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

Conner Dailey, Erik Schnetter, Niayesh Afshordi

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 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.

Introduction

Recently we described a method for simulating black hole echoes in numerical relativity by imposing reflecting boundary conditions (BCs) near the horizon of a black hole in spherical symmetry. In that work, we defined reflecting BCs on a scalar field based on conservation laws and implemented it using modern numerical methods, including Summation By Parts (SBP) finite differencing operators and BCs applied using Simultaneous Approximation Terms (SATs), to ensure the system remains numerically stable. In an effort to apply this approach to spacetimes with no symmetries, we consider here a generalization to three spatial dimensions.

The motivation to characterize and demonstrate stable initial boundary value problems (IBVPs) in numerical relativity comes from our interest in simulating black hole echoes. Additionally, this is of direct interest in the numerical relativity community in the wake of the first demonstration of Cauchy characteristic matching (CCM), a technique that combines Cauchy evolution and characteristic evolution to provide gravitational wave (GW) scattering feedback to the sources in the Cauchy domain. It was shown that it is possible to shrink the Cauchy domain while still obtaining accurate GW forms at null infinity by demonstrating several simulations of black holes perturbed by GWs. Shrinking the Cauchy domain can dramatically save on the computational cost of simulations, as most of this cost comes from the Cauchy evolution. Simulating black hole echoes and evolving CCM necessitate a numerically stable way of evolving dynamical spacetimes with boundaries in the strong gravity regime.

Abandoning spherical symmetry adds a considerable amount of complexity to the BCs and the manner in which they are applied. We have previously derived and demonstrated reflecting BCs based on the Misner–Sharp mass, which happens to be locally conserved in spherical symmetry. However, in general, local conservation laws do not exist in general relativity. Instead, we have to rely on quasi-local conservation laws to define what it means to reflect gravitational radiation and matter waves. For a review of local and quasi-local conservation laws. Although we plan to develop BCs based on controlling the flux of a conserved quasi-local quantity, the focus in this work will be on the adjusted evolution equations, BC framework, and numerical implementation that we develop to make them feasible.