Module 2 - Relativity
Part I
I.
Conflict Between Mechanics and E&M
A.
Mechanics
Galilean relativity states that it is impossible for an observer to experimentally
distinguish between uniform motion in a straight line and absolute rest. Thus, all
states of uniform motion are equal! It is possible to distinguish between
accelerated and non-accelerated motion (see Chapter 15 Vol. I of Feynam Lecture
Series).
B.
E&M
InitiallyThe initial interpretation of the speed of light in Maxwell's theory was this c was
the speed of light seen by observers in absolute rest with respect to the eather. In
other reference frames, the speed of light would be different from c and could be
obtained by the Galilean transformation.
ProblemIt would now be possible for an observer to distinguish between different states of
uniform motion by measuring the speed of light or doing other electricity,
magnetism, and optics experiments.
C.
Possible Solutions
1.
Maxwell's theory of electricity and magnetism was flawed. It was
approximately 20 years old while Newton's mechanics was approximately
200 years old.
2.
Galilean relativity was incorrect. You can detect absolute motion!
3.
Something else was wrong with mechanics (ie Galilean transformation).
D.
Experimental Results
Most physicists felt that Maxwell's equations were probably in error. Numerous
experiments were performed to detect the motion of the earth through the eather
wind. Every one of these experiments failed! The most famous of these
experiments was the Michelson-Morley experiment. Because of the tremendous
precision of their interferometer, it was impossible for Michelson and Morley to
miss detecting the effect of the earth's motion through the eather unless mechanics
was flawed!
II.
Michelson-Morley Experiment
A.
Setup
Full Mirror
L2
Half Mirror
Full Mirror
L1
u
Screen
Earth's velocity with
respect to the eather
The Michelson-Morley experiment is a race between light beams. The incoming
light beam is split into two beams by a half-silvered mirror. The beams follow
perpendicular paths reflecting off full mirrors before recombining back at the half
mirror. Time differences are seen in the interference pattern on the screen.
B.
Purpose - To detect the relative motion of the earth through the eather.
C.
Theory We will simplify the calculations by assuming that L1 = L2 = L and neglecting
the difficulty in measuring these lengths. A more complete discussion is provided
in Modern Physics by Bernstein, Fishbane, and Gasiorowicz.
The time required to complete path 1 (horizontal path) is given by
T1 
L
L

cu cu
where we have used the Galilean transformation and velocity = distance/time.
T1 
L
c  u  c  u   22Lc 2
2
c u
c u
2




2L 
1

T1 
c   u 2 
1    
  c  
Since u/c = 10-4, we can use the binomial approximation
2
2L   u  
T1 
1    
c   c  
We can determine the time required to complete path 2 (vertical path) using the
distance diagram below:
T
c 2
2
c
L
u T2
T2
2
u T2
Using the Pythagorean theorem, we have
2
 T2 
2
2
c 2   L  u T2 


c 2 T2  L2  u 2 T2
2
c
T2 
2
2

 u 2 T2  L2
L2

c2  u 2

2

2
L2
  u 2 
c 1    
  c  
2
T2 
2L
2
u
c 1  
c
Again, using the binomial approximation
2L
T2 
c
 1  u 2 
1    
 2  c  
Thus, the time difference for the two paths is approximately
L 1  u 
ΔT  T1 - T2  2   
c  2  c 
2
 L  u 2
  
 c  c 
We can now calculate the phase shift in terms of wavelengths as follows:
f λc
λ
c
T
λ  cT
Thus, the phase shift in terms of a fraction of a wavelength is given by
u
 λ  c T  L  
c
λ Lu
  
λ
λc
2
2
Using a sodium light,  = 590 nm, and a interferometer with L = 11 m, we have
 4
Δλ 
11 m
 10
 
