Free Evolution of Nonlinear Scalar Field Collapse in Double-null Coordinates

We study numerically the fully nonlinear spherically-symmetric collapse of a selfgravitating, minimally-coupled, massless scalar field. Our numerical code is based on double-null coordinates and on free evolution of the metric functions and the scalar field. The numerical code is stable and second-order accurate. We use this code to study the late-time asymptotic behavior at fixed r (outside the black hole), along the event horizon, and along future null infinity. In all three asymptotic regions we find that, after the decay of the quasi-normal 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 (neutral) self-gravitating scalar field, and find the same type of behavior—i.e., quasi-normal modes followed by inverse power-law tails, with the same indices as in the uncharged case.