Mathematical Issues in a Fully-Constrained Formulation of Einstein Equations

TL;DR

This paper analyzes the mathematical structure of a new 3+1 formulation of Einstein equations, focusing on hyperbolic and elliptic systems, boundary conditions, and implications for black hole simulations.

Contribution

It provides a preliminary mathematical analysis of the coupled elliptic-hyperbolic system in a fully-constrained Einstein equations formulation, emphasizing boundary conditions and gauge choices.

Findings

The system is strongly hyperbolic under Dirac's gauge.

Boundary conditions for the shift vector are crucial for well-posedness.

Insights into boundary prescriptions for black hole excision methods.

Abstract

Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak [Phys. Rev. D {\bf 70}, 104007 (2004)] proposed a new formulation for 3+1 numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Dirac's gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depends on the boundary conditions for the shift vector in the elliptic subsystem. This interplay between the hyperbolic and elliptic parts of the complete evolution system is used to assess the prescription of inner boundary conditions for the hyperbolic part when using an excision approach to black hole spacetime evolutions.