Dirty black holes: Spacetime geometry and near-horizon symmetries

TL;DR

This paper demonstrates that the geometry and stress-energy tensor near static black hole horizons exhibit high symmetry, which constrains the near-horizon physics and may relate to black hole entropy via conformal symmetry.

Contribution

It shows, using geometric methods, that the curvature and stress-energy tensors near static black hole horizons are highly symmetric and invariant under conformal deformations.

Findings

Stress-energy tensor becomes block-diagonal near the horizon.

Transverse components of the stress-energy tensor are proportional to the transverse metric.

Symmetries may underpin near-horizon conformal symmetry and black hole entropy.

Abstract

We consider the spacetime geometry of a static but otherwise generic black hole (that is, the horizon geometry and topology are not necessarily spherically symmetric). It is demonstrated, by purely geometrical techniques, that the curvature tensors, and the Einstein tensor in particular, exhibit a very high degree of symmetry as the horizon is approached. Consequently, the stress-energy tensor will be highly constrained near any static Killing horizon. More specifically, it is shown that -- at the horizon -- the stress-energy tensor block-diagonalizes into ``transverse'' and ``parallel'' blocks, the transverse components of this tensor are proportional to the transverse metric, and these properties remain invariant under static conformal deformations. Moreover, we speculate that this geometric symmetry underlies Carlip's notion of an asymptotic near-horizon conformal symmetry controlling the entropy of a black hole.