Quantization of the Schwarzschild black hole: a Noether symmetry approach

TL;DR

This paper applies Noether symmetry to the canonical formalism of Schwarzschild black holes, deriving classical solutions and quantizing the model, which suggests potential singularity avoidance.

Contribution

It introduces a Noether symmetry approach to quantize the Schwarzschild black hole within a canonical framework, linking classical solutions with quantum wave functions.

Findings

Classical Schwarzschild solutions derived from the $r$-Hamiltonian.

Quantization yields solutions to the Wheeler-DeWitt equation.

Wave function analysis indicates possible singularity avoidance.

Abstract

We study the canonical formalism of a spherically symmetric space-time. In the context of the 3+1 decomposition with respect to the radial coordinate $r$, we set up an effective Lagrangian in which a couple of metric functions play the role of independent variables. We show that the resulting $r$-Hamiltonian yields the correct classical solutions which can be identified with the Schwarzschild black hole. The Noether symmetry of model is then investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. According to the Noether symmetry approach, we also quantize the model and show that the existence of a Noether symmetry yields a general solution to the Wheeler-DeWitt equation where exhibits a good correlation with the classical regime. We use the resulting wave function in order to (qualitative) investigate the possibility of the avoidance of classical singularities.