Spherical Evolution of the Generalized Harmonic Gauge Formulation of General Relativity on Compactified Hyperboloidal Slices

TL;DR

This paper demonstrates successful numerical evolution of the generalized harmonic gauge formulation of general relativity on compactified hyperboloidal slices, including black hole formation and scalar field dynamics, with verified convergence and physical accuracy.

Contribution

It introduces a dual-foliation approach for hyperboloidal evolution in spherical symmetry, including methods with height functions and eikonal equations, and applies these to scalar field and black hole spacetimes.

Findings

Achieved stable numerical evolutions with convergence verification.

Recovered expected quasinormal modes and tail decay rates.

Successfully modeled black hole formation from scalar collapse.

Abstract

We report on the successful numerical evolution of the compactified hyperboloidal initial value problem in general relativity using generalized harmonic gauge. We work in spherical symmetry, using a massless scalar field to drive dynamics. Our treatment is based on the dual-foliation approach, proceeding either by using a height function or by solving the eikonal equation to map between frames. Both are tested here with a naive implementation and with hyperboloidal layers. We present a broad suite of numerical evolutions, including pure gauge perturbations, constraint violating and satisfying data with and without scalar field matter. We present calculations of spacetimes with a regular center. For black hole spacetimes we use excision to remove part of the black hole interior. We demonstrate both pointwise and norm convergence at the expected rate of our discretization. We present evolutions in which the scalar field collapses to form a black hole. Evolving nonlinear scalar field perturbations of the Schwarzschild spacetime, we recover the expected quasinormal frequencies and tail decay rates from linear theory.