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 $\rightarrow$ B is characterized by

$$\begin{eqnarray*} \lefteqn{ k_{AB} \equiv }\\ & & \frac{\textrm{(frequency of a... ...\textrm{(frequency of atomic clock at A, but observed at B)}} ~, \end{eqnarray*}$$

while the reverse transmission process B $\rightarrow$ A is characterized by

$$\begin{eqnarray*} \lefteqn{ k_{BA} \equiv }\\ & & \frac{\textrm{(frequency of a... ...{\textrm{(frequency of atomic clock at B, but observed at A)}}~. \end{eqnarray*}$$

The numbers $k_{AB}$ and $k_{BA}$ 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 $k_{AB}=k_{BA}$, while for the latter one has $k_{AB}=1/k_{BA}$.

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 $\rightarrow$ B $\rightarrow$ A for monochromatic radiation. There one has

$$\begin{eqnarray*} k_{A\rightarrow B\rightarrow A}\equiv k_{A\rightarrow B}k_{B\r... ...textrm{uniform collinear acc'n} \end{array} \end{array}\right. \end{eqnarray*}$$

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.