Spherical symmetry as a test case for unconstrained hyperboloidal evolution

TL;DR

This paper explores hyperboloidal evolution in numerical relativity under spherical symmetry, demonstrating stable simulations of black hole-like systems including null infinity using adapted equations and constraint damping.

Contribution

It introduces a stable unconstrained evolution scheme for spherically symmetric spacetimes with null infinity, addressing regularization challenges at the boundary.

Findings

Achieved stable numerical evolutions with generalized BSSN and Z4 equations.

Developed an appropriate evolution equation for the lapse function.

Adapted constraint damping to handle null infinity effectively.

Abstract

We consider the hyperboloidal initial value problem in numerical relativity, motivated by the goal to evolve radiating compact objects such as black hole binaries with a numerical grid that includes null infinity. Unconstrained evolution schemes promise optimal efficiency, but are difficult to regularize at null infinity, where the compactified Einstein equations are formally singular. In this work we treat the spherically symmetry case, which already poses nontrivial problems and constitutes an important first step. We have carried out stable numerical evolutions with the generalized BSSN and Z4 equations coupled to a scalar field. The crucial ingredients have been to find an appropriate evolution equation for the lapse function and to adapt constraint damping terms to handle null infinity.