Black Hole Boundary Conditions and Coordinate Conditions

TL;DR

This paper develops boundary and coordinate conditions for simulating black holes in 3+1D Einstein equations, proposing a well-posed elliptic system that ensures stable numerical evolution.

Contribution

It introduces prescribed curvature boundary conditions for black hole horizons, forming a well-posed elliptic system compatible with maximal slicing in 3+1D Einstein equations.

Findings

The boundary conditions lead to a well-posed elliptic system.

The approach ensures stable and globally well-behaved numerical simulations.

Boundary conditions without boundary values are emphasized.

Abstract

This paper treats boundary conditions on black hole horizons for the full 3+1D Einstein equations. Following a number of authors, the apparent horizon is employed as the inner boundary on a space slice. It is emphasized that a further condition is necessary for the system to be well posed; the ``prescribed curvature conditions" are therefore proposed to complete the coordinate conditions at the black hole. These conditions lead to a system of two 2D elliptic differential equations on the inner boundary surface, which coexist nicely to the 3D equation for maximal slicing (or related slicing conditions). The overall 2D/3D system is argued to be well posed and globally well behaved. The importance of ``boundary conditions without boundary values" is emphasized. This paper is the first of a series. This revised version makes minor additions and corrections to the previous version.