# Light clock with duration space

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, a distance L 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, the independent quantity is distance, and duration is a dependent quantity. This is key to understanding the experiment properly.

Figure (a) 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 2L, the pace is 1/c, and the duration of one round trip is

T = (L/c) + (L/c) = 2L/c.

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

Figure (b) shows an observer moving with lenticity w transversely to the light clock. In this case there are two components of duration: longitudinal and transverse. These components are independent of one another since they are in different dimensions. The transverse component has no effect on the longitudinal component. The longitudinal motion is the same as the stationary case above: the total distance is 2L and the total duration is

T = (L/c) + (L/c) = 2L/c.

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

Figure (c) shows an observer moving with lenticity w longitudinally to the light clock. Here the Galilean duration transformation applies: t′ = twx, where x is the longitudinal axis. Since the distance L is independent, it does not change from observer to observer. The duration of the first leg is (L/c + wL), and the duration of the second leg is (L/cwL). The duration of one round trip is

T = (L/c + wL) + (L/cwL) = 2L/c.

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

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

Revised 2023-03-03.