Spacetimes with no position drift

TL;DR

This paper compares three criteria for zero position drift in spacetimes, analyzing their relations and applying them to specific metrics like Stephani and Szekeres, revealing conditions for drift-free observations.

Contribution

It clarifies the relationships among three different criteria for zero position drift and applies these to specific spacetime metrics, identifying necessary conditions for drift-free spacetimes.

Findings

HP criterion is necessary for KK to apply

Zero Weyl tensor is necessary for KK in perfect fluid spacetimes

Szekeres metrics are drift-free only in Friedmann limit

Abstract

This paper compares three criteria for a spacetime to be free of position drift: those by Hasse and Perlick (HP), Krasi\'nski and Bolejko (KB) and Korzy\'nski and Kopi\'nski (KK). A spacetime having no position drift means that every observer sees all light sources in unchanging directions. The following is shown: (1) The HP criterion is a necessary condition for the KK criterion to apply. (2) If the spacetime metric obeys the Einstein equations with a perfect fluid source, then another necessary condition for the KK criterion is the Weyl tensor being zero. (3) Result (2) points to the Stephani metric, so it is shown that this metric obeys an equation which is still one more necessary condition for the KK criterion. (4) The general Szekeres metrics become drift-free by the KK criterion only in the Friedmann limit. (5) The HP and KB criteria coincide, and the HP zero-drift condition imposes on the Stephani metric the same restriction as found by Krasi\'nski and Bolejko (KB). The relations between the three criteria are displayed and compared in a diagram.