Black Holes: The Legacy of Hilbert's 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.