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

TL;DR

This paper rigorously analyzes the differences between Schwarzschild and Hilbert-Droste solutions of Einstein's equations, clarifies their topological distinctions, and details the maximal extension of the Hilbert-Droste solution with historical context.

Contribution

It provides a rigorous mathematical comparison of Schwarzschild and Hilbert-Droste solutions, establishing their topological differences and deriving the maximal extension with detailed topology analysis.

Findings

Schwarzschild and Hilbert-Droste solutions have different manifold topologies.

The maximal extension of the Hilbert-Droste solution (Kruskal-Szekeres) is explicitly derived.

The embedding of the Hilbert-Droste solution in a vector manifold is clarified.

Abstract

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.