How to Avoid Artificial Boundaries in the Numerical Calculation of Black Hole Spacetimes

TL;DR

This paper introduces a numerical scheme for simulating asymptotically flat black hole spacetimes using conformal Einstein equations, avoiding artificial boundaries and enabling stable, boundary-free computations.

Contribution

It derives new first order evolution equations in symmetric hyperbolic and flux-conservative forms for boundary-free black hole spacetime simulations.

Findings

Stable numerical implementation of conformal Einstein equations.

Elimination of artificial boundaries in black hole spacetime calculations.

Framework applicable to a range of asymptotically flat spacetimes.

Abstract

This is the first of a series of papers describing a numerical implementation of the conformally rescaled Einstein equation, an implementation designed to calculate asymptotically flat spacetimes, especially spacetimes containing black holes. In the present paper we derive the new first order time evolution equations to be used in the scheme. These time evolution equations can either be written in symmetric hyperbolic or in flux-conservative form. Since the conformally rescaled Einstein equation, also called the conformal field equations, formally allow us to place the grid boundaries outside the physical spacetime, we can modify the equations near the grid boundaries and get a consistent and stable discretisation. Even if we calculate spacetimes containing black holes, there is no need for introducing artifical boundaries in the physical spacetime, which then would complicate, influence, or even exclude the computation of certain spacetime regions.