A mathematical theory of gravitational collapse

We study the asymptotic behaviour, as the retarded time u tends to infinity, of the solutions of Einstein's equations in the spherically symmetric case with a massless scalar field as the material model. We prove that when the final Bondi mass M ₁ is different from zero, as u → ∞, a black hole forms of mass M ₁ surrounded by vacuum. We find the rate of decay of the metric functions and the behaviour of the scalar field on the horizon.