Scalar field quantization on the 2+1 dimensional black hole background

TL;DR

This paper studies scalar field quantization on a 2+1 dimensional Anti de Sitter black hole, calculating Green's functions, stress-energy tensor, and back-reaction effects, revealing horizon shifts and singularity formation.

Contribution

It provides an explicit calculation of the Green's function and back-reaction effects for a scalar field in this black hole background, including the horizon shift and singularity development.

Findings

Green's function matches Hartle-Hawking state

Stress-energy tensor is regular at horizon, diverges at singularity

Back-reaction causes horizon shift and singularity formation

Abstract

The quantization of a massless conformally coupled scalar field on the 2+1 dimensional Anti de Sitter black hole background is presented. The Green's function is calculated, using the fact that the black hole is Anti de Sitter space with points identified, and taking into account the fact that the black hole spacetime is not globally hyperbolic. It is shown that the Green's function calculated in this way is the Hartle-Hawking Green's function. The Green's function is used to compute $\langle T^\mu_\nu \rangle$, which is regular on the black hole horizon, and diverges at the singularity. A particle detector response function outside the horizon is also calculated and shown to be a fermi type distribution. The back-reaction from $\langle T_{\mu\nu} \rangle$ is calculated exactly and is shown to give rise to a curvature singularity at $r=0$ and to shift the horizon outwards. For $M=0$ a horizon develops, shielding the singularity. Some speculations about the endpoint of evaporation are discussed.