The rest frame in stationary space-times with axial symmetry

Abstract

A rest frame in a stationary, axially symmetric space-time is defined as a synchronizable frame which is as nearly Killing as possible. This is a possible generalization of the Newtonian rest frame. A kinematical theorem giving the condition for the existence of a rest frame whose time vector is a linear combination of the Killing vectors is proved. The frame is also unique. The condition is shown to be weaker than the assumption of orthogonal transitivity. The surfaces of simultaneity of the rest frame are the surfaces of constancy of a particular Killing time coordinate, and its time vector is the component of the time Killing vector orthogonal to the angular Killing vector. Some properties of the rest frame are then discussed; it is shown that the frame is well-behaved down to the event horizon, where its time vector becomes null. Under a suitable condition on the event horizon, the time vector coincides with a Killing vector there. The gravitational redshift relation in the rest frame is derived. There is a dependence on the angular momentum of the geodesic. Furthermore, the event horizon is shown to be an infinite redshift surface for the rest frame observers. Finally, the three-vector potential of Landau and Lifshitz is interpreted and shown to be closely related to the rest frame, and a corresponding four-vector potential is invariantly defined.