  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. The extended field
equations, when supplemented by suitable coordinate conditions, determine the
time evolution of all these variables without any constraint. Einstein's
solutions are recovered when the additional four-vector vanishes, so that the
energy and momentum constraints hold true. The extended system is well posed
when using the natural extension of either harmonic coordinates or the harmonic
slicing condition in normal coordinates
