A general relativistic space-time universe is considered together with a radiation such as the microwave background radiation. It is shown that if certain reasonable conditions are satisfied, then the presence of such a radiation would imply space-time singularities in the sense that all time-like curves will be incomplete in the past. The considerations provide an upperbound to the age of the universe, which is consistent with present data.

The usual definition of a black hole is modified to make it applicable in a globally hyperbolic space-time. It is shown that in a closed globally hyperbolic universe the surface area of a black hole must eventually decrease. The implications of this breakdown of the black hole area theorem are discussed in the context of thermodynamics and cosmology. A modified definition of surface gravity is also given for non-stationary universes. The limitations of these concepts are illustrated by the explicit example of the Kerr-Vaidya metric.