Rindler Coordinates - The Fermat Metric

The Fermat Metric

The fact that in the Rindler chart, the projections of null geodesics into any spatial hyperslice for the Rindler observers are simply semicircular arcs can be verified directly from the general solution just given, but there is a very simple way to see this. A static spacetime is one in which a vorticity-free timelike Killing vector field can be found. In this case, we have a uniquely defined family of (identical) spatial hyperslices orthogonal to the corresponding static observers (who need not be inertial observers). This allows us to define a new metric on any of these hyperslices which is conformally related to the original metric inherited from the spacetime, but with the property that geodesics in the new metric (note this is a Riemannian metric on a Riemannian three-manifold) are precisely the projections of the null geodesics of spacetime. This new metric is called the Fermat metric, and in a static spacetime endowed with a coordinate chart in which the line element has the form

the Fermat metric on is simply

(where the metric coeffients are understood to be evaluated at ).

In the Rindler chart, the timelike translation is such a Killing vector field, so this is a static spacetime (not surprisingly, since Minkowski spacetime is of course trivially a static vacuum solution of the Einstein field equation). Therefore, we may immediately write down the Fermat metric for the Rindler observers:

But this is the well-known line element of hyperbolic three-space H3 in the upper half space chart! This is closely analogous to the well known upper half plane chart for the hyperbolic plane H2, which is familiar to generations of complex analysis students in connection with conformal mapping problems (and much more), and many mathematically minded readers already know that the geodesics of H2 in the upper half plane model are simply semicircles (orthogonal to the circle at infinity represented by the real axis).