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The subject of this article is inertially expanding reference frames. The theme is their establishment as a one-way windows into uniformly and linearly accelerated reference frames. To execute this task, this article constructs both kinds of frames using the basic properties of Doppler radar and pulse radar and then points out a radar-generated mapping between the two frames.

The construction requires three ingredients. First one constructs ``Fourier-compatible'' geometrical clocks. Each one is characterized by a single number, the frequency shift factor between the moving ends of a one-dimensional moving cavity that traps an e.m. pulse. Its bouncing action provides the ticking of the clock.

Second one introduces geometrical clocks. In general, the ticking rates of these clocks are non-uniform relative to an atomic clock comoving with one end of the cavity, or the other. Thus the necessity of comparing one geometrical clock with another is fulfilled by considering the ratio of theses rates. This leads to the concept ``commensurability'' as applied to geometrical clocks. They are commensurable whenever one can choose one of them as a representative with its pair of properties (separation between successive ticks and separation between cavity ends) serving as a standard: the properties of all other clocks can be related to it numerically. Thus ``commensurability'' is a relation which implies and is implied by a standard of measurement. Moreover, ``commensurability'' is an equivalence relation. It divides clocks into mutually exclusive sets, which in mathematics are called equivalence classes and in physics are called reference frames, accelerated ($I$ or $II$ as in Figure 1) or inertially expanding ($F$ or $P$ as in Figure 1).

It is valuable to take note of the importance of commensurability as a general concept and as a special concept when applied to geometrical clocks in particular. It is the connecting link between nature (i.e. reality, existence) and the observer's mind. This is because the observer can choose one of these clocks as a standard to which he can quantitatively relate all others (by taking ratios). Then, by referring to merely a single representative clock the observer can grasp the corresponding equivalence class of all possible commensurable clocks, a particular spacetime coordinate frame (accelerated or inertially expanding).

Without a measurement process of some sort, there would not be a commensurability criterion, there would not be an equivalence relation, hence no equivalence class, i.e. no concept. The concept of time and of place consists of the set of measurement results which the observer obtains when he relates the tickings and the size of his standard clock to any event occurring in his reference frame. Establishing these relations is what he means by measuring (the time and place of) these events.

The third ingredient consists of specifying some sort of measurement process. Just as a one-dimensional array of calibrated graduation marks on a measuring rod facilitates measuring the length of any specific object, so a lattice of calibrated graduation events in a spacetime coordinate frame facilitates measuring the time and position of any specific event. The calibration process can be performed by the radar method, which is based on having a single standard clock control the emission and the reception of radar pulses, or by the common (non-radar) method, which is based on counting synchronized identically constructed clocks (i.e. copies of the clock chosen as a standard) and their ticks. Even though the equivalence of these two methods extends to accelerated as well as inertially exspanding clocks, the introduction of radar does not make identically constructed clocks obsolete or useless.

On the contrary. Suppose one applies the essential aspect of being ``identically constructed'' to two clocks separated by an event horizon between them. ``Identical construction'' means that, even though one clock is accelerated while the other is inertially expanding, their cavities have identical eigenfrequency spectra, and that, as a consequence, the timing pulses emitted by the accelerated clock cause the inertially expanding clock to tick in perfect synchrony with their arrival at this clock.

Thus the first useful aspect of two ``identically constructed'' clocks, one accelerated the other inertially expanding, is that they lend themselves to being (one-way) synchronized even though they are separated by an event horizon.

The second and more important aspect is that a physicist in the inertially expanding frame can ``look'' into the accelerated frame on the other side of the event horizon and ``see'' the spacetime trajectories of sources in that frame. This is because the two clocks serve to (one-way) transfer radar images across the event horizon. The elements (pixels) of a radar image are in the form of the amplitudes of the pulses reflected by a scatterer located in the frame of the accelerated clock. Taking advantage of its transponder capability, the accelerated clock forwards these pulses to the identically constructed inertially expanding clock. There the pulses are used to reconstruct a spacetime image of what the accelerated clock sees. For example, suppose the radar controlled by this clock measures that a localized scatterer has the history of a hyperbolic world line in boost-invariant sector $I$. Once the pixels of this radar image have been sent across the event horizon, the inertially expanding clock reconstructs them into a straight timelike line in boost-invariant sector $F$. As a second example, the pixels of a linear array of simultaneous scattering events in $I$, upon transmission across the event horizon, get reconstructed as a spacelike hyperbola in $F$.

Thus, by using an accelerated and an inertially expanding clock which are ``identically constructed'', an inertial observer can verify by radar whether the dipole source in the augmented Larmor formula is accelerated in a uniform and linear way.

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Ulrich Gerlach 2003-02-25