Michelson-Morley re-examined

There are many expositions of the famous Michelson-Morley experiment (for example here) but they all assume the independent variable is time, which is not the case. As we shall see, distance is the independent variable, and so the experiment is temporo-spatial (1+3). Let us examine the original experiment as it should have been done:

The configuration diagrammed above is as follows: the apparatus is presumed to travel with pace w relative to the aether. In it a light source travels with pace k = 1/c to a beam splitter, whereupon part of it travels a distance L longitudinally and is reflected back, whereas another part travels a distance transversely and is reflected back. Part of the light is sent to an observer, who looks for an interference pattern.

Since the distance L is fixed, distance is the independent variable. In the stationary frame the round trip longitudinal distance between the beam splitter and the mirror is 2L. Let the time that light travels longitudinally from the beam splitter to the mirror be T1, and let the time for the return be T2. Then

T1 = L (k + w) and T2 = L (kw).

The stationary observer measures the total longitudinal time, T, as

T = T1 + T2 = L (k + w) + L (kw) = 2kL = 2L/c

This is confirmed by calculating the pace of motion as

T / L = 2kL / (2L) = k = 1/c

The diagram above also shows the transverse motion, where there are two components of motion: the transverse y-axis and the longitudinal x-axis. These components are independent of one another since they are in different dimensions. The y-axis is the same as the stationary case above: the total time is 2T and the total distance is 2cT. The mean speed is 2cT/(2T) = c.

The longitudinal axis is simply the Galilean transform: t′ = t + wx = t + x/v. If the light clocks coincide at t = 0, this is t′ = wx = x/v. which is what was given.

Thus the distance, time, and mean speed are the same for both observers and independent of their relative velocity, which is what the Michelson-Morley experiment found.

Let us now compare the corresponding spatio-temporal (3+1) experiment.

The configuration diagrammed above is as follows: the apparatus is presumed to travel with speed v relative to the aether. In it a light source travels with speed c to a beam splitter, whereupon part of it travels a time interval T longitudinally and is reflected back, whereas another part travels a time interval transversely and is reflected back. Note: in practice this is impossible.

The beam splitter sends part of the light to an observer, who looks for an interference pattern. Since the time interval T is fixed, time is the independent variable. In the stationary frame the round trip longitudinal time interval between the beam splitter and the mirror is 2T. Let the distance that light travels longitudinally from the beam splitter to the mirror be L1, and let the distance for the return be L2. Then

L1 = T (c + v) and L2 = T (cv).

The stationary observer measures the total longitudinal distance, L, as

L = L1 + L2 = T(c + v) + T(cv) = 2cT.

This is confirmed by calculating the speed of motion as

L / T = 2cT / (2T) = c

The diagram above also shows the transverse motion, where there are two components of motion: the transverse y-axis and the longitudinal x-axis. These components are independent of one another since they are in different dimensions. The y-axis is the same as the stationary case above: the total time is 2T and the total distance is 2cT. The mean speed is 2cT/(2T) = c.

The longitudinal axis is simply the Galilean transform: x′ = x + vt. If the light clocks coincide at x = 0, this is x′ = vt, which is what was given.

Thus the distance, time, and mean speed would be the same for both observers and independent of their relative velocity.