LATE-TIME EVOLUTION OF NONLINEAR GRAVITATIONAL COLLAPSE

We study numerically the fully nonlinear gravitational collapse of a s elf-gravitating, minimally coupled, massless scalar field in spherical symmetry. Our numerical code is based on double-null coordinates and on free evolution of the metric functions: The evolution equations are integrated numerically, whereas the constraint equations are only mon itored. The numerical code is stable (unlike recent claims) and second -order accurate. We use this code to study the late-time asymptotic be havior at fixed r (outside the black hole), along the event horizon, a nd along future null infinity. In all three asymptotic regions we find that, after the decay of the quasinormal modes, the perturbations are dominated by inverse power-law tails. The corresponding power indices agree with the integer values predicted by linearized theory. We also study the case of a charged black hole nonlinearly perturbed by a (ne utral) self-gravitating scalar field, and find the same type of behavi or-i.e., quasinormal modes followed by inverse power-law tails, with t he same indices as in the uncharged case. [S0556-2821(97)03024-5].