标题： 重新审视爱因斯坦方程的施瓦茨希尔德解和希尔伯特-多斯特解以及后者的最大延伸 显示英文标题

标题： Revisiting the Schwarzschild and the Hilbert-Droste Solutions of Einstein Equation and the Maximal Extension of the Latter

摘要： 在本教学笔记中，通过基于流形的扭曲积概念的严格数学方法，仔细研究了爱因斯坦方程的施瓦茨希尔德解和希尔伯特-德罗斯特解之间的差异。将证明这些解确实是不同的，因为与它们对应的流形的拓扑结构是不同的。在无可置疑地确立这一事实之后，详细推导了希尔伯特-德罗斯特解（克鲁斯卡尔-泽克雷斯时空）的最大扩展，并将其拓扑结构与其施瓦茨希尔德解和希尔伯特-德罗斯特解的拓扑结构进行比较。我们还研究了希尔伯特-德罗斯特解在向量流形中的嵌入问题，希望澄清卡斯纳和弗龙斯达尔在该主题上的工作。在附录中，我们对爱因斯坦-罗森桥进行了严格的讨论。最后给出了涉及我们工作的历史论文的全面参考文献列表。 显示英文摘要

摘要： In this pedagogical note, the differences between the Schwarzschild and the Hilbert-Droste solutions of Einstein equation are scrutinized through a rigorous mathematical approach, based on the idea of warped product of manifolds. It will be shown that those solutions are indeed different because the topologies of the manifolds corresponding to them are different. After establishing this fact beyond any doubt, the maximal extension of the Hilbert-Droste solution (the Kruskal-Szekeres spacetime) is derived with details and its topology compared with the ones of the Schwazschild and the Hilbert-Droste solution. We also study the problem of the imbedding of the Hilbert-Droste solution in a vector manifold, hopefully clarifying the work of Kasner and Fronsdal on the subject. In an Appendix, we present a rigorous discussion of the Einstein-Rosen Bridge. A comprehensive bibliography of the historical papers involved in our work is given at the end.