Spacetimes with continuous linear isotropies I: spatial rotations

TL;DR

This paper establishes that local rotational symmetry in Petrov type D spacetimes is characterized by local spatial rotation invariance of the Riemann tensor and its derivatives, extending to conformally flat cases with specific conditions.

Contribution

It proves the equivalence of Goode and Wainwright's LRS criterion to Riemann tensor invariance and clarifies conditions for conformally flat spacetimes, including accelerated perfect fluids.

Findings

LRS criterion is equivalent to Riemann tensor invariance.

Conformally flat spacetimes mostly follow the same criterion.

Three curvature derivatives are needed for accelerated perfect fluids.

Abstract

The weakest known criterion for local rotational symmetry (LRS) in spacetimes of Petrov type D is due to Goode and Wainwright (1986). Here it is shown, using methods related to the Cartan-Karlhede procedure, to be equivalent to local spatial rotation invariance of the Riemann tensor and its first derivatives. Conformally flat spacetimes are similarly studied and it is shown that for almost all cases the same criterion ensures LRS. Only for conformally flat accelerated perfect fluids are three curvature derivatives required to ensure LRS, showing that Ellis's original condition for that case is necessary as well as sufficient.