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 2*L*. Let the time that light travels longitudinally from the beam splitter to the mirror be *T*_{1}, and let the time for the return be *T*_{2}. Then

*T*_{1} = *L* (*k* + *w*) and *T*_{2} = *L* (*k* − *w*).

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

*T* = *T*_{1} + *T*_{2} = *L* (*k* + *w*) + *L* (*k* − *w*) = 2*kL* = 2*L*/*c*

This is confirmed by calculating the pace of motion as

*T* / *L* = 2*kL */ (2*L*) = *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 2*T* and the total distance is 2*cT*. The mean speed is 2*cT*/(2*T*) = *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 2*T*. Let the distance that light travels longitudinally from the beam splitter to the mirror be *L*_{1}, and let the distance for the return be *L*_{2}. Then

*L*_{1} = *T* (*c* + *v*) and *L*_{2} = *T* (*c* − *v*).

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

*L *= *L*_{1} + *L*_{2} = *T*(*c* + *v*) + *T*(*c* − *v*) = 2*cT*.

This is confirmed by calculating the speed of motion as

*L* / *T* = 2*cT */ (2*T*) = *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 2*T* and the total distance is 2*cT*. The mean speed is 2*cT*/(2*T*) = *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.