David Hilbert and the origin of the "Schwarzschild solution"

TL;DR

This paper examines the historical development of the Schwarzschild solution, highlighting differences between Schwarzschild's original manifold and Hilbert's version, and discusses their implications for understanding black hole singularities.

Contribution

It provides a detailed analysis of the historical and mathematical differences between Schwarzschild's and Hilbert's solutions, emphasizing the topological and singularity issues.

Findings

Hilbert's manifold has a non-physical arrow of time

Hilbert's solution admits an intrinsic singularity in the interior

Extensions like Kruskal-Szekeres address the arrow of time issue

Abstract

The very early dismissal of Schwarzschild's original solution and manifold, and the rise, under Schwarzschild's name, of the inequivalent solution and manifold found instead by Hilbert, are scrutinised and commented upon, in the light of the subsequent occurrences. It is reminded that Hilbert's manifold suffers from two defects, that are absent in Schwarzschild's manifold. It does not admit a consistent drawing of the arrow of time, and it allows for an invariant, local, intrinsic singularity in its interior. The former defect is remedied by the change of topology of the extensions proposed by Synge, Kruskal and Szekeres. The latter persists unaffected in the extensions, since it is of local character.