Formulations of the 3+1 evolution equations in curvilinear coordinates

TL;DR

This paper modifies hyperbolic formulations of the 3+1 Einstein equations to ensure all auxiliary quantities are true tensors, enabling their use in curvilinear coordinates like spherical or cylindrical coordinates, with practical numerical examples.

Contribution

It introduces tensor-compatible modifications to the NOR and BSSN formulations for use in curvilinear coordinates, including spherical symmetry and regularity considerations.

Findings

Modified BSSN formulation works in spherical symmetry

Ensures auxiliary quantities are true tensors

Numerical examples demonstrate effectiveness

Abstract

Following Brown, in this paper we give an overview of how to modify standard hyperbolic formulations of the 3+1 evolution equations of General Relativity in such a way that all auxiliary quantities are true tensors, thus allowing for these formulations to be used with curvilinear sets of coordinates such as spherical or cylindrical coordinates. After considering the general case for both the Nagy-Ortiz-Reula (NOR) and the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulations, we specialize to the case of spherical symmetry and also discuss the issue of regularity at the origin. Finally, we show some numerical examples of the modified BSSN formulation at work in spherical symmetry.