Toward a Midisuperspace Quantization of LeMaitre-Tolman-Bondi Collapse Models

TL;DR

This paper develops a canonical quantization framework for LeMaître-Tolman-Bondi models of dust collapse, enabling the study of quantum gravitational effects during stellar collapse and the formation of black holes or naked singularities.

Contribution

It introduces a canonical reduction and quantization of inhomogeneous dust collapse models, deriving a Wheeler-DeWitt equation for analyzing quantum effects in gravitational collapse.

Findings

Derived a simple Wheeler-DeWitt equation for dust collapse models.

Provided solutions to the quantum constraint, enabling analysis of quantum gravity effects.

Facilitated future studies on quantum effects in black hole and naked singularity formation.

Abstract

LeMa\^\i tre-Tolman-Bondi models of spherical dust collapse have been used and continue to be used extensively to study various stellar collapse scenarios. It is by now well-known that these models lead to the formation of black holes and naked singularities from regular initial data. The final outcome of the collapse, particularly in the event of naked singularity formation, depends very heavily on quantum effects during the final stages. These quantum effects cannot generally be treated semi-classically as quantum fluctuations of the gravitational field are expected to dominate before the final state is reached. We present a canonical reduction of LeMa\^\i tre-Tolman-Bondi space-times describing the marginally bound collapse of inhomogeneous dust, in which the physical radius, $R$, the proper time of the collapsing dust, $\tau$, and the mass function, $F$, are the canonical coordinates, $R(r)$, $\tau(r)$ and $F(r)$ on the phase space. Dirac's constraint quantization leads to a simple functional (Wheeler-DeWitt) equation. The equation is solved and the solution can be employed to study some of the effects of quantum gravity during gravitational collapse with different initial conditions.