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## Fourier Compatibility

A fundamental property of every pair of radar units is their compatibility with respect to frequency measurements. More precisesly, one has the following definition:

Two radar units are said to be Fourier compatible if and only if a continuous wave train emitted by one radar unit produces a return signal which has a sharp Fourier spectrum at the second radar unit. If the return signal is not spectrally sharp (within prespecified bounds), then the two units are said to be Fourier incompatible.

A pair of Fourier compatible radar units, say A and B, is characterized by two frequency shift factors. The transmission process A B is characterized by

while the reverse transmission process B A is characterized by

The numbers and are, of course, the familiar Doppler frequency shift factors if A and B are freely floating units, and the pseudo-gravitational frequency shift factors if A and B are uniformly and collinearly accelerated units. For the former one has , while for the latter one has .

These frequency shifts (Fourier compatibility factors'') are strictly kinematical aspects of A and B. They involve neither the inertia nor the dynamics of material particles. Nevertheless, they do distinguish between free-float and acceleration. Indeed this distinction is encoded into the the relation between the frequency shifts associated with the reflection process A B A for monochromatic radiation. There one has

For example, it is clear that all freely floating (inertial'') units are Fourier compatible. So are the units which are linearly and uniformly accelerated and have the same future and past event horizons. However, an accelerated unit and one in a state of free float are not Fourier compatible. Neither are two units if one of them undergoes non-uniform acceleration. Such units measure a Doppler chirp instead of a constant Doppler shift when they receive the wave train reflected by the other.

Next: Geometrical Clocks Up: GEOMETRICAL CLOCKS Previous: Radar Units   Contents
Ulrich Gerlach 2003-02-25