The black-hole limits of the spherically symmetric and static relativistic polytrope solutions

Abstract

We examine the black-hole limits of the family of static and spherically symmetric solutions of the Einstein field equations for polytropic matter, that was presented in a previous paper. This exploration is done in the asymptotic sub-regions of the allowed regions of the parameter planes of that family of solutions, for a few values of the polytropic index n, with the limitation that \(n>1\). These allowed regions were determined and discussed in some detail in another previous paper. The characteristics of these limits are examined and analyzed. We find that there are different types of black-hole limits, with specific characteristics involving the local temperature of the matter. We also find that the limits produce a very unexpected but specific type of spacetime geometry in the interior of the black holes, which we analyze in detail. Regarding the spatial part of the interior geometry, we show that in the black-hole limits there is a general collapse of all spatial distances to zero. Regarding the temporal part, there results an infinite overall red shift in the limits, with respect to the flat space at radial infinity, over the whole interior region. The analysis of the interior geometry leads to a very surprising connection with quantum-mechanical studies in the background metric of a naked Schwarzschild black hole. The nature of the solutions in the black-hole limits leads to the definition of a new type of singularity in General Relativity. We argue that the black-hole limits cannot actually be taken all the way to their ultimate conclusion, due to the fact that this would lead to the violation of some essential physical and mathematical conditions. These include questions of consistency of the solutions, questions involving infinite energies, and questions involving violations of the quantum behavior of matter. However, one can still approach these limiting situations to a very significant degree, from the physical standpoint, so that the limits can still be considered, at least for some purposes, as useful and simpler approximate representations of physically realizable configurations with rather extreme properties.