STRUCTURE AND ENERGY OF SINGULAR RELATIVISTIC POLYTROPES

The thick mass-shell model with a two-dimensional singular inner surface introduced by de Felice and Yu (1993) to describe a static incompressible fluid arbitrarily close to a black-hole degeneracy, is extended here to sources with a polytropic equation of state. We find that the properties of the mass-shell solutions change considerably from small to large polytropic indices. The definition of gravitational energy given by Lynden-Bell and Katz (1985) is found to be applicable to our solutions and, in particular, we show that the remarkable implication of that definition, namely that 'all the energy remains outside the black hole' is consistent with the behaviour of the matter and field energies inside our shell configurations as they shrink to the black-hole limit.