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).

A 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, not one-dimensional time. This is key to understanding the experiment properly.

Figure 1. Light clock

Figure (1a) on the left 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 1/c, and the duration Zǁ of one round trip is

Zǁ = (D/c) + (D/c) = 2D/c.

The mean pace is (2D/c)/(2D) = 1/c, which is a harmonic mean speed of c.

Figure (1b) 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 (D/c + wǁD), and the duration of the second leg is (D/cwǁD). The duration of one round trip is

(D/c + wǁD) + (D/cwǁD) = 2D/c = Zǁ

Again, the mean pace is (2D/c)/(2D) = 1/c, which is a harmonic mean speed of c.

Figure (1c) shows an observer moving with lenticity w transverse to the light clock. Here the independent distance D′ is longer than D. [note: but we do not say that D′ is a dilated form of D because they are simply two different values of the independent distance.]

The dependent durations are in two dimensions: the transverse duration Z and the longitudinal duration Zǁ. The transverse duration is the boosted duration:

Z′ = Z − wD

The transverse duration has no effect on the longitudinal duration, which is the unboosted light clock: Zǁ′ = Zǁ. That is, the moving light clock has the same duration as the light clock at rest.

Thus the 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.