STRUCTURE AND ENERGY OF SINGULAR RELATIVISTIC POLYTROPES

The thick mass-shell model with a two-dimensional singular inner surfa ce introduced by de Felice and Yu (1993) to describe a static incompre ssible 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 g iven by Lynden-Bell and Katz (1985) is found to be applicable to our s olutions and, in particular, we show that the remarkable implication o f that definition, namely that 'all the energy remains outside the bla ck hole' is consistent with the behaviour of the matter and field ener gies inside our shell configurations as they shrink to the black-hole limit.