New first-order formulation for the Einstein equations

We derive a new first-order formulation for Einstein’s equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the $3+1$ decomposition with arbitrary lapse and shift. In the reduction to first-order form only eight particular combinations of the 18 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of a symmetric hyperbolic system in 14 unknowns, namely, the components of the extrinsic curvature perturbation and the eight new variables, from whose solution the metric perturbation can be computed by integration.