Weyl Curvature Evolution System for GR

TL;DR

This paper reformulates the equations of General Relativity into a simple, first-order evolution system using a chiral connection and Weyl curvature, enabling better numerical constraint management.

Contribution

It introduces a novel, simplified evolution system for GR based on chiral variables, extending Maxwell's equations and improving constraint handling.

Findings

The evolution equations are first order in time and space.

The system naturally generalizes chiral Maxwell equations.

Constraint violations can be propagated and removed numerically.

Abstract

Starting from the chiral first-order pure connection formulation of General Relativity, we put the field equations of GR in a strikingly simple evolution system form. The two dynamical fields are a complex symmetric tracefree 3x3 matrix Psi, which encodes the self-dual part of the Weyl curvature tensor, as well as a spatial SO(3,C) connection A. The right-hand sides of the evolution equations also contain the triad for the spatial metric, and this is constructed non-linearly from the field Psi and the curvature of the spatial connection A. The evolution equations for this pair are first order in both time and spatial derivatives, and so simple that they could have been guessed without a computation. They are also the most natural generalisations of the equations one obtains in the case of the chiral description of Maxwell's theory. We also determine the modifications of the evolution system needed to enforce the "constraint sweeping", so that any possible numerical violation of the constraints present becomes propagating and gets removed from the computational grid.