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### Normal Modes

The complementary, but equivalent (via Fourier synthesis), perspective on this resonance is to note that the two clocks have identical normal mode spectra. More explicitly, the cavities have their ends moving in such a way that the normal modes, which are governed by the wave equation

vibrate (as a function of ) at the same respective rates in the two cavities. For two accelerated clock cavities AB and CD this equality is achieved by the condition
 (19)

because it yields

For the circumstance of two inertially expanding clock cavities AB and CD this is achieved by the condition

 (20)

because it yields

The first condition is precisely the conditions for clocks AB and CD in to be identically constructed, the second one for clocks in . Indeed, using Eq.(7), the definition of , one sees that Eq.(22) reads
 (21)

which coincides with Eq.(21). Similarly, using the definition
 (22)

for an inertially expanding cavity in spacetime sector , one sees that Eq.(23) reads
 (23)

which again coincides with Eq.(21).

The results expressed by Eqs.(24) and (26) can therefore be summarized by the simple statement: Identically constructed clocks are those with cavities having identical eigenvalue spectra. This means that the frequencies of the field oscillators in one cavity coincide with the frequencies of those in the other.

If there is a weak mutual interaction between the cavities (i.e. the reflectors at the cavity ends are slightly transmissive), then there is a coupling among each pair of normal modes (field oscillators), one in each of the two cavities. If cavity AB starts out with all the field energy, then this coupling mediates the excitation of the field oscillators in CD at their respective frequencies. They will start oscillating in sympathy with those of AB.

The sum of all the (normal mode) amplitudes of these field oscillators forms a bouncing pulse in CD. The fact that the sympathetic resonance makes these amplitudes increase with time implies that the bouncing pulse in CD does the same.

To summarize: An analysis in terms of bouncing pulses or in terms of normal modes leads to the same conclusion: The physical process of the transfer of time (a train of clock ticks) between identically constructed clocks is the process of sympathetic resonance between their cavities.

Next: TRANSFER OF TIME ACROSS Up: IDENTICALLY CONSTRUCTED CLOCKS AS Previous: Travelling Pulses   Contents
Ulrich Gerlach 2003-02-25