Radar Distance in The Large
Even in flat spacetime, it turns out that accelerating observers (even linearly accelerating observers; see Rindler coordinates) can employ various distinct but operationally significant notions of distance. Perhaps the simplest of these is radar distance.
Consider how a static observer at R=0 might determine his distance to a ring riding observer at R = R0. At event C he sends a radar pulse toward the ring, which strikes the world line of a ring-riding observer at A′ and then returns to the central observer at event C″. (See the right hand diagram in the figure at right.) He then divides the elapsed time (as measured by an ideal clock he carries) by two. It is not hard to see that he obtains for this distance simply R0 (in the cylindrical chart), or r0 (in the Born chart).
Similarly, a ring-riding observer can determine his distance to the central observer by sending a radar pulse, at event A toward the central observer, which strikes his world line at event C′ and returns to the ring-riding observer at event A″. (See the left hand diagram in the figure at right.) It is not hard to see that he obtains for this distance (in the cylindrical chart) or (in the Born chart), a result which is somewhat smaller than the one obtained by the central observer. This is a consequence of time dilation: the elapsed time for a ring riding observer is smaller by the factor than the time for the central observer. Thus, while radar distance has a simple operational significance, it is not even symmetric.
Just to drive home this crucial point, let us compare the radar distances obtained by two ring-riding observers with radial coordinate R = R0. In the left hand diagram at the figure to the left, we can write the coordinates of event A as
and we can write the coordinates of event B′ as
Writing the unknown elapsed proper time as, we now write the coordinates of event A″ as
By requiring that the line segments connecting these events be null, we obtain an equation which in principle we can solve for Δ s. It turns out that this procedure gives a rather complicated nonlinear equation, so we simply present some representative numerical results. With R0 = 1, Φ = π/2, and ω = 1/10, we find that the radar distance from A to B is about 1.308, while the distance from B to A is about 1.505. As ω tends to zero, both results tend toward .
Despite these possibly discouraging discrepancies, it is by no means impossible to devise a coordinate chart which is adapted to describing the physical experience of a single Langevin observer, or even a single arbitrarily accelerating observer in Minkowski spacetime. Pauri and Vallisneri have adapted the Märzke-Wheeler clock synchronization procedure to devise adapted coordinates they call Märzke-Wheeler coordinates (see the paper cited below). In the case of steady circular motion, this chart is in fact very closely related to the notion of radar distance "in the large" from a given Langevin observer.
Read more about this topic: Born Coordinates
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