GRAVITATIONAL COLLAPSE OF AN INHOMOGENEOUS DUST SPHERE IN HIGHER-DIMENSIONAL SPACETIME

A class of interior solutions for a spherically inhomogeneous dust distribution in multidimensional space-time is obtained. This generalizes to (n + 2)-dimension, the well-known solution of Tolman-Bondi in the sense that when n = 2 one recovers the Tolman-Bondi spacetime. The dynamical behavior of the model is studied, and it is observed that there are some differences from the analogous four-dimensional case. Unlike the standard four-dimensional case where, in general, the metric coefficients are obtained in parametric forms only, we here get the integrals of the field equations in closed form. Two important aspects, shell crossing and shell focusing singularities which are generally associated with this type of inhomogeneous dust collapse, are also discussed. It is found that depending on the degree of inhomogeneity of the collapsing matter it may, in some cases, lead to a naked singularity. This offers counterexamples to the cosmic censorship hypothesis. A metric in the empty space which is continuous with the line element in a noncomoving coordinate system in the interior is also presented. Under suitable transformations, our spacetime also reduces to the higher dimensional analog of the Einstein-de Sitter model. It is further observed that a recollapsing inhomogeneous model may admit of spaces that are not closed (not finite in spatial extent).