Classical and Quantum Equations of Motion for a BTZ Black String in AdS Space

TL;DR

This paper studies the classical and quantum dynamics of a BTZ black string in AdS space, revealing that quantum effects can eliminate the classical singularity and introduce non-local behavior near the origin.

Contribution

It derives the classical Hamiltonian for the black string and analyzes quantum effects, showing singularity removal and non-locality near the origin for the first time.

Findings

Quantum effects remove the classical singularity at the origin.

Near the horizon, quantum effects do not alter classical infalling observer conclusions.

Quantum behavior near the origin exhibits non-local effects depending on the domain wall's energy density.

Abstract

We investigate gravitational collapse of a $(3+1)$-dimensional BTZ black string in AdS space in the context of both classical and quantum mechanics. This is done by first deriving the conserved mass per unit length of the cylindrically symmetric domain wall, which is taken as the classical Hamiltonian of the black string. In the quantum mechanical context, we take primary interest in the behavior of the collapse near the horizon and near the origin (classical singularity) from the point of view of an infalling observer. In the absence of radiation, quantum effects near the horizon do not change the classical conclusions for an infalling observer, meaning that the horizon is not an obstacle for him/her. The most interesting quantum mechanical effect comes in when investigating near the origin. First, quantum effects are able to remove the classical singularity at the origin, since the wave function is non-singular at the origin. Second, the Schr\"odinger equation describing the behavior near the origin displays non-local effects, which depend on the energy density of the domain wall. This is manifest in that derivatives of the wavefunction at one point are related to the value of the wavefunction at some other distant point.