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

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 $〈{T}_{\ensuremath{\nu}}^{\ensuremath{\mu}}〉$, 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 $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ is calculated exactly and is shown to give rise to a curvature singularity at $r=0$ and to shift the horizon outward. For $M=0$ a horizon develops, shielding the singularity. Some speculations about the end point of evaporation are discussed.