Simple Nature - Light and Matter

Exercises

Exercise 7A: The Michelson-Morley Experiment

In this exercise you will analyze the Michelson-Morley experiment, and find what the results

should have been according to Galilean relativity and Einstein’s theory of relativity. A beam of

light coming from the west (not shown) comes to the half-silvered mirror A. Half the light goes

through to the east, is reflected by mirror C, and comes back to A. The other half is reflected

north by A, is reflected by B, and also comes back to A. When the beams reunite at A, part

of each ends up going south, and these parts interfere with one another. If the time taken

for a round trip differs by, for example, half the period of the wave, there will be destructive

interference.

The point of the experiment was to search for a difference in the experimental results between
the daytime, when the laboratory was moving west relative to the sun, and the nighttime, when
the laboratory was moving east relative to the sun. Galilean relativity and Einstein’s theory
of relativity make different predictions about the results. According to Galilean relativity, the
speed of light cannot be the same in all reference frames, so it is assumed that there is one special
reference frame, perhaps the sun’s, in which light travels at the same speed in all directions;
in other frames, Galilean relativity predicts that the speed of light will be different in different
directions, e.g., slower if the observer is chasing a beam of light. There are four different ways
to analyze the experiment:

• Laboratory’s frame of reference, Galilean relativity. This is not a useful way to analyze
  the experiment, since one does not know how fast light will travel in various directions.


• Sun’s frame of reference, Galilean relativity. We assume that in this special frame of
  reference, the speed of light is the same in all directions: we call this speedc. In this
  frame, the laboratory moves with velocityv, and mirrors A, B, and C move while the light
  beam is in flight.


• Laboratory’s frame of reference, Einstein’s theory of relativity. The analysis is extremely
  simple. Let the length of each arm beL. Then the time required to get from A to either
  mirror isL/c, so each beam’s round-trip time is 2L/c.


• Sun’s frame of reference, Einstein’s theory of relativity. We analyze this case by starting
  with the laboratory’s frame of reference and then transforming to the sun’s frame.

Exercises 465