Modelling quantum particles falling into a black hole: the deep interior limit

TL;DR

This paper develops a solvable quantum model of a black hole's interior with scalar field excitations, using loop quantum gravity techniques to explore the deep interior region and its quantum dynamics.

Contribution

It introduces a solvable toy model for black hole interiors incorporating loop quantum gravity, focusing on the deep interior and scalar field effects.

Findings

Exact classical and quantum solutions in the deep interior regime

Quantization of the black hole mass with infinite degeneracy

Continuum limit of polymer dynamics reproduces Schrödinger evolution

Abstract

In this paper we construct a solvable toy model of the quantum dynamics of the interior of a spherical black hole with falling spherical scalar field excitations. We first argue about how some aspects of the quantum gravity dynamics of realistic black holes emitting Hawking radiation can be modelled using Kantowski-Sachs solutions with a massless scalar field when one focuses on the deep interior region $r\ll M$ (including the singularity). Further, we show that in the $r\ll M$ regime, and in suitable variables, the KS model becomes exactly solvable at both the classical and quantum levels. The quantum dynamics inspired by loop quantum gravity is revisited. We propose a natural polymer-quantization where the area $a$ of the orbits of the rotation group is quantized. The polymer (or loop) dynamics is closely related with the Schroedinger dynamics away from the singularity with a form of continuum limit naturally emerging from the polymer treatment. The Dirac observable associated to the mass is quantized and shown to have an infinite degeneracy associated to the so-called $\epsilon$-sectors. Suitable continuum superpositions of these are well defined distributions in the fundamental Hilbert space and satisfy the continuum Schroedinger dynamics.