Necessity of Singularities in the Solution of the Field Equations of General Relativity

The Friedmann solution of the field equations of general relativity predict the expansion of the universe from a singular instant in time. This paper considers the behavior of universes which are less symmetric than the Friedmann model, and which have more general fields that carry stress and energy and produce gravitation. We treat the nonsymmetric problem as one treats the symmetric problem, via a "co-moving coordinate system" such that in it free infinitesimal test particles once at rest remain at rest. In such a coordinate system and with standard assumptions about the stress energy tensor we establish that the solutions of the field equations of the general theory of relativity necessarily have singularities at finite time. The considerations are independent of the symmetry, topology or boundary conditions assumed for the space-like three-dimensional hypersurfaces.