# Invariance of round-trip speed

The mean round-trip speed, as in simple harmonic motion, is Galilean invariant. There are two senses in which this is the case: (1) the time is the same in both directions, and (2) the distance covered is the same in both directions. In the first case, the mean round-trip speed equals the arithmetic mean of the two speeds. In the second case, the mean round-trip speed equals the harmonic mean of the two speeds. In both cases, the mean round-trip speed is Galilean invariant.

The mean round-trip speed in the first case is the arithmetic mean:

$\frac{u_1&space;+&space;u_2}{2}&space;=\frac{1}{2}&space;\left&space;(\frac{\mathrm{d}x_1}{\mathrm{d}&space;t}&space;+\frac{\mathrm{d}x_2}{\mathrm{d}&space;t}&space;\right&space;)$

In the second case the mean round-trip speed is the harmonic mean:

$\frac{u_1&space;\boxplus&space;u_2}{2^{-1}}&space;=&space;\left&space;(\frac{u_1^{-1}&space;+&space;u_2^{-1}}{2}&space;\right&space;)^{-1}&space;=&space;2\left&space;(\left&space;(\frac{\mathrm{d}x}{\mathrm{d}t_1}&space;\right&space;)^{-1}+\left&space;(\frac{\mathrm{d}x}{\mathrm{d}t_2}&space;\right&space;)^{-1}&space;\right&space;)^{-1}$

In order to apply the Galilean transformation, start with the standard configuration in which motion is along the X and X′ axes in S and S′ respectively with velocity v in S′ relative to S.

In the first case, the speed in S′ relative to the forward direction is:

$u_{1}'=&space;\frac{\mathrm{d}x'}{\mathrm{d}&space;t'}&space;=&space;\frac{\mathrm{d}x'}{\mathrm{d}&space;t}&space;=&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;t}\left&space;(x-vt&space;\right&space;)&space;=&space;\frac{\mathrm{d}x}{\mathrm{d}&space;t}-v&space;=&space;u_{1}-v.$

And the speed relative to the return direction is:

$u_{2}'=&space;\frac{\mathrm{d}x'}{\mathrm{d}&space;t'}&space;=&space;\frac{\mathrm{d}x'}{\mathrm{d}&space;t}&space;=&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;t}\left&space;(x+vt&space;\right&space;)=&space;\frac{\mathrm{d}x}{\mathrm{d}&space;t}+v&space;=u_{2}+v.$

Then the mean speed is:

$\frac{u_{1}'+u_{2}'}{2}&space;=&space;\frac{(u_{1}-v)&space;+(u_{2}+v&space;)}{2}&space;=&space;\frac{u_{1}+u_{2}}{2}$

Thus the mean round-trip speed is Galilean invariant.

In the second case, the speed in S′ relative to the forward direction is (with the overall inverse left to the end):

$u_{1}'=&space;\frac{\mathrm{d}t'}{\mathrm{d}&space;x'}&space;=&space;\frac{\mathrm{d}t'}{\mathrm{d}&space;x}&space;=&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}(t-\frac{x}{v})&space;=&space;\frac{\mathrm{d}t}{\mathrm{d}&space;x}-\frac{1}{v}&space;=&space;\frac{1}{u_{1}}-\frac{1}{v}.$

And the speed relative to the return direction is:

$u_{2}'=&space;\frac{\mathrm{d}t'}{\mathrm{d}&space;x'}&space;=&space;\frac{\mathrm{d}t'}{\mathrm{d}&space;x}&space;=&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}(t+\frac{x}{v})&space;=&space;\frac{\mathrm{d}t}{\mathrm{d}&space;x}+\frac{1}{v}&space;=&space;\frac{1}{u_{2}}+\frac{1}{v}.$

Then the harmonic mean speed is:

$\left&space;(\frac{1/u_{1}'+1/u_{2}'}{2}&space;\right&space;)^{-1}&space;=&space;\left&space;(\frac{(1/u_{1}-1/v)&space;+(1/u_{2}+1/v&space;)}{2}&space;\right&space;)^{-1}&space;=&space;\left&space;(\frac{1/u_{1}+1/u_{2}}{2}&space;\right&space;)^{-1}$

This applies to the speed of light, c, which is Galilean invariant since only the mean round-trip speed can be measured. For a light clock with a nominal speed of c in each direction and an observer with longitudinal speed v, the mean speed of light is

$\frac{(&space;c&space;+&space;v&space;)&space;+&space;(&space;c&space;-&space;v&space;)}{2}&space;=&space;\frac{2c}{2}&space;=&space;c$

An observer with transverse speed has no effect on the measurement of the speed of light c.

Revised 2023-07-29.