Strongly hyperbolic second order Einstein's evolution equations

TL;DR

This paper proves that a class of Einstein's evolution equations used in numerical relativity, specifically BSSN-type equations with densitized lapse, are strongly hyperbolic, ensuring well-posedness of the initial value problem.

Contribution

It demonstrates that a pseudo-differential first order reduction of BSSN-type equations yields strongly hyperbolic systems without adding extra equations or constraints.

Findings

BSSN-type equations are strongly hyperbolic after reduction.

Densitized Arnowitt-Deser-Misner equations are weakly hyperbolic.

The reduction preserves arbitrary lapse and shift functions.

Abstract

BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformal-traceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudo-differential first order reduction of these equations is strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner evolution equations are found to be weakly hyperbolic. In both cases, the positive densitized lapse function and the spacelike shift vector are arbitrary given fields. This first order pseudodifferential reduction adds no extra equations to the system and so no extra constraints.