Hamiltonian Formalism for Black Holes and Quantization

TL;DR

This paper develops a Hamiltonian formalism for black holes using the Einstein equations, enabling a Wheeler--DeWitt quantization approach that treats the Schwarzschild mass as a canonical variable.

Contribution

It introduces a novel canonical formalism in the radial variable for static spherically symmetric spacetimes, facilitating quantum analysis of black holes.

Findings

Schwarzschild mass is represented as a commuting canonical function.

General solutions are expressed as superpositions of mass eigenfunctions.

The formalism is applicable inside the Schwarzschild horizon.

Abstract

Starting from the Lagrangian formulation of the Einstein equations for the vacuum static spherically symmetric metric, we develop a canonical formalism in the radial variable $r$ that is time--like inside the Schwarzschild horizon. The Schwarzschild mass turns out to be represented by a canonical function that commutes with the $r$--Hamiltonian. We investigate the Wheeler--DeWitt quantization and give the general representation for the solution as superposition of eigenfunctions of the mass operator.