Weyl curvature evolution system for GR

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 matrix , which encodes the self-dual part of the Weyl curvature tensor, as well as a spatial connection . 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 and the curvature of the spatial connection . 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 the most natural spin two generalisations of Maxwell’s spin one equations. 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.