Arbitrary concentrations of matter and the Schwarzschild singularity

Abstract

A nonstatic solution of the Einstein gravitational equations representing a spherically symmetric cluster of radially moving particles in an otherwise empty space is obtained. While it has been presumed by Einstein that the Schwarzschild singularity is physically unattainable as matter cannot be concentrated arbitrarily, the present solution seems to show that there is no theoretical limit to the degree of concentration, and that the Schwarzschild singularity has no physical reality as it occurs only in some particular coordinate systems. Incidentally, it is shown that in case of spherical symmetry the condition of conservation of gravitational energy of an isolated system of fluid material is equivalent to the vanishing of pressure at the boundary.