Entropic principles

We discuss the evolution of radiation and Bekenstein-Hawking entropies in e xpanding isotropic universes. We establish a general relation which shows w hy it is inevitable that there is currently a huge difference in the numeri cal values of these two entropies. Some anthropic constraints on their valu es are given and other aspects of the cosmological 'entropy gap' problem ar e discussed. The coincidence of the classical and quantum entropies for bla ck holes with Hawking lifetime equal to the age of the universe, and hence of radius equal to the proton size, is shown to be identical to the conditi on that we observe the universe at the main sequence lifetime. (C) 1999 Els evier Science B.V. All rights reserved.