Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations

TL;DR

This paper investigates how boundary conditions derived from the Einstein tensor influence constraint propagation in initial-boundary value problems for Einstein's equations, comparing ADM and Einstein-Christoffel formulations.

Contribution

It clarifies the relationship between Einstein tensor boundary conditions and constraint preservation in two formulations of Einstein's equations.

Findings

Normal projection of Einstein tensor relates to constraint propagation.

Boundary conditions are linear combinations of evolution equations and constraints.

Connection to constraint-preserving boundary conditions is established.

Abstract

We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein equations with vanishing shift vector: the ADM formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as non-trivial boundary conditions are linear combinations of the evolution equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint-preserving'' boundary conditions becomes apparent.