On the well posedness of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations

TL;DR

This paper establishes a well-posed initial value formulation for the BSSN formulation of Einstein's equations, ensuring mathematical stability and consistency for numerical relativity simulations.

Contribution

It introduces a first order symmetric hyperbolic system for BSSN with gauge conditions, including boundary conditions, enhancing the theoretical foundation of numerical relativity.

Findings

The formulation is well posed with fixed gauge conditions.

Boundary conditions are derived for artificial boundaries.

Dynamical gauges lead to strongly hyperbolic, well-posed systems.

Abstract

We give a well posed initial value formulation of the Baumgarte-Shapiro-Shibata-Nakamura form of Einstein's equations with gauge conditions given by a Bona-Masso like slicing condition for the lapse and a frozen shift. This is achieved by introducing extra variables and recasting the evolution equations into a first order symmetric hyperbolic system. We also consider the presence of artificial boundaries and derive a set of boundary conditions that guarantee that the resulting initial-boundary value problem is well posed, though not necessarily compatible with the constraints. In the case of dynamical gauge conditions for the lapse and shift we obtain a class of evolution equations which are strongly hyperbolic and so yield well posed initial value formulations.