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## 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 -axis such that A and D are the outer pair, and B and C the inner pair, as in Figures 7 and 8

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

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.

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

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

to establish that the rate of clock CD normalized to that of clock AB,
 (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 AB
2. AB CD implies CD AB
3. AB CD together with CD EF implies AB 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.

Next: MEASURING EVENTS VIA RADAR Up: COMMENSURABILITY Previous: Commensurable Accelerated Clocks   Contents
Ulrich Gerlach 2003-02-25