Dirty black holes: Symmetries at stationary non-static horizons

TL;DR

This paper demonstrates that near the horizon of any stationary, non-static black hole, the Einstein tensor exhibits a high degree of symmetry, which constrains matter behavior and may impact black hole entropy calculations.

Contribution

It extends previous static black hole results to stationary, non-static horizons, showing Einstein tensor symmetry and implications for matter and entropy.

Findings

Einstein tensor near the horizon is block-diagonal with symmetric transverse components.

Symmetries constrain matter fields near the horizon.

Supports the idea that horizon symmetries influence black hole entropy calculations.

Abstract

We establish that the Einstein tensor takes on a highly symmetric form near the Killing horizon of any stationary but non-static (and non-extremal) black hole spacetime. [This follows up on a recent article by the current authors, gr-qc/0402069, which considered static black holes.] Specifically, at any such Killing horizon -- irrespective of the horizon geometry -- the Einstein tensor block-diagonalizes into ``transverse'' and ``parallel'' blocks, and its transverse components are proportional to the transverse metric. Our findings are supported by two independent procedures; one based on the regularity of the on-horizon geometry and another that directly utilizes the elegant nature of a bifurcate Killing horizon. It is then argued that geometrical symmetries will severely constrain the matter near any Killing horizon. We also speculate on how this may be relevant to certain calculations of the black hole entropy.