Evolution systems for non-linear perturbations of background geometries

TL;DR

This paper develops a background geometry approach for the Einstein equations' initial value problem, expressing the equations in terms of deviations from a known background to improve numerical relativity methods.

Contribution

It introduces a specialized background geometry formulation that rewrites Einstein equations in terms of non-linear deviations, enhancing numerical stability and analysis.

Findings

Reformulation of Einstein equations using background geometry.

Equations expressed in first order form linked to Einstein-Christoffel system.

Potential numerical advantages due to altered source structure.

Abstract

The formulation of the initial value problem for the Einstein equations is at the heart of obtaining interesting new solutions using numerical relativity and still very much under theoretical and applied scrutiny. We develop a specialised background geometry approach, for systems where there is non-trivial a priori knowledge about the spacetime under study. The background three-geometry and associated connection are used to express the ADM evolution equations in terms of physical non-linear deviations from that background. Expressing the equations in first order form leads naturally to a system closely linked to the Einstein-Christoffel system, introduced by Anderson and York, and sharing its hyperbolicity properties. We illustrate the drastic alteration of the source structure of the equations, and discuss why this is likely to be numerically advantageous.