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Figure 11: One-way commensurability between inertially expanding clock AB and accelerated clock BC, each containing its pulse bouncing back and forth. Depicted in this diagram is a sequence of 5 high intensity pulses coming from A and impinging on B which are matched by a sequence of 4 low intensity pulses coming across the event horizon from C. The ratio $m/n=5/4$ is the ticking rate of AB normalized to that of CD. The fact that this ratio stays constant throughout the history of the two clocks makes them one-way commensurable.

Commensurability is a more basic property than the property of clocks being synchronized. Before one tries to synchronize two clocks, one must first ascertain that they are commensurable.

Furthermore, two commensurable clocks cannot be synchronized unless there is a two-way interaction between them. In the context of an inertially expanding or an accelerated coordinate frame (Figure 9 or 10) such an interaction consists of a radar (to and fro) signal between each pair of clocks, say AB and CD as in Figures 7 or 8. Such a radar signal accommodates a two-way transfer of time: AB transmits its tick number to CD, and CD sends via the return pulse its own tick number back to AB. With this mutual knowledge the two clocks can be relabelled, if necessary, to give synchronized time.

However, if there is an event horizon between clocks AB and CD, then qualitatively new considerations enter.

On one hand, at most only a one-way transfer of time is possible. The establishment of a time synchronous to both of them is out of the question.

On the other hand, that event horizon brings with it a pleasant surprise: an accelerated clock and an inertially expanding clock, which at first sight seem to be incommensurable, are in fact commensurable when there is an event horizon between them. In particular one clock can (via sympathetic resonance) exert a one-way control over the other. Here is why:

As one can see from Figure 1 there is an event horizon that separates the clocks in spacetime sector $I$ from those in spacetime sector $F$. But the problem with taking advantage of a one-way transfer of time from CD in $I$ to AB in $F$ seems to be that the clock in $I$ is accelerated while the one in $F$ is inertially expanding. At first sight there seems to be no way that the two are commensurable as defined on page [*] in section IV+.1667emB. One must note, however, that that definition was based on a two-way transfer of time (``AB is radar-visible to CD''). This was necessary. Indeed, the definition of boost-invariant sector $I$ as well as $F$ (``equivalence classes of geometrical clocks that can be synchronized'') depended on it.

To accommodate the context of an event horizon as a one-way membrane between clocks AB and CD, we enlarge the concept ``commensurability'' by defining the concept ``one-way commensurability''. This is done by dropping the requirement that radar units B and C be in two-way contact, and by saying that one-way contact, say from C to B, is good enough. The result of doing this is illustrated in Figure 11.

Accelerated clock CD moves along the line of sight of inertially expanding clock AB. This clock is characterized by Doppler shift factor $k_{AB}$. Clock CD, whose radar units are accelerated with constant accelerations $1/\xi_C$ and $1/\xi_D$ to the right, is characterized by the pseudo-gravitational frequency shift factor


between them. As shown in Figure 11, clock CD sends pulses on a one-way journey to AB. There are no return pulses. Nevertheless, one can compare a sequence of $a$ pulses at B from A with a matched sequence of $c$ pulses from CD. The result is
\frac{a}{c}= \frac{(1/\log k_{AB})}{(1/\log k_{CD})}~.
\end{displaymath} (24)

This is the ticking rate of inertial clock AB normalized relative to accelerated clock CD. This ticking rate is a constant independent of the starting time of the two matched pulse sequences. Consequently, inertial clock AB is one-way commensurable with accelerated clock BC.

Figure 12: One-way transfer of time from clock CD to clock AB. This transfer is achieved by the ticking of CD causing sympathetic tickings in clock AB. This process is brought about by e.m. pulses travelling from CD across the future event horizon to AB (solid thin 45$^\circ $ lines). They strike B at precisely the same rate and with the same phase as AB's clock pulse (heavy zigzag line) bouncing repeatedly off B.
Equation (27) is a remarkable result for a number of reasons. First of all, there is its constancy. Contrast this with the tickings of the comoving atomic clocks at radar units B, which is floating freely, and C, which is accelerated. They yield

\frac{\textrm{(\char93  of ticks of atomic clock C but observed at
B)}}{\textrm{(\char93  of ticks of atomic clock B)}} ~,

a corresponding rate which is a Doppler chirp towards the red as seen by a physicist comoving with the free-float atomic clock at B. By contrast, the constancy of Eq.(27) expresses the fact that the slowdown in the proper ticking rate of geometrical clock AB compensates precisely for the slowdown in the proper rate of pulses arriving at B from C.

Second, if $n/m=1$, i.e.

\end{displaymath} (25)

then the process of transferring a train of clock pulses from across its future event horizon to clock AB (``one way transfer of time'') is a process of tickings in cavity CD bringing about sympathetic tickings in cavity AB. The implementation of this transfer is depicted in Figure 12. Thus, following the discussion in Section VI, one concludes that, even though CD is accelerated while AB is expanding inertially, (i) the two cavities are identically constructed from perspective of their normal mode spectra, and that (ii) AB and CD are ticking at the same rate as measured at B.

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Ulrich Gerlach 2003-02-25