Symmetries at stationary Killing horizons

TL;DR

This paper investigates the symmetry properties of the Einstein tensor near stationary black hole horizons, providing an elegant argument for its constrained form, which may be relevant to understanding black hole entropy.

Contribution

It offers a geometrical argument supporting the symmetric form of the Einstein tensor at stationary horizons, highlighting a potential link to black hole entropy.

Findings

Einstein tensor exhibits a block-diagonal form near horizons

The argument is elegant but not fully rigorous due to a loophole

Symmetries may be relevant to black hole entropy understanding

Abstract

It has often been suggested (especially by Carlip) that spacetime symmetries in the neighborhood of a black hole horizon may be relevant to a statistical understanding of the Bekenstein-Hawking entropy. A prime candidate for this type of symmetry is that which is exhibited by the Einstein tensor. More precisely, it is now known that this tensor takes on a strongly constrained (block-diagonal) form as it approaches any stationary, non-extremal Killing horizon. Presently, exploiting the geometrical properties of such horizons, we provide a particularly elegant argument that substantiates this highly symmetric form for the Einstein tensor. It is, however, duly noted that, on account of a "loophole", the argument does fall just short of attaining the status of a rigorous proof.