special theory of relativity ii

8/13/2019 Special Theory of Relativity II

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Lecture 04

Special Theory of

Relativity II


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Maxwells Equations

Maxwells equations DO NOT have the sameform under a Galilean transformation.

In Maxwells theory the speed of light, in termsof the permeability and permittivity of free

space, was given by

Thus the velocity of light between movingsystems must be a constant.

1

v = c = 1 / 00


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Light Speed Contradiction

Based on Maxwells theory,

But classical mechanics tells us that

Which one is correct?

c = c '

c = c '+ v

2


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A Preferred Reference System

Ether was proposed as an absolute referencesystem in which the speed of light was this

constant and from which other measurementscould be made.

The Michelson-Morley experiment was anattempt to show the existence of ether.

Earth

Ether

v

v 10-4c

3


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The Michelson Interferometer

AC is parallel to the motionof the Earth.

Light from source S is splitby mirror A

and travels

to mirrors Cand D in mutually

perpendiculardirections.

After reflection the beamsrecombine at A slightly outof phase as viewed by

telescope E.

4


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Interferometer Fringe Pattern

Beam-splitter

Inputbeam

Mirror

Mirror

z

x

Earth

Drift

5


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Wave

6

(x,t) =Asin(kxt+ 0)

whereAis amplitude,

=kx t+ 0 is phase, and 0 is the initial phase,

kis wave number: k=2

, where is wave length,

is angular velocity: = 2T

, where Tis period,

f =1

Tis frequency, the phase speed v =f =

T=

k.


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Wave Interference

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Wave Interference

8

Sources S1 and S2 are coherent wave sources

Constructive Interferences: r2 r

1= m (m = 0,1,2,3,...)

Destructive Interferences: r2 r

1= (m + 0.5) (m = 0,1,2,3,...)


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Wave Interference

9

Sources S1 and S2 are coherent wave sources

Constructive Interferences: = 2m (m = 0,1,2,3,...)

Destructive Interferences: = (2m +1) (m = 0,1,2,3,...)


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Wave Interference: Double-Slit

10

At the center, = 0


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Interferometer Fringe Pattern

Beam-splitter

Inputbeam

Mirror

Mirror

z

x

Earth

Drift

The fringe pattern should shift if

the apparatus is rotated by 90.

11

At the center, = t = t2f

n=

'

2=

(

t'

t)f


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Galilean Velocity Transformation

P

v

vP =

v 'P+

v

12

vP

v

v 'P

vP

vv 'P

vP

v

v 'P

v '

P = v

p v

v 'P = v

p+ v

v 'P = v

P

2 v

2


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The Analysis

Time t1 from A to C and back:

Assuming the Galilean Transformation

13

v

l1

l2

AC

D

Parallel and

anti-parallelpropagation

Time t2from A to D and back:

Perpendicular

propagation

t2 =

2l2

c2v

2

=

2l2

c

1

1v2/ c

2


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The Analysis

So that the change in time is:

Assuming the Galilean Transformation

14

t' = t'2 t'

1 =

2

c

l2

1 v2 / c2 l1

1 v2 / c2

Upon rotating the apparatus, the optical path lengths 1and 2are interchanged producing a different change in

time: (note the change in denominators)

t = t2 t

1 =

2

c

l2

1 v2 / c2 l11 v2 / c2


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The Analysis

and upon a binomial expansion, assumingv

/c


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Michelson-Morley Experiment

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Michelson and Morley's results

from A. A. Michelson, Studies in

Optics

Interference fringes showed nochangeas the interferometer

was rotated.

The Michelson interferometer

shouldve revealed a fringe shift as

it was rotated with respect to the

ether velocity. MM expected 0.4periods of shift (11 m for l1and l2) and

could resolve 0.005 periods.

They saw none!Their apparatus

Nobel Prize in 1907


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In several repeats andrefinements with

assistance fromEdward Morley, he

always saw a null

result.

He concluded that thehypothesis of the

stationary ether must

be incorrect.

Thus, ether seemsnot to exist!

Michelsons Conclusion

Edward Morley

(1838-1923)

Albert Michelson

(1852-1931)

18

Nobel Prize in 1907: One of the most famous failed expt.


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The Lorentz-FitzGerald Contraction

Another hypothesis proposedindependently by both H. A. Lorentzand G. F. FitzGerald suggested thatthe length in the direction of the

motion was contracted by a factorof

Therefore

20

t =t'=

2

c

l2

1 v2 / c2

l1

1 v2 / c2

George F. FitzGerald

(1851-1901)

Hendrik A. Lorentz

(1853-1928)

This, however, was an ad hoc assumption that could not be

experimentally tested.


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Assignments

Reading: Ch. 2.3, 2.4.

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special theory of relativity ii

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