An axisymmetric evolution code for the Einstein equations on hyperboloidal slices

TL;DR

This paper introduces a stable numerical evolution scheme for Einstein's equations on hyperboloidal slices, enabling accurate simulations extending to null infinity, with successful long-term evolution of Schwarzschild spacetime and gravitational perturbations.

Contribution

It presents the first stable dynamical evolution of Einstein's equations on hyperboloidal slices using a conformally rescaled metric, incorporating regularization techniques at null infinity.

Findings

Achieved long-term stable evolutions of Schwarzschild spacetime.

Successfully computed the Bondi news function at null infinity.

Demonstrated regularization of singular equations at null infinity.

Abstract

We present the first stable dynamical numerical evolutions of the Einstein equations in terms of a conformally rescaled metric on hyperboloidal hypersurfaces extending to future null infinity. Axisymmetry is imposed in order to reduce the computational cost. The formulation is based on an earlier axisymmetric evolution scheme, adapted to time slices of constant mean curvature. Ideas from a previous study by Moncrief and the author are applied in order to regularize the formally singular evolution equations at future null infinity. Long-term stable and convergent evolutions of Schwarzschild spacetime are obtained, including a gravitational perturbation. The Bondi news function is evaluated at future null infinity.