Quantum Structure of Space Near a Black Hole Horizon

TL;DR

This paper develops a quantum model of space near black hole horizons using a midi-superspace approach, revealing a discrete spatial topology influenced by scale-invariant quantum mechanics and Bohr quantization.

Contribution

It introduces a novel quantization scheme for black hole horizons that results in a discrete spatial structure and connects quantum mechanics with black hole geometry.

Findings

Discrete spatial topology outside the horizon

Spectrum approaches continuum asymptotically

Quantum mechanics described by $1/X^2$ potential

Abstract

We describe a midi-superspace quantization scheme for generic single horizon black holes in which only the spatial diffeomorphisms are fixed. The remaining Hamiltonian constraint yields an infinite set of decoupled eigenvalue equations: one at each spatial point. The corresponding operator at each point is the product of the outgoing and ingoing null convergences, and describes the scale invariant quantum mechanics of a particle moving in an attractive $1/X^2$ potential. The variable $X$ that is analoguous to particle position is the square root of the conformal mode of the metric. We quantize the theory via Bohr quantization, which by construction turns the Hamiltonian constraint eigenvalue equation into a finite difference equation. The resulting spectrum gives rise to a discrete spatial topology exterior to the horizon. The spectrum approaches the continuum in the asymptotic region.