Strong hyperboloidal compactification for the spherical DF-GHG formulation of GR

TL;DR

This paper develops a robust hyperboloidal compactification method for the spherical DF-GHG formulation of GR, enabling accurate numerical simulations of black hole perturbations at null infinity.

Contribution

It introduces a new choice of constraint addition, gauge, and reduction fields to minimize singular terms in hyperboloidal compactification of the Einstein equations.

Findings

Successfully implemented aggressive compactification with minimal singular terms.

Numerical results match linear theory predictions for black hole perturbations.

Accurately extracted gravitational wave signals at null infinity.

Abstract

The use of compactified hyperboloidal coordinates for metric formulations of the Einstein field Equations introduces formally singular terms in the equations of motion whose numerical treatment requires care. In this paper we study a particular choice of constraint addition, choice of gauge and reduction fields in order to minimize the number of these terms in a spherically symmetric reduction of the Dual-Foliation Generalized Harmonic Gauge formulation of General Relativity. We proceed to the numerical implementation of a more aggressive compactification, as compared to our previous work. With the present setup there is a direct analogy with conformal compactification used in other approaches to the use of hyperboloidal coordinates. We present numerical results of constraints violating and satisfying perturbations on top of a Schwarzschild black hole. For small perturbations we recover the expected physics from linear theory, corresponding to quasi normal mode ringing and tail decay for a scalar field, both extracted directly at future null infinity from our numerical data.