How to avoid artificial boundaries in the numerical calculation of black hole 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. 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 obtain a consistent and stable discretization. Even if we calculate spacetimes containing black holes, there is no need to introduce artificial boundaries in the physical spacetime, which then would complicate, influence or even exclude the computation of certain spacetime regions.