  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.
