Einstein boundary conditions for the Einstein equations in the conformal-traceless decomposition

TL;DR

This paper derives boundary conditions for the BSSN formulation of Einstein equations, ensuring well-posedness and proper constraint propagation in numerical relativity simulations.

Contribution

It explicitly formulates boundary conditions from Einstein tensor projections and analyzes their role in constraint propagation within the BSSN framework.

Findings

Boundary conditions derived from Einstein tensor projections.

Identification of incoming constraints at the boundary.

Characterization of prescribed and arbitrary boundary fields.

Abstract

In relation to the BSSN formulation of the Einstein equations, we write down the boundary conditions that result from the vanishing of the projection of the Einstein tensor normally to a timelike hypersurface. Furthermore, by setting up a well-posed system of propagation equations for the constraints, we show explicitly that there are three constraints that are incoming at the boundary surface and that the boundary equations are linearly related to them. This indicates that such boundary conditions play a role in enforcing the propagation of the constraints in the region interior to the boundary. Additionally, we examine the related problem for a strongly hyperbolic first-order reduction of the BSSN equations and determine the characteristic fields that are prescribed by the three boundary conditions, as well as those that are left arbitrary.