Distant Clocks

To compare the operation of two distant clocks, AB and CD, note that they have four different radar units. Assume them to be moving collinearly along the $z$-axis such that A and D are the outer pair, and B and C the inner pair, as in Figures 7 and 8

Figure 7: Two distant inertially expanding geometrical clocks. Clock CD can be calibrated in terms of AB because CD and AB are commensurable: Taking into account the ticking rate of the intermediate clock BC, one sees that the ratio of their rates is a constant, namely 1:1. Because this ratio happens to be unity, these two commensurable clocks are said to have the additional property of being identically constructed.

One says that two distant (nonadjacent) clocks AB and CD are commensurable, or more briefly

$$AB \approx CD~,$$

if and only if

(i)

Radar units A and B are visible for all times to radar units C and D and

(ii)

AB is commensurable with BC, and BC is commensurable with CD.

Being ``visible'' means that, by using its pulse radar, C can always see B on its radar screen, i.e. BC forms a geometrical clock. Thus two clocks AB and CD are commensurable if the clock formed by radar units B and C is commensurable with both of its neighbors, AB and CD.

Figure 8: Two distant accelerated geometrical clocks. Clock CD can be calibrated in terms of AB because CD and AB are commensurable: Taking into account the ticking rate of the intermediate clock BC, one sees that the ratio of their rates is $\frac {1}{3}\times \frac {3}{1}=\frac {1}{1}$ in this figure. Because this ratio happens to be unity, these two commensurable clocks are said to have the additional property of being identically constructed.

According to this definition, one uses the constancy of the rate of clock CD normalized to that of clock BC,

$$\frac{(1/\log k_{CD})}{(1/\log k_{BC})}=const~,$$

and the constancy of the rate of clock BC normalized to that of clock AB,

$$\frac{(1/\log k_{BC})}{(1/\log k_{AB})}=const~,$$

to establish that the rate of clock CD normalized to that of clock AB,

$$\frac{(1/\log k_{CD})}{(1/\log k_{AB})}=const~,$$ (9)

is also a constant, and therefore that CD is commensurable with AB. One sees from Eqs.(6) and (9) that this criterion for commensurability holds for both inertial and accelerated clocks, as is depicted in Figures 7 and 8.

Commensurability of distant clocks subsumes that of adjacent clocks as a special case. This follows from simply letting the space between clocks AB and CD in Figures 7 and 8 shrink to zero so that the final result is two adjacent clocks as in Figures 5 and 6. The commensurability is readily preserved throughout this limiting process.

Commensurability is a relation which satisfies the following three properties:

1. AB $\approx$ AB
2. AB $\approx$ CD implies CD $\approx$ AB
3. AB $\approx$ CD together with CD $\approx$ EF implies AB $\approx$ EF

A physicist can choose one of these commensurable clocks as his primary standard. It is a dual function device : It represents a temporal standard and a spatial standard at the same time. The spatial extent of the clock is determined uniquely by its ticking rate, a light pulse bouncing back and forth between the clock's two ends.