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.

我们研究了多方物质的爱因斯坦场方程的静态和球对称解族的黑洞极限，这在之前的论文中已经介绍过。对于多方指数 n 的几个值，这种探索是在该解族的参数平面的允许区域的渐近子区域中完成的，限制为 \（n>1\）。这些允许的区域在之前的另一篇论文中进行了一些详细的确定和讨论。检查和分析了这些限制的特性。我们发现有不同类型的黑洞极限，其特定特征涉及物质的局部温度。我们还发现，这些极限在黑洞内部产生了一种非常出乎意料但特殊的时空几何类型，我们对此进行了详细分析。关于内部几何的空间部分，我们表明在黑洞极限中，所有空间距离普遍坍缩为零。关于时间部分，相对于径向无穷大处的平坦空间，在整个内部区域上，极限会产生无限的整体红移。对内部几何学的分析导致了与裸露的史瓦西黑洞背景度量中的量子力学研究非常令人惊讶的联系。黑洞极限中解的性质导致了广义相对论中一种新型奇点的定义。我们认为，黑洞极限实际上不能一直得出最终结论，因为这会导致违反一些基本的物理和数学条件。 这些包括解的一致性问题、涉及无限能量的问题以及涉及违反物质量子行为的问题。然而，从物理学的角度来看，人们仍然可以在很大程度上处理这些限制情况，因此至少出于某些目的，这些限制仍然可以被视为具有相当极端属性的物理可实现配置的有用且简单的近似表示。