Revised 2022-08-23.
There are many expositions of the famous Michelson-Morley experiment (for example here) but they all assume the variable in common is time, which is not the case. In fact, distance is the variable in common, 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:
A light source travels with velocity c to a beam splitter, whereupon part of it travels a displacement L longitudinally and is reflected back as −L, whereas another part travels a distance transversely and is reflected back. Meanwhile, the apparatus is thought to travel with velocity v relative to the aether; let it be longitudinal to the apparatus. Part of the light is sent to an observer, who looks for an interference pattern.
Since the distance L is in fixed, distance is the variable in common. 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 (1/c + 1/v) and T2 = −L (−1/c + 1/v) = L (1/c − 1/v).
The stationary observer measures the total longitudinal time, T, as
T = T1 + T2 = L (1/c + 1/v) + L (1/c − 1/v) = 2L/c
This is confirmed by calculating the space velocity as
T / L = (2L/c) / (2L) = 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 dual Galilean transform: t′ = t + x/v. If the light clocks coincide at t = 0, this is t′ = 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 in theory.
The configuration diagrammed above is as follows: the apparatus is presumed to travel with time speed v relative to the aether. In it a light source travels with time velocity c [which is not known] to a beam splitter, whereupon part of it travels a distimement T longitudinally and is reflected back as −T, whereas another part travels a distimement 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 variable in common. 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 (−c + v) = T (c − v).
The stationary observer measures the total longitudinal distance, L, as
L = L1 + L2 = T(c + v) + T(c − v) = 2cT.
This is confirmed by calculating the time velocity 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.
This opens up the prospect of physics without the postulate of the constancy of the velocity of light.