Black Holes: The Legacy of Hilbert's Error

TL;DR

The paper argues that Hilbert's derivation of the Schwarzschild solution contains an error, which leads to the incorrect interpretation of the black hole as an artifact of this mistake, challenging the traditional understanding of black holes.

Contribution

It identifies a fundamental error in Hilbert's derivation of the Schwarzschild metric, impacting the conceptual foundation of black hole solutions.

Findings

Hilbert's assertion about the coordinate r* is invalid due to a misinterpretation of the invariant C(0+)

This error causes the boundary at r* = a to be a two-sphere instead of a point

The maximal extension of the space-time contains a black hole as an artifact of this error

Abstract

The historical postulates for the point mass are shown to be satisfied by an infinity of space-times, differing as to the limiting acceleration of a radially approaching test particle. Taking this limit to be infinite gives Schwarzschild's result, which for a point mass at x = y = z = 0 has C(0+) = a^2, where a = 2m and C(r) denotes the coefficient of the angular terms in the polar form of the metric. Hilbert's derivation used the variable r* =[C(0+)]^1/2. For Hilbert, however, C was unknown, and thus could not be used to determine r*(0). Instead he asserted, in effect, that r* = (x^2 + y^2 + z^2)^1/2, which places the point mass at r* = 0. Unfortunately, this differs from the value (a) obtained by substituting Schwarzschild's C into the expression for r*(0), and since C(0+) is a scaler invariant, it follows that Hilbert's assertion is invalid. Owing to this error, in each spatial section of Hilbert's space-time, the boundary (r* = a) corresponding to r = 0 is no longer a point, but a two-sphere. This renders his space-time analytically extendible, and as shown by Kruskal and Fronsdal, its maximal extension contains a black hole. Thus the Kruskal-Fronsdal black hole is merely an artifact of Hilbert's error.