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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.
\includegraphics[scale=.5]{two_inertial_clocks}

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

\begin{displaymath}
\Delta \xi
\end{displaymath}

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

\begin{displaymath}
k_{AB}\Delta \xi
\end{displaymath}

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,

\begin{displaymath}
\frac{k_{AB}\Delta \xi}{\Delta \xi}=k_{AB}
\end{displaymath}

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

\begin{displaymath}
\xi,\, k_{AB}^2 \xi, \cdots,\, k_{AB}^{2a} \xi.
\end{displaymath}

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

\begin{displaymath}
\xi,\,k_{BC}^2 \xi, \cdots, \, k_{BC}^{2c} \xi.
\end{displaymath}

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,

\begin{displaymath}
k_{AB}^{2a} \xi-\xi=k_{BC}^{2c} \xi-\xi,
\end{displaymath}

or
\begin{displaymath}
\frac{c}{a}= \frac{(1/\log k_{BC})}{(1/\log k_{AB})}.
\end{displaymath} (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.


next up previous contents
Next: Commensurable Accelerated Clocks Up: Adjacent Clocks Previous: Adjacent Clocks   Contents
Ulrich Gerlach 2003-02-25