)(

) v(

2v xtt

zz

yy

txx

c

is

which looks like the original, except that “primed” and “unprimed”

variables are interchanged, and v is replaced by -v

Note:

deviations from Galilean transformations become large when

1~c

v

Our every-day experience involves c v

satellite ain 2x10 ~ v

planejet ain 10~v5-

-6

c

c

Some implications of what this means:

Length Contraction K frame: unprimedK’ frame: primed

Consider a rigid rod of length 120 xxL at rest in frame K’

What is the length in frame K? 12 xxL

where x2 and x1 are the positions of the ends of the rodmeasured at the same time t in the K frame.

L

xx

xxL

)( 12

120

€

L = 1− v 2

c2( )1/ 2

L0

The rod appears shorter by a factor

€

1

γ= 1− v 2

c2( )1/ 2

This is symmetric:

K sees a rod which appears shorter by a factor of 1/γ

K’ would see the same rod held up by an observer in the K frame contracted by 1/γ

This is because the observers in the 2 frames would not agree that the ends of the stick were measured at the same time in both frames.

Time Dilation

Suppose a clock at rest at the origin of the K’ frame measures an interval of time

120 ttT

What time interval does an observer in frame K measure?

)( 12

12

tt

ttT

In K’ we have x’=0, so the time interval measured in K is

0 TT 21

1

The moving clock appears to have slowed down.

Similarly, K’ will think K’s clocks have slowed down.

Relativity of Simultaneity

Events which are simultaneous in one frame are NOT simultaneousin any other frame.

All the clocks in your inertial frame seem to be going at the same rate,but clocks in the moving frame differ from one another dependingon their LOCATION.

€

t = γ ′ t + vc2 ′ x ( )

So at a given t in the K frame, xt 2cv must be some fixed

value for all clocks in the K’ frame

Thus the greater x’ is, the smaller t’ is.

This effect is the explanation for most “paradoxes” in SR.

To illustrate, Imagine we have two clocks in the K’ frame at positionsA and B, separated by length L’.

A flashbulb, located exactly in the middle, goes off.

A and B are instructed to set their clocks to t’=0 when the flash reaches them, which in the K’ frame, happens at exactly the same time

K’

L’

What does K see?

What does K see?

At t=0, the flash goes off, and the wavefront starts to expand

At t=tA, it reaches Point A, and Aresets his clock at t’ = 0

K

Sometime later, at t=tB, thewavefront reaches B, and he resetshis clock, also at t’ = 0

t=0

t=tA

t=tB

A B

vv

A

A

B

B

Thus, to an observer in the K frame,A has reset his clock BEFORE B reset his clock.... whereas in K’, the clocks were reset SIMULTANEOUSLY.

Cosmic-Ray Mesons: Observed time dilation

Bruno Rossi & D.B. Hall 1941, Phys Rev, 59, 223

Cosmic Rays impinging on the top of the Earth’s atmosphere produce μ-mesons which decay via

21 ewith some half-life.

neutrinos

• The mesons travel down through the Earth’s atmosphere at v ~ c

• From the half-life for decay, one can estimate how many should be seen at sea-level vs. high in the atmosphere, given the travel time.

• Many more are seen at sea level than expected their radioactive “clock” appears to be running slow.

GPS: Observed time dilation

• GPS system consists of a network of 24 satellites in high orbits around the Earth.

• Each satellite has an orbital period of about 12 hours, and an orbital speed of about 14,000 km/hour

• The satellite orbits are arranged so that at least 4 and sometimes as many as 12 satellites are visible from any point on Earth.

• Each satellite carries an atomic clock which “ticks” with an accuracy of 1 nanosecond. (Actually they each carry 2 cesium clocks and 2 rubidium clocks).

• A GPS receiver determines your position by comparing time of arrival signals from a number of different satellites, and thus figuring out how far you are from each satellite -- “triangulation.”

• The inexpensive hand-held GPS receivers can determine your position to 5 or 10 meters. To achieve this accuracy you need to know the clock ticks to an accuracy of 20-30 nanoseconds. Military applications and airliners have more accurate GPS receivers.

• Special relativity: the clocks on board the satellites will run slower than atomic clocks on the ground by 7 microseconds per day because of time dilation.

• General relativity: The clocks on board the satellites run FASTER than atomic clocks on the ground by 45 microseconds a day because the curvature of space-time from the Earth’s gravitational field is LESS farther from the surface.

• If you don’t take these two effects into account then you would be as much as 10 km off from the build up of errors each DAY.

• To solve this basically the number of cesium atom “ticks” per second is redefined for the clocks on the satellites so that “one second” on the satellite is the same as “one second” on the ground. (on the ground a CS-133 clock frequency is 9,192,631,770 Hz)

TRANSFORMATION OF VELOCITIES

Suppose a particle has velocity ),,( zyx uuuu

in frame K’

What is its velocity ),,( zyx uuuu

in frame K?

x x’

y y’

v

K K’

)(

) v(

2v xtt

zz

yy

txx

c

LorentzTransform

Differentials

)(

)d v(

2v xdtddt

zddz

yddy

txddx

c

so...

22

v1

v

v

v

cu

u

cxd

td

tdxd

dt

dxu

x

xx

2

v1

cu

uu

x

yy

2

v1

cu

uu

x

zz

Note: These formulae can be used to add velocities

e.g. Suppose ccux 5.0 vand 5.0

In Galilean transforms, you expect

c v xx uu

But, using the Lorentz-transform formulae

cccc

ux

5

4

5.01

5.05.02

sums of velocities are ALWAYS < c

TRANSFORMATION OF ACCELERATIONS

Similarly, one can ask how accelerations transform from K to K’

velocity u

2

v1

v

c

u

xx

x

uu

so

2

22

2

2

v

22

22

v1

v1

1

v

v1

v

v1

2

2

cu

du

du

cu

c

du

c

u

u

cu

dudu

x

x

x

x

c

x

x

x

x

xx

Recalltd

c

u

c

xdtddt x

22

v1 v

So3

23 v

1

c

udt

dua

x

tddu

xx

x

3

23 v

1

cu

aa

x

xx

Similarly, can show

3

22

2

2

22 v

1

v

v

1

cu

ac

u

cu

aa

x

xy

x

yy

So ax and ay are related to ax’ and ay’ by some fairly complicatedexpressions involving not only v, but also ux’ and uy’

If a body is instantaneously at rest in the K’ frame, thenux’ = uy’ = 0 and ax is diminished by γ3

ay is diminished by γ2

Transformation of Angles (Aberration of Star Light)

We saw how ux, uy, uz transformed to ux’, uy’, u’z for v = relatively velocity of frames along x-axis.

We could also write u in terms of the components parallel and perpendicular to v, where v is in some arbitrary direction with respect to x,y

Then

2||

||||

v1

v

c

u

uu

2||v

1

c

u

uu

x

y

x’

y’

||u

usinu u

uu’ cosu || u

The angle

€

tanθ =u⊥u||

=u⊥

′

γ 1+vu||

′

c 2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1+

vu||′

c 2

u||′ + v

= u⊥

′

γ u||′ + v( )

=′ u sin ′ θ

γ ′ u cos ′ θ + v( )where uu

The interesting case is u=u’=c then

cv

cos

sintan