General-covariant evolution formalism for numerical relativity

A general-covariant extension of Einstein’s field equations is considered with a view to numerical relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector $Z_{\mu }.$ Einstein’s solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition $Z_{\mu }=0.\mathrm{}$ The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing.