Quantum Wavefunction for a Schwarzschild Black Hole

TL;DR

This paper explores the quantum wavefunction of a Schwarzschild black hole, showing it vanishes at the singularity and discussing implications for quantum mechanics and gravity, with a focus on Hamiltonian quantization.

Contribution

It introduces a Hamiltonian-based quantization approach for black hole wavefunctions and analyzes their behavior at singularities, highlighting limitations and future directions.

Findings

Wavefunction vanishes at the Schwarzschild singularity

Provides conditions for wavefunction vanishing based on quantum numbers

Highlights limitations of the current mathematical approach

Abstract

A pseudo-Riemannian manifold contains an inherent Hamiltonian structure within the symplectic manifold in the cotangent bundle corresponding to the metric. Using this structure, it is possible to define a Hamiltonian, which can be quantized, and substituted into The Schrodinger Equation. The wave function, despite having a very complicated structure, has the very interesting property where it vanishes at the Schwarzschild $(r=0)$ singularity. This demonstrates an apparent violation of Quantum Mechanics, in which information is required to be conserved, and gives further evidence of a wave function vanishing at the singularity point. In addition, we give conditions for this wave function vanishing in terms of quantum numbers. NOTE: It has been graciously pointed out that the techniques of transforming the Hamiltonian to an equivalent metric tensor is only valid for a Riemannian manifold, not the pseudo-Riemannian manifold as required by General Relativistic theories. Although, the specific mathematics used is completely valid in this paper, the correct approach would be to use the ADM techniques for Quantum Gravity. Updated version to follow!