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 +) = α2, where α = 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(r)]1/2, which transforms the coordinate location of the point mass to [Formula: see text]. For Hilbert, however, C was unknown, and thus could not be used to determine [Formula: see text]. Instead he asserted, in effect, that r* = (x2 + y2 + z2)1/2, which places the point mass at r* = 0. Unfortunately, this differs from the value (α) obtained by substituting Schwarzschild's C into the expression for [Formula: see text], and since C(0 +) is a scalar 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* = α) 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.