Einstein boundary conditions for the 3+1 Einstein equations

TL;DR

This paper derives explicit boundary conditions for the 3+1 Einstein equations in a specific gauge, crucial for well-posed initial-boundary value problems in numerical relativity.

Contribution

It explicitly calculates boundary equations from Einstein tensor projections and demonstrates their role in prescribing boundary data within a hyperbolic formulation.

Findings

Derived four boundary conditions for Einstein equations.

Showed how to prescribe incoming characteristic fields.

Identified conditions on outgoing fields.

Abstract

In the 3+1 framework of the Einstein equations for the case of vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the normal to the boundary surface of the initial-boundary value problem. Such conditions take the form of evolution equations along (as opposed to across) the boundary for certain components of the extrinsic curvature and for certain space-derivatives of the intrinsic metric. We argue that, in general, such boundary conditions do not follow necessarily from the evolution equations and the initial data, but need to be imposed on the boundary values of the fundamental variables. Using the Einstein-Christoffel formulation, which is strongly hyperbolic, we show how three of the boundary equations should be used to prescribe the values of some incoming characteristic fields. Additionally, we show that the fourth one imposes conditions on some outgoing fields.