Covariant formulations of BSSN and the standard gauge

TL;DR

This paper reformulates the BSSN and standard gauge equations in a covariant tensor form, enabling coordinate-independent evolution and comparison of initial data across different spatial coordinate systems.

Contribution

It introduces a covariant tensor formulation of BSSN and standard gauge equations, allowing for coordinate transformations and improved flexibility in numerical relativity.

Findings

Two variants of covariant BSSN equations are proposed.

The tensor formulation simplifies the comparison of results across coordinate systems.

The approach enhances the robustness of numerical simulations in general relativity.

Abstract

The BSSN and standard gauge equations are written in covariant form with respect to spatial coordinate transformations. The BSSN variables are defined as tensors with no density weights. This allows us to evolve a given set of initial data using two different coordinate systems and to relate the results using the familiar tensor transformation rules. Two variants of the covariant equations are considered. These differ from one another in the way that the determinant of the conformal metric is evolved.