Gravitational collapse with non-vanishing tangential stresses: a generalization of the Tolman-Bondi model

The gravitational dynamics of anisotropic elastic spheres supported only by tangential stresses and satisfying an equation of state is analysed, and a fairly large class of non-static, spherically symmetric solutions of the Einstein field equations is found by quadratures. The solutions contain three arbitrary functions. Two such functions are immediately recognized as the initial distributions of mass and energy, familiar from the Tolman-Bondi (dust) models, while the third is the elastic internal energy per unit volume. If this function is a constant, the energy density becomes proportional to the matter density and therefore the metric reduces to the Tolman-Bondi one. In the general case, however, the solutions contain oscillating models as well as finite-bouncing models.