This post was inspired by Chandru Iyer’s post *here*.

Consider a light ray sent a certain *distance* *s* that is immediately reflected back. According to Newtonian mechanics if a light ray travels at speed *c*, then for a body moving at speed *v *relative to the stationary frame, the light ray should travel at the speed *c − v* one way and at speed *c + v* the other way.

The *total distance* of the light ray is 2*s*. The *total time* of the light ray is

Then the mean speed is

However, according to Einstein’s relativity theory, the mean speed of light is a constant, *c*. So the above speed needs to be multiplied by the gamma factor squared, *γ*². As Iyer notes, this is accomplished by contracting the moving rulers by the factor (1/*γ*) and dilating the moving clocks by the factor *γ*.

**But that approach is wrong.** The correct approach follows.

Consider a light ray sent a certain *distance* *s* that is immediately reflected back, for a total distance of 2*s*. It is important to note that distance is the independent variable and time is a dependent variable, and so the speed is *inverse speed*, which is added by *reciprocal addition*. With *c* as the inverse speed of light, the total time is 2*s*/*c*.

The inverse speed of a longitudinally moving observer is *v*. For a body moving at inverse speed *v *relative to the stationary frame, the light ray should travel at the inverse speed 1/(1/*c − 1/v*) one way and at the inverse speed 1/(1/*c + 1*/*v*) the other way. Then the total time is

which is the same as that observed from the rest frame. The mean speed is

If time *were* the independent variable, the result would be the same:

Consider a light ray sent a certain *time interval* *t* that is immediately reflected back, for a total time of 2*t*. According to Newtonian mechanics if a light ray travels at time speed *c*, then for a body moving at speed *v *relative to the stationary frame, the light ray should travel at the time speed *c − v* one way and at time speed *c + v* the other way.

The *total time* of the light ray is 2*t*. The *total distance* of the light ray is

Again, this is the same as that observed from the rest frame. The mean speed is