Light clock in motion

This post builds on the post about the Michelson-Morley experiment here. Compare the light clock in the “Derivation of time dilation” (e.g., here).

Linear Light Clock

A linear light clock is a thought experiment in which a light beam reflects back and forth between two parallel mirrors that are a distance D apart (see figure below). When the light beam returns to the first mirror, one unit of duration passes (“the clock ticks”). Since the distance between the mirrors is set by the experimenter, distance is the independent quantity, and the dependent quantity is multidimensional duration z, not one-dimensional time. This is key to understanding the experiment properly.

Figure 1. Linear light clock

Case 1 in Figure 1 shows a light clock at rest, with a light beam reflecting longitudinally back and forth between two mirrors. In this frame the round trip longitudinal distance between the two mirrors is 2D, the pace is κ, and the duration Z1 of one round trip is

z1 = (κD) + (κD) = 2κD.

The mean pace is (2κD)/(2D) = κ, which is the inverse of the harmonic mean speed of light c.

Case 2 in Figure 1 shows an observer moving with lenticity wǁ longitudinally to the light clock. Here the Galilean duration transformation applies: z′ = zwǁs, where s is the longitudinal axis. Since the distance D is independent, it does not change from observer to observer. The duration of the first leg is (κD + wǁD), and the duration of the second leg is (κD − wǁD). The duration of one round trip is

 z2 = (κD + wǁD) + (κD − wǁD) = 2κD

Again, the mean pace is (2κD)/(2D) = κ, which is the inverse of the harmonic mean speed of light c.

In Case 3 an observer is moving transversely to the light clock in the duration frame. Because transverse motion is independent of longitudinal motion in any frame of reference system, the duration of one round trip is the same as for Case 1:

z3 = (κD) + (κD) = 2κD.

Thus the arithmetic mean paces and harmonic mean speeds are the same for all observers and are independent of their relative velocity, which is essentially what the Michelson-Morley experiment found, though didn’t expect.

Published 2021-07-07, revised 2023-03 & 2023-06.