Relativity II
I.
Henri Poincare's Relativity Principle
In the late 1800's, Henri Poincare proposed that the principle of Galilean relativity
be expanded to include all physical phenomena and not just mechanics. This view
was supported experimentally by the Michelson-Morely experiment. Poincare's
work was very important because it had a great influence upon young Albert
Einstein.
The laws of physics are identical in all inertial frames of reference.
equivalently
All reference frames in uniform linear motion are equivalent. You can not
physically determine absolute motion.
II.
Albert Einstein's Postulates
We have previously seen that H.A. Lorentz developed the basic equations of
special relativity based upon a theory of the electron and determining the
transformation equations under which Maxwell's equations were invariant. We
now consider the man who transformed physics in the twentieth century.
Einstein considered the problem in a radically different way. First, he began with
two fundamental postulates:
1)
The laws of physics are the identical in all inertial reference frames.
2)
The speed of electromagnetic radiation in vacuum is constant, independent
of any motion of the source.
To Lorentz, the fact that all observers measure the speed of light was a
consequence (result) of his theory based on the electron. It had no deeper
meaning.
To Einstein, this fact about light was too unique to be a coincidence. Thus, it was
the underlying principle upon which nature operated. Likewise, Einstein felt that
Poincare's relativity principle was supported by experimental evidence and must
be a fundamental principle upon which to build a theory. Lorentz theory was
based upon unverified properties of electrons and the ether while Einstein saw no
need for the ether at all. His theory was based upon experimentally verified
facts!
Einstein obtained the same equations as Lorentz but they had a far different
meaning. The interferometer didn't contract due to being built out of matter.
Space itself is contracted. It makes no sense to talk about the true length of an
object. It is the length of the object as seen in this reference frame. Our very
concepts about the nature of space and time must be modified!
III.
Events
A physical event is defined by spatial and time coordinates (x,y,z,t) or
equivalently (x',y',z',t').
IV.
Simultaneity
Events that appear simultaneous to one observer may not be simultaneous to a
second observer!
Example: Consider two light beams emitted in opposite directions from the middle of a
train traveling at 0.3 c as shown below. According to a person on the train, the light
beams strike the detectors at the end of the train simultaneously at t = L/c. However, a
person standing by the train track believes the beam at the back of the train strikes first!
0.3 c
Space-Time Diagram
ct
ct'
x'
x
The green line shows that the train observer sees the two black events
simultaneously. However, simultaneous events according to the track observer
occur along the purple line with the line flowing toward the upper right corner for
increasing time. Black dashed line denotes the light cone.
V.
General Space Time Problems
In Newtonian Mechanics, space and time are independent. Einstein showed that
time and space are intertwined. Thus, it is not possible to separate space and time
in most problems. (Problems in Chapter 6 of Schaum’s Outline should help
clarify this.)
VI.
Classical Doppler Shift (See Freshman Physics Textbook)
Anyone who has watched auto racing on TV is aware of the Doppler shift. As a
race car approaches the camera, the sound of its engine increases in pitch
(frequency). After the car passes the camera, the pitch of the car’s engine
decreases. We could use this pitch to determine the relative speed of the car. This
technique is used in many real world applications including ultrasound imaging,
Doppler radar, and to determine the motion of stars.
A.
Moving Observer
Assume that we have a stationary audio source that produces sound waves of
frequency f and speed v. A stationary observer shown below sees the time
between each wave as
y
v
Source
u

Observer
Time 
Distance
Speed
1 λ
T 
f v
If the observer is now moving at a velocity u relative to the source then the speed
of the waves as seen by the observer is
x
Speed  v  u
where the positive sign is when the observer is moving toward the source. The
time between waves is now
T' 
1
λ

.
f ' vu
Taking the ratio of our two results we get that
T f ' vu
 
.
T' f
v
 vu 
f 'f 

 v 
B.
Moving Source
We now consider the case in which the source is moving toward the observer.
In this case, the wave's speed is unchanged but the distance between wave fronts
(wavelength) is reduced (increased) for the source moving toward (away) from
the observer as shown below:
y
u
uT
'
Source
v
Observer

From the diagram, we find the new wavelength as
λ'λ  u T
x
λ' λuT
T

1  u
λ
λ
λ
λ'
u
1 
λ
λf
λ'
u vu
1  
λ
v
v
c λ' v  u

λc
v
f vu

f'
v
Thus, the frequency seen by the observer for a moving source is given by
 v 
f ' f 
.
 vu 
Note: The motion of the observer and source create different effects. For sound, this
difference is explained due to motion relative to the preferred reference frame! This
preferred frame is the reference frame stationary to the medium propagating the sound
(air)!
Problem: Our classical derivation would imply that by measuring the Doppler shift of
light, you could determine if the observer or source was in motion (ie measure absolute
motion). Thus, our work is not consistent with relativity that requires the effects due to
motion by the observer and source to be the same. This problem caused Einstein to
discard the ether theory of light and thereby influenced his development of Relativity.
VII.
Relativistic Doppler Shift
Since light has no medium, there should be no difference between moving the
source and the observer. The problem with our previous derivation when dealing
with light was that we didn't considered that space and time coordinates are
different for the source and observer. Thus, we must account for the contraction
of space and dilation of time due to motion. After accounting for differences in
time and space, we get the following result for both moving source and moving
observer
f '
cu
f
cu
See Chapter 3 of Modern Physics by Bersetin, Fishbane, and Gasiorowicz for the
proof.
VIII. Newton Second Law and Linear Momentum
A.
Newton Second Law
Newton's Second Law has the same form in Special Relativity that it does in
Classical Physics. This guarantees that Special relativity produces the same
results as Newtonian mechanics at slow speeds.


dp
 Fext 
dt
B.
Classical Linear Momentum
According to the first postulate of relativity, the laws of physics are the same for
all inertial reference frames. Thus, all inertial observers should agree that linear
momentum is conserved in collisions!
Einstein discovered that for collisions the classical linear momentum,



dr
p  mo v  mo
, was not conserved for all observers under the Lorentz
dt
transformation!! Thus, there was a problem with calculating the linear
momentum!
Einstein discovered that the following quantity was conserved during collisions:


dr
p  mo
where to is the proper time.
dt o
However, this formula doesn't conceptually make sense as it involves the
measurement of position and time by different observers. Surely, the linear
momentum of an object doesn't require two different observers to have
reality!!
Using Calculus, we can rearrange the equation as follows:


d r dt

p  mo
 mo v
dt dt o

p








mo
1-  v 
c
2








1
v
1-  
c
2
.


v  mv
Thus, you could consider that it was the mass and not the velocity whose
calculation had to be modified!! Einstein actually only considered the concept of
rest mass to be useful. I have found that this philosophical debate has no
pedagogical value at the undergraduate level. Most undergraduate students are
better served both in understanding the connection between relativity and classical
physics and in solving practical engineering problems by considering m to be the
true mass. Thus, I will teach the material with this philosophical approach.
II.
Rest Mass and Relativistic Mass
The mass of an object measured by an observer depends on the motion of the
object relative to the observer in relativity.
A.
If the object is at rest with respect to an observer, its mass will be the lowest and
is called its rest mass, mo.
This is an intrinsic property of the object and is the mass that we used in
classical physics.
B.
An object moving at a speed v with respect to an observer will have a larger
apparent mass. This relativistic mass, m, is related to the object's rest mass by
the following formula for objects with non-zero rest masses:
m
mo
1  v 
c
C.
2
Linear Momentum
Newton's Second Law is still correct but you must use the correct mass
(relativistic mass) in calculating the linear momentum.