Scalar field quantization on the 2+1 dimensional black hole background
G. Lifschytz, M. Ortiz
TL;DR
This work analyzes a massless, conformally coupled scalar on the 2+1 BTZ black hole background by constructing the exact Hartle-Hawking Green's function via the method of images from AdS$_3$. The authors compute the renormalized $\langle\phi^2\rangle$ and $\langle T_{\mu}^{\ \nu}\rangle$, showing regular behavior on the horizon and a curvature singularity at $r=0$, with the horizon shifted outward by back-reaction. They also evaluate the response of a stationary particle detector outside the horizon, finding a fermion-like distribution and, for $M=0$, no particle detection despite a nonzero energy-momentum tensor. The back-reaction analysis indicates horizon formation for the $M=0$ case and a possible endpoint of evaporation that may lie beyond the classical $M=0$ geometry, highlighting the need for a quantum-gravity treatment near the singular region.
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 $\langle T^μ_ν\rangle$, 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 $\langle T_{μν} \rangle$ 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.