7
λ
5.90
x
10
m




2
 0.2
This was a very large shift (20%) and couldn't have been over looked.
D.
Result - No shift was ever observed regardless of when the experiment was
performed or how the interferometer was orientated!
E.
Fitzgerald's Solution
The Irish physicist Fitzgerald proposed a solution to the Michelson-Morely
experiment. Fitzgerald suggested that the arm of the interferometer parallel to the
motion of the earth through the eather had contracted! Fitzgerald reasoned that in
order to make T1 = T2 in our previous work, the length of the arm in path 1 must
2
u
be changed to L   L 1    .
c
Most physicists were unconvinced by Fitzgerald's argument since it appeared to
have no physical basis.
F.
Lorentz's Idea
H.A. Lorentz suggested that the contraction of the interferometer's arm could be
a property of the newly discovered electron. Since rulers were also composed of
electrons, a moving ruler would also be contracted. Thus, it would be impossible
to detect the contraction effect by measuring the interferometer with a ruler!
III.
Lorentz and the Lorentz Transformation
A.
Lorentz's Method
One of the main reasons physicists accepted H.A. Lorentz's work more seriously
than Fitzgerald's is that Lorentz attempted to base the theory upon a more solid
physical basis (property of newly discovered electron).
Lorentz was able to provided additional physical support by developing an
electromagnetic theory of the phenomena from the following facts:
1)
Michelson-Morely experiment appears to validated Maxwell's theory of
electricity and magnetism
2)
Galilean transformation is incorrect because Maxwell's equations are not
invariant under a Galilean transformation.
Lorentz needed to find a set of transformation equations
1)
that reduced to the Galilean transformation at slow speeds
2)
under which Maxwell's equations were invariant.
B.
Lorentz Transformation
These equations that Lorentz found are called the Lorentz transformation
equations and are given below for 1-D motion in the x-direction. These equations
replace the flawed Galileo transformation equations in relating the
measurements of two different observers in uniform motion relative to each
other.
Lorentz Transformation
x'  γ x  vt 
y'  y
z'  z
 xv 
t'  γ  t  2 
 c 
1
γ
1 β2
β
C.
v
c
Addition of Velocities
In Lesson 1, we reviewed how the "addition of velocities" was developed as a
consequence of the definition of velocity and the Galileo transformation
equations. Since the Galileo transformation is only a low speed approximation of
the Lorentz transformation, we must find the correct formula for "adding
velocities."
We start by taking the derivative with respect to t' of the first Lorentz
transformation equation:
dx'
dt 
 dx
 γ v .
dt'
dt' 
 dt'
From Calculus, we have that
dx'

dt'
dt 
 dx dt
γ
v 
dt' 
 dt dt'
dx'

dt'
 dx
 dt
γ   v .
 dt
 dt'
Using the definition of velocity, we have that
u '  γ u  v
dt
dt'
Differentiating the fourth Lorentz equation with respect to t, we obtain
dt'  dt v dx 
γ  2

dt
 dt c dt 
dt' 
v
 γ 1  2
dt
 c
 γ
u    c 2  u v 

 c2 
dt
c2
1


dt'  2
γ c  u v  γ1  u v 

 

 c2 
We now substitute this result into our velocity equation to obtain
u '
uv

uv
1
c2
 uv 

c
 c  βu 
where u is velocity of object in x dir. seen by unprimed observer
u ' is velocity of object in x dir. seen by primed observer
v is vel. in x dir. of primed observer as seen by unprimed ob.
The results above only deal with the velocity component of the object being
measured in the direction of the relative motion of the two observers (i.e. the xcomponent based upon our setup). We could also develop a similar relationship
for the y & x components of the object’s velocity. In these cases, time is still
effected, but not the y or z-component (i.e. the denominator is still changed but
not the numerator).
u y '
EX:
uy
uz '
 u v
γ1  x 
c2 

uz
 u v
γ1  x 
c2 

Show that both observers agree on the speed of light regardless of there relative
speed.
Soln: We determine the speed u' as seen by the primed observer for a beam of light u = c
seen by the un-primed observer. Using the velocity addition formula we have:
u '
IV.
cv
cv  cv


