Reflecting boundary conditions in numerical relativity as a model for black hole echoes

TL;DR

This paper develops a numerical framework to model reflecting boundary conditions for scalar fields in dynamical black hole spacetimes, enabling the study of black hole echoes and near-horizon physics.

Contribution

It introduces a novel method for implementing reflecting boundary conditions in numerical relativity simulations of black holes.

Findings

Successful numerical implementation of reflecting boundaries near black hole horizons

Demonstrated convergence of wave evolution with boundary close to the horizon

Enables future studies of exotic near-horizon phenomena

Abstract

Recently, there has been much interest in black hole echoes, based on the idea that there may be some mechanism (e.g., from quantum gravity) that waves/fields falling into a black hole could partially reflect off of an interface before reaching the horizon. There does not seem to be a good understanding of how to properly model a reflecting surface in numerical relativity, as the vast majority of the literature avoids the implementation of artificial boundaries, or applies transmitting boundary conditions. Here, we present a framework for reflecting a scalar field in a fully dynamical spherically symmetric spacetime, and implement it numerically. We study the evolution of a wave packet in this situation and its numerical convergence, including when the location of a reflecting boundary is very close to the horizon of a black hole. This opens the door to model exotic near-horizon physics within full numerical relativity.