Formulating the complete initial boundary value problem in numerical relativity to model black hole echoes

TL;DR

This paper develops a generalized, stable numerical framework in relativity to simulate black hole echoes, including complex boundary conditions, advancing the modeling of quantum gravity effects near black hole horizons.

Contribution

It introduces a generalized harmonic formulation with an SBP scheme and embedded boundary framework for simulating black hole echoes in non-symmetric spacetimes.

Findings

Successfully implemented a stable numerical scheme for complex boundaries

Demonstrated gravitational wave scattering simulations near black hole horizons

Paved the way for modeling quantum gravity effects in black hole mergers

Abstract

In an attempt to simulate black hole echoes (generated by potential quantum-gravitational structure) in numerical relativity, we recently described how to implement a reflecting boundary outside of the horizon of a black hole in spherical symmetry. Here, we generalize this approach to spacetimes with no symmetries and implement it numerically using the generalized harmonic formulation. We cast the evolution equations and the numerical implementation into a Summation By Parts (SBP) scheme, which seats our method closer to a class of provably numerically stable systems. We implement an embedded boundary numerical framework that allows for arbitrarily shaped domains on a rectangular grid and even boundaries that evolve and move across the grid. As a demonstration of this framework, we study the evolution of gravitational wave scattering off a boundary either inside, or just outside, the horizon of a black hole. This marks a big leap toward the goal of a generic framework to obtain gravitational waveforms for behaviors motivated by quantum gravity near the horizons of merging black holes.