cc
cv
v cv
1
1
c
c2
Lorentz-Fitzgerald (Length) Contraction
Fitzgerald's length contraction is now a direct consequence of the Lorentz
transformation. Consider two different observers in relative motion who measure
the length of a box as shown below:
v
v
x1
x1'
x2
observer sees box moving at v
x2'
observer sees stationary box
At a single instant of time t, the unprimed observer measures the box's length as
L = x 2 - x1
while the unprimed observer measures the box's proper length as
Lo = x2' - x1'
Using the Lorentz transformation equation for x', we have that
Lo  γ x 2  vt 2  γ x1  vt1 
Lo  γ x 2  x1  γv vt 2  vt1  γ L  γv vt 2  vt1 
In order for L to be the length of the box, observer #1 must measure both x1 and
x2 at the same time (i.e. t1= t2). Therefore, we have that
Lo  γ L
Q.E.D.
For two observers in uniform motion relative to each other, the measurements of
an object's length in the direction of motion are related by the Lorentz-Fitzgerald
contraction. The observers agree on all object lengths that are perpendicular to the
direction of relative motion.
Lorentz-Fitzgerald (Length) Contraction
v
L  Lo 1   
c
2
where L is the improper length
Lo is the proper length
v is the relative speed between the observers
The question that must be answered is "who is the proper observer of length?" It
depends on the length that is being measured! Use the steps below to ensure that
you correctly identify this observer.
The Two Steps to Success
1.
2.
Mentally nail a ruler on the length or distance you wish to measure.
If an observer can grab the ruler during the problem without getting a splinter
(ie the observer sees the ruler as stationary) then they are a proper observer of
length otherwise they are an improper observer of length!
Warning: Your everyday use of English terms can get you in trouble in this material.
The term "proper" is not meant to indicate that its the "correct" or "right" answer. Two
observers might measure the length of a train as 10 m when they see the train as
stationary. If one observer rides in the train as it moves down the track, they will measure
the train's length as 10 m while an observer standing by the track sees the train as shorter
than 10 m. Both Observers Are CORRECT!!! The question "What is the length of
the train?" is meaningless!! You must specify the frame. There is no such thing as
absolute distance anymore!
V.
Time Dilation
Another interesting aspect of relativity is that moving clocks always run slow.
This is not an artifact of the clock! It is a consequence of the nature of time itself
according to Einstein. All clocks will behave this way include your biological
processes!
Let us consider a "Light Clock" in which one unit of time corresponds to the time
it takes a light pulse to travel between two mirrors a distance L apart. We will
place the clock on a train traveling at speed u with the unprimed observer.
Mirror
L
Light Path As Seen By Train Observer
For the unprimed observer on the train, the time it takes the light pulse to make a
single tick is given by
t
L
c
The primed observer standing by the railroad track see the train and clock pass
with a speed of u. Thus, the path of the light beam appears to follow the diagonal
path shown below:
ct'
L
v t'
From the Pythagorean theorem, t' is given by
c t' 2  v t'2  L2
c 2 t' 2  v 2 t' 2  L2
c
2
 v 2  t' 2  L2
L2
t'  2
c  v 2  
L2
2
t' 
 v2 
c 1    
 c 


2
L
v
c 1  
c
2
γ
L
c
We now plug in our results for the unprimed observer and we find that
t'  γ t
Thus, our observers do not agree upon time!
Time Dilation Equation
T '
where To is the proper time
T is the improper time
To
v
1  
c
2
v
Possible Methods for Determining the Proper Time Observer:
1.
If an observer is the observer of proper length then they are NOT the
observer of proper time.
2.
The person at the start and finish line of a race is the proper observer of
time as long as they have not accelerated!
The time dilation equation also comes from the Lorentz equations. The person
measuring the proper time between two events is the observer who sees the events
at the same position.
x v 
x v

t'2  t'1  γ  t 2  22   γ  t1  12 
c  
c 

t'2  t'1  γt 2  t1   γ
v
x  x 
c2 2 1
If observer #1 sees both events happening at the same location (i.e. x2 = x1) then
they observe the proper time (i.e. t2 – t1 = To). Thus, we have
t'2  t'1  γTo   γ
v
0
c2
T  γ To
Q.E.D.
VI.
Loose Ends
It is important to realize that the Lorentz transformation is only required when
comparing measurements made by observers in different reference frames. If all
the information is obtained by a single observer then you don't need
transformation equations.