Physics 2102A/B Lecture 2: Ch2-Special_Relativity
1 May 2016
School
Department
Course
Professor
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Chapter 2. Special Relativity
Notes:
• Some material presented in this chapter is taken “The Feynman Lectures on
Physics, Vol. I” by R. P. Feynman, R. B. Leighton, and M. Sands, Chap. 15 (1963,
Addison-Wesley).
2.1 The Ether and the Michelson-Morley Experiment
As we saw at the end of Chapter 1, the prediction and experimental verification of the
propagation of electromagnetic waves in space led physicists to the conclusion that there
should exist a medium, the ether, permeating all of space on which these waves travelled.
Of course, the wave equation we derived from Maxwell’s equations (see equation (1.24)
in Chapter 1) does not at all require that. It does, in fact, imply that such waves could
propagate in vacuum. But this notion of wave propagation without a medium was so
counterintuitive to physicists at the time that they elected to postulate the existence of the
ether. On the other hand, this hypothesis had the advantage of being testable through
experiments.
This is exactly what American physicists Albert Michelson (1852-1931) and Edward
Morley (1838-1923) sought out to do in a famous experiment (i.e., the Michelson-
Morley experiment) in 1887. To do so they used an apparatus (now called a Michelson
interferometer) as shown in the schematic of Figure 1. In a nutshell, the experiment
consists of sending a light signal (from Source A in the figure) and splitting it over two
mutually orthogonal paths (at the plate B), each propagating through a distance
L
to a
mirror (mirrors C and E in the up-down and left-right directions, respectively) where they
are reflected back and recombined (“below” the plate B).
Source
A
B B0
C C0
D D0
E E0
F F 0
u
L
L
λ
∆x
Waves
in phase
Waves out
of phase
Figure 1 - Schematic diagram of the Michelson-
Morley experiment (from The Feynman Lectures
on Physics, Vol. I).
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Michelson and Morley surmised that if the earth is moving relative to the ether at a speed
u
in the B-E direction, for example, then the electromagnetic wave from the light source
travelling along that direction should make its round trip from plate B to mirror E and
back to plate B in a different amount of time than the wave propagating along the B-C-B
path. More precisely, since the apparatus is moving in the B-E direction the
corresponding velocity of the light wave relative to the interferometer should be
(according the Newtonian mechanics) equal to
c−u
. It follows that the time needed for
the light wave to go from B to E is
t1=L
c−u
.
(2.1)
Once the wave reflects on mirror E, its velocity relative to the apparatus becomes equal to
c+u
and the time needed to travel from E to B is
t2=L
c+u
.
(2.2)
We then find that the time necessary for the round trip is
tE=t1+t2
=2L c
1−u2c2
( )
.
(2.3)
For the light wave going from plate B to mirror C and back the speed relative to the
interferometer is simply
c
, since its velocity is perpendicular to that of the apparatus. The
distance the light goes through when going from B to C is given by
d3=ut3
( )
2+L2,
(2.4)
where
t3
is the time needed for the wave to travel that distance. But since we also have
t3=d3c
, we can write
ct3
( )
2=ut3
( )
2+L2,
(2.5)
or
t3=L c
1−u2c2
.
(2.6)
It follows that the round trip B-C-B is
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tC=2L c
1−u2c2
.
(2.7)
According to the ether hypothesis, the time difference between the two paths should have
been
Δt=tC−tE
=2L c
1−u2c2
1−1 1 −u2c2
( )
≠0.
(2.8)
This non-zero time difference
Δt
predicted that Michelson and Morley would observe
that, when recombined, the two light waves should not be in phase with each other. That
is, if when they left the plate B at the initial time
t0
the waves had the same amplitude
E t0
( )
=Acos
φ
0
( )
,
(2.9)
then, when recombined, the total amplitude of the signal should had been
ETt
( )
=Acos
ω
tC+
φ
0
( )
+Acos
ω
tE+
φ
0
( )
=Acos
ω
tC+
φ
0
( )
+cos
ω
tC−
ω
Δt+
φ
0
( )
⎡
⎣⎤
⎦
=2Acos
ω
Δt
2
⎛
⎝
⎜⎞
⎠
⎟cos
ω
tC+
φ
0−
ω
Δt
2
⎛
⎝
⎜⎞
⎠
⎟,
(2.10)
where we used the identity
cos
θ
1
( )
cos
θ
2
( )
=cos
θ
1−
θ
2
( )
+cos
θ
1+
θ
2
( )
⎡
⎣⎤
⎦2
.
However, the signal Michelson and Morley detected corresponded to
ETt
( )
=2Acos
ω
tC+
φ
0
( )
,
(2.11)
which implied that
Δt=0
! In other words, the time taken by the light waves to travel
their respective paths was the same, just as if the speed of light was the same in both
orientations. This result was directly at odds with the idea that electromagnetic waves
propagated on the hypothetical ether.
It interesting to note the observation of Lorentz who suggested that the result could be
explained if bodies (like the B-E leg of the Michelson interferometer) contracted
according to
LE=L1−u2c2,
(2.12)
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Document Summary
Notes: some material presented in this chapter is taken the feynman lectures on. I by r. p. feynman, r. b. leighton, and m. sands, chap. Of course, the wave equation we derived from maxwell"s equations (see equation (1. 24) in chapter 1) does not at all require that. It does, in fact, imply that such waves could propagate in vacuum. But this notion of wave propagation without a medium was so counterintuitive to physicists at the time that they elected to postulate the existence of the ether. On the other hand, this hypothesis had the advantage of being testable through experiments. This is exactly what american physicists albert michelson (1852-1931) and edward. Morley (1838-1923) sought out to do in a famous experiment (i. e. , the michelson- To do so they used an apparatus (now called a michelson interferometer) as shown in the schematic of figure 1. Figure 1 - schematic diagram of the michelson-
