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 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.