Einstein's Equations with Asymptotically Stable Constraint Propagation

TL;DR

This paper proposes a modified form of Einstein's equations embedded in a larger hyperbolic system, ensuring that the constraint surface acts as an attractor, which could improve numerical solutions in general relativity.

Contribution

It introduces a new hyperbolic formulation of Einstein's equations with stable constraint propagation, potentially enhancing numerical relativity simulations.

Findings

Constraint surface is an attractor in the linearized system.

Extended system reproduces Einstein's dynamics on the constraint surface.

Potential for improved numerical stability in simulations.

Abstract

We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, non-linear gravity.