The Black-Hole Limits of the Spherically Symmetric and Static Relativistic Polytrope Solutions

We examine the black-hole limits of the family of static and spherically
symmetric exact 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 radial
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 nature of the solutions in the black-hole limits leads to the definition of a new type of singularity in General Relativity. The analysis of the interior geometry leads to a very surprising connection with
quantum-mechanical studies of naked Schwarzschild black holes.
  
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.