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:

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

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

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

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:

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

And the speed relative to the return direction is:

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

Then the mean speed is:

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

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):

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

And the speed relative to the return direction is:

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

Then the harmonic mean speed is:

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

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

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

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

Revised 2023-07-29.

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