Collapsing Shells and Black Holes: a quantum analysis

TL;DR

This paper performs a quantum analysis of collapsing null shells and black holes, incorporating phase-space noncommutativity, and finds that noncommutative effects cause the wavefunction to vanish at singularities and horizons.

Contribution

It extends the quantization of null shells to include noncommutative quantum mechanics and analyzes the resulting wavefunctions in this novel framework.

Findings

Wavefunction oscillates inside the shell in the commutative case

Wavefunction vanishes at singularity and horizon in the noncommutative case

Provides a quantum description of black hole boundary conditions

Abstract

The quantization of a spherically symmetric null shells is performed and extended to the framework of phase-space noncommutative (NC) quantum mechanics. The encountered properties are investigated making use of the Israel junction conditions on the shell, considering that it is the boundary between two spherically symmetric spacetimes. Using this method, and considering two different Kantowski-Sachs spacetimes as a representation for the Schwarzschild spacetime, the relevant quantities on the shell are computed, such as its stress-energy tensor and the action for the whole spacetime. From the obtained action, the Wheeler-deWitt equation is deduced in order to provide the quantum framework for the system. Solutions for the wavefunction of the system are found on both the commutative and NC scenarios. It is shown that, on the commutative version, the wave function has a purely oscillatory behavior in the interior of the shell. In the NC setting, it is shown that the wavefunction vanishes at the singularity, as well as, at the event horizon of the black hole.