Commensurable Inertial Clocks

Consider a pair of clocks AB and BC, where all three radar units A, B, and C are freely floating, and radar unit B is common to AB and BC, as in Fig. 5.

Figure 5: Two adjacent geometrical clocks consisting of inertially expanding cavities AB and BC, each containing a pulse bouncing back and forth. Depicted in this diagram are sequences of 2 pulses coming from A and impinging on B which are matched by corresponding sequences of 3 pulses coming from C. The ratio $c/a=3/2$ is the ticking rate of BC normalized to that of AB. The fact that this ratio stays constant throughout the history of the two clocks makes them commensurable.

What is the ratio $c/a$ of two matched pulse sequences impinging on radar unit B and coming from radar units A and C? This ratio is determined by the following mini-calculation:

Let A emit two pulses separated by

$$\Delta \xi$$

as measured by atomic clock A. Due to the relative motion of A and B these two pulses, once received at B, have time separation

$$k_{AB}\Delta \xi$$

as measured by atomic clock B. Here $k_{AB}$ is a positive (``Doppler'') factor whose magnitude expresses the motion of A relative to B. There are now two time intervals: the one between the emitted pulses and the one between the received pulses. These intervals are proportional to the wavelengths of emitted and received monochromatic radiation. Their ratio,

$$\frac{k_{AB}\Delta \xi}{\Delta \xi}=k_{AB}$$

is the Doppler shift factor. The two radar units are understood to be at rest relative to each other whenever $k_{AB}=1$. They are receding (resp. approaching) each other whenever $k_{AB}>1$ (resp. $k_{AB}<1$), which expresses a Doppler red (resp. blue) shift. It is clear that this Doppler shift of clock AB controls the rate at which the back-and-forth bouncing pulse produces ticks at radar unit B. In fact, the pulse arrival times of $a$ consecutive pulses coming from A are

$$\xi,\, k_{AB}^2 \xi, \cdots,\, k_{AB}^{2a} \xi.$$

Similarly, the arrival times of $c$ consecutive pulses coming from radar unit C, which is part of clock BC, are

$$\xi,\,k_{BC}^2 \xi, \cdots, \, k_{BC}^{2c} \xi.$$

These two pulse sequences have coincident initial pulse arrival times $\xi$. If these two sequences are ``matched'', then their final pulse arrival times, $k_{AB}^{2m}\xi=k_{BC}^{2n}\xi$, also coincide. Under this circumstance the length of these two pulse sequences as measured by atomic clock B are the same. Consequently,

$$k_{AB}^{2a} \xi-\xi=k_{BC}^{2c} \xi-\xi,$$

or

$$\frac{c}{a}= \frac{(1/\log k_{BC})}{(1/\log k_{AB})}.$$ (5)

This is the ticking rate of clock CB normalized relative to clock AB. This ticking rate is a constant independent of the starting time $\xi$ of the two matched pulse sequences. Consequently, clock AB is commensurable with BC.