Quantization of Black Holes in the Wheeler-DeWitt Approach

TL;DR

This paper explores black hole quantization using the Wheeler-DeWitt approach, linking boundary degrees of freedom with thermodynamics, and applies the framework to the BTZ black hole in 2+1 dimensions.

Contribution

It provides a Hamiltonian formulation including surface terms and demonstrates how thermodynamical properties emerge from quantum equations in black hole spacetimes.

Findings

Boundary degrees of freedom relate to black hole thermodynamics

Quantum equations reproduce thermodynamical properties via WKB approximation

Analysis of BTZ black hole in 2+1 dimensions

Abstract

We discuss black hole quantization in the Wheeler-DeWitt approach. Our consideration is based on a detailed investigation of the canonical formulation of gravity with special considerations of surface terms. Since the phase space of gravity for non-compact spacetimes or spacetimes with boundaries is ill-defined unless one takes boundary degrees of freedom into account, we give a Hamiltonian formulation of the Einstein-Hilbert-action as well as a Hamiltonian formulation of the surface terms. It then is shown how application to black hole spacetimes connects the boundary degrees of freedom with thermodynamical properties of black hole physics. Our treatment of the surface terms thereby naturally leads to the Nernst theorem. Moreover, it will produce insights into correlations between the Lorentzian and the Euclidean theory. Next we discuss quantization, which we perform in a standard manner. It is shown how the thermodynamical properties can be rediscovered from the quantum equations by a WKB like approximation scheme. Back reaction is treated by going beyond the first order approximation. We end our discussion by a rigorous investigation of the so-called BTZ solution in 2+1 dimensional gravity.