Scaling Limit of the Noncommutative Black Hole

TL;DR

This paper investigates the scaling limit of a quantum black hole wave operator in noncommutative spacetime, revealing classical geometry with quantum effects near the horizon, including a frequency-dependent skin layer and finite redshift.

Contribution

It introduces a natural scaling limit at the event horizon in noncommutative spacetime, connecting quantum wave operators to classical geometry with quantum gravity remnants.

Findings

Finite redshift at the horizon for positive frequency modes.

Presence of a frequency-dependent skin layer near the horizon.

Analysis using Bessel and hypergeometric functions confirms propagation features.

Abstract

We show that the recent `quantum' black hole wave operator in the $\kappa$-Minkowski or bicrossproduct model quantum spacetime has a natural scaling limit $\lambda_p\to 0$ at the event horizon. Here $\lambda_p$ is the Planck time and the geometry at the event horizon in Planck length is maintained at the same time as the limit is taken, resulting in a classical theory with quantum gravity remnants. Among the features is a frequency-dependent `skin' of some $\omega\over\nu$ Planck lengths just inside the event horizon for $\omega>0$ and just outside for $\omega<0$, where $\nu$ is the frequency associated to the Schwarzschild radius. We use bessel and hypergeometric functions to analyse propagation through the event horizon and skin in both directions. The analysis confirms a finite redshift at the horizon for positive frequency modes in the exterior.