Boundary conditions for the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations

TL;DR

This paper develops boundary conditions for the BSSN formulation of Einstein's equations, ensuring constraint preservation, controlling gravitational degrees of freedom, and maintaining well-posedness for numerical relativity simulations.

Contribution

It introduces nine boundary conditions for the BSSN system that preserve constraints, control incoming gravitational waves, and ensure well-posedness in weak gravity regimes.

Findings

Boundary conditions preserve all constraints during evolution.

They control incoming gravitational degrees of freedom via Weyl scalar Psi_0.

The problem remains well-posed in the weak gravity limit.

Abstract

We discuss the initial-boundary value problem for the Baumgarte-Shapiro-Shibata-Nakamura evolution system of Einstein's field equations which has been used extensively in numerical simulations of binary black holes and neutron stars. We specify nine boundary conditions for this system with the following properties: (i) they impose the momentum constraint at the boundary, which is shown to preserve all the constraints throughout evolution, (ii) they approximately control the incoming gravitational degrees of freedom by specifying the Weyl scalar Psi_0 at the boundary, (iii) they control the gauge freedom by requiring a Neumann boundary condition for the lapse, by setting the normal component of the shift to zero, and by imposing a Sommerfeld-like condition on the tangential components of the shift, (iv) they are shown to yield a well-posed problem in the limit of weak gravity. Possible numerical applications of our results are also discussed briefly.