Physics
Scientific paper
Sep 1989
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989cajph..67..919a&link_type=abstract
Can. J. Phys., Vol. 67, No. 9, p. 919 - 926
Physics
13
Black Holes
Scientific paper
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 r0* = [C(0+)]1/2. For Hilbert, however, C was unknown, and thus could not be used to determine r0*. 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 r0*, and since C(0+) is a scalar invariant, it follows that Hilbert's assertion is invalid. Owing to this erorr, 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.
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