Symmetric hyperbolicity and consistent boundary conditions for second-order Einstein equations

TL;DR

This paper develops symmetric hyperbolic formulations of Einstein's equations with characteristic variables and energy conservation, proposing boundary conditions that preserve constraints and are suitable for well-posed initial-boundary value problems.

Contribution

It introduces two families of second-order Einstein equations formulations with complete characteristic sets and constraint-preserving boundary conditions, advancing the mathematical understanding of Einstein's equations.

Findings

Formulations admit complete characteristic variables.

Energy conservation expressed in characteristic variables.

Boundary conditions preserve constraints and are applicable for smooth boundaries.

Abstract

We present two families of first-order in time and second-order in space formulations of the Einstein equations (variants of the Arnowitt-Deser-Misner formulation) that admit a complete set of characteristic variables and a conserved energy that can be expressed in terms of the characteristic variables. The associated constraint system is also symmetric hyperbolic in this sense, and all characteristic speeds are physical. We propose a family of constraint-preserving boundary conditions that is applicable if the boundary is smooth with tangential shift. We conjecture that the resulting initial-boundary value problem is well-posed.