Reflected motion

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 2s. The total time of the light ray is

\frac{s}{c-v}+\frac{s}{c+v} = \frac{s(c+v)+s(c-v)}{c^2-v^2}=\frac{2sc}{c^2-v^2}=2s \gamma^2 /c

Then the mean speed is

\frac{2s}{\frac{s}{c-v}+\frac{s}{c+v}}=\frac{2s}{\frac{2sc}{c^2-v^2}}=\frac{c^2-v^2}{c}=c(1-(v^2/c^2))= c/\gamma^2

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, γ². 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 is not the correct approach.

Consider a light ray sent a certain distance s that is immediately reflected back, for a total distance of 2s. Since distance is the independent variable and time is a dependent variable, the speed is space speed, which is added by harmonic addition. With c as the space speed of light, the total time is 2s/c.

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

s\: /\left ( c^{-1}-v^{-1} \right )^{-1} + s\: /\left ( c^{-1}+v^{-1} \right )^{-1} =s(c^{-1}-v^{-1}+c^{-1}+v^{-1})=2sc^{-1}

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

\frac{2s}{2s/c}=c

If time is the independent variable, the result is similar:

Consider a light ray sent a certain time interval t that is immediately reflected back, for a total time of 2t. 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 2t. The total distance of the light ray is

t(c-v)+t(c+v) = 2ct

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

\frac{2ct}{2t}=c