Quantum mechanics of the interior of radiating 2D black holes

Abstract

We study the homogeneous sector of the RST model describing the gravitational dynamics, including back-reaction, of radiating 2-d black holes. We find the exact solutions both in conformal gauge and in time-parametrized form, isolate the black hole sector of the classical phase space and quantize the near singularity dynamics in conformal gauge. We show that different choices of measure and different self-adjoint extensions can lead to inequivalent quantum theories, all of which resolve the singularity. For a specific range of extension parameters, the Hamiltonian spectrum admits bound states that correspond physically to stable remnants. Finally, we argue that our work provides a good starting point for quantization of the full homogeneous theory using both reduced and Dirac quantization.

Quantum mechanics of the interior of the Russo-Susskind-Thorlacius black hole

“…Quantizing the scalars yields the usual trace anomaly [11], which we add to the above action in a local form that was first introduced by Hayward [12]. We use the conventions and notations of [7]. The local form of the action that forms the basis of our analysis is…”

“…where 1/ ( ≡ ∇ 2 ) refers to the scalar Green's function. Substituting (7) back into the z z term in the action (2) gives the usual non-local form R 1 R of the Polyakov action. A more careful analysis [12] verifies that this heuristic process does indeed work.…”

“…In this case there is a Killing horizon at finite θ = θ H where the metric component e 2ρ H = 0 and the curvature is finite. It was shown in [7] that the solution can be analytically extended past this point, and a suitable radial coordinate r defined in the exterior region such that r → ∞ corresponds to the asymptotic exterior region of the black hole. It was shown that this solution corresponds to the RST solution [3] for P = 0 and mass M = θ 0 λ √ κ, which, as noted by Birnir and…”

“…They argued that the fields would tend to homogeneity in this limit, and showed that the resulting quantum theory resolved the singularity as required. More recently Gegenberg et al [7] applied an analysis similar to that of [6] to the homogeneous interior of black holes in the RST model. In particular, they performed a complete analysis of the dynamics and the space of solutions, identifying the singularity and isolating the black hole sector.…”

“…The paper is organized as follows: In the next Section we review the RST model and the analysis in [7]. Section 3 presents the Hamiltonian describing the dilaton dynamics and constructs the corresponding time dependent Schrödinger equation.…”