Free hyperboloidal evolution in spherical symmetry

TL;DR

This paper develops a stable numerical scheme for hyperboloidal evolution in spherical symmetry within Numerical Relativity, enabling accurate simulation of spacetime reaching null infinity and handling black hole formation.

Contribution

It introduces a regularized numerical approach using BSSN and Z4 formulations with a conformal factor, addressing divergence issues at null infinity in spherical symmetry.

Findings

Stable evolutions achieved with regular and black hole initial data.

Black hole formation results in a different final state from initial data.

The method effectively handles the treatment of null infinity in simulations.

Abstract

We address the hyperboloidal initial value problem in the context of Numerical Relativity, motivated by its evolution on hyperboloidal slices: smooth spacelike slices that reach future null infinity, the "location" in spacetime where radiation is to be extracted. Our approach uses the BSSN and Z4 formulations and a time-independent conformal factor. The resulting system of PDEs includes formally diverging terms at null infinity. Here we discuss a regularized numerical scheme in spherical symmetry. A critical ingredient are the gauge conditions, which control the treatment of future null infinity. Stable numerical evolutions have been performed with regular and black hole initial data on a hyperboloidal slice. A sufficiently large scalar field perturbation will create a black hole, whose final stationary state is different from the trumpet initial data derived here.