Evolving Einstein's field equations with matter: The "hydro without hydro" test

We include matter sources in Einstein's held equations and show that our recently proposed 3 + 1 evolution scheme can stably evolve strong-field solutions. We insert in our code known matter solutions, namely the Oppenheimer-Volkoff solution for a static star and the Oppenheimer-Snyder solution for homogeneous dust sphere collapse to a black hole, and evolve the gravitational field equations. We fmd that we ran evolve stably static, strong-field stars for arbitrarily long times and can follow dust sphere collapse accurately well past black hale formation. These tests are useful diagnostics for fully self-consistent, stable hydrodynamical simulations in 3 + 1 general relativity. Moreover, they suggest a successive approximation scheme for determining gravitational waveforms from strong-field sources dominated by longitudinal fields, such as binary neutron stars: approximate quasi-equilibrium models can serve as sources for the transverse field equations, which can be evolved without having to re-solve the hydrodynamical equations ("hydro without hydro"). [S0556-2821(99)03118-5].