Einstein's evolution equations as a system of balance laws

The evolution system in the space plus time (3+1) decomposition of Einstein's field equations is explicitly written as a system of balance laws. This is achieved by demanding that the time coordinate be harmonic (harmonic synchronization) and the space coordinate lines be normal to the constant-time hypersurfaces. No symmetry nor special form of the metric has been assumed, so that the equations may be used as a part of a three-dimensional numerical code for general relativity. The particular case of spherical symmetry is also considered and a numerical test of this case is provided by using a nonstandard form of the Schwarzschild line element.