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Summary by Stephen Crothers

Not only is the Newtonian mass introduced ad hoc and a posteriori into Hilbert's corruption (the so-called "Schwarzschild solution") of Schwarzschild's true solution, the Newtonain potential, so introduced by legerdemain, contains the quantity 'r' it its denominator. In Newton's theory "r" denotes the radial distance from the centre of mass, but in the "Schwarzschild solution" the quantity 'r' is not a distance of any kind in the metric manifold - it is in fact the inverse square root of the Gaussian curvature of the spatial section, and only in that sense is it a radius - the radius of Gaussian curvature. Thus, the quantity 'r' in the "Schwarzschild solution" determines the Gaussian curvature at any point, because that space has a non-Euclidean geometry - it does not denote any distance at all in the manifold. In Newton's space, the radius of Gaussian curvature of a sphere and the radius of that sphere are identical (because Newton's space is Euclidean). This is not true in Einstein's non-Euclidean (his pseudo-Riemannian) geometry. One only need to recall that some 2-surfaces embedded in a 3-space have constant negative Gaussian curvatures, e.g. the tractroid, and so the radius of Gaussian curvature is a complex number, but the magnitude of the radial vector to the 2-surface from the origin of coordinates of the 3-space it is embedded in is real and non-negative.  Moreover, in a 2-surface, the usual notion of radius is meaningless because the surface is a manifold on its own. Gaussian curvature is intrinsic to any such surface (Gauss' Theorema Egregium).
The proponents of the black hole are entirely ignorant of Gaussian curvature. In addition, in their "Schwarzschild solution" they assert that only g_00 and g_11 are modified by the presence of their "matter", 'm'. They claim that neither g_22 nor g_33 are affected by their "matter", 'm'. That is nonsense too. All the components of the metric tensor for a Schwarzschild space are modified by the constant alpha. And since Ric = 0 is devoid of matter, this constant alpha is responsible for the curvature of the manifold, making a pure geometry, a system of kinematics only, not a system of dynamics, as mentioned by Prof. Evans. One can only wonder at the claims of the black holers for a gravitational field in the absence of matter. And even if their "m" was admissible (but it isn't) as a source of their alleged gravitational field, it is a lone mass in an otherwise empty Universe (since Ric = 0). There is, I submit, no meaning to such a gravitational field either, because it can act on nothing physical because there is no other matter or electromagnetic radiation in the manifold. Gravitation is an interactive phenomenon. One needs at least two gravitationally coupled physical entities to describe it, in principle. There are no known solutions to the EH field equations for two or more bodies, and there is no existence theorem by which the EH field equations might contain latent solutions for such configurations of matter. In order to rightly obtain a Newtonian approximation one must first find a solution to the EH field equations for the interaction of two bodies and then show that in a very weak field this solution reduces to the Newtonain field. Nobody even knows how to write the energy-momentum tensor for the interaction of two discrete pieces of matter (let alone more than two discrete bodies), and simply guessing a relevant metric is not very likely (in which case the energy-momentum tensor would then have to be calculated from that guessed metric and examined for physical validity).

The "Schwarzschild solution" is asymptotically Minkowski spacetime, which is a pure geometry into which matter is assumed to be able to be inserted, just as matter is assumed to be able to be inserted into Newton's space, a priori. However, Ric = 0 is a definite statement that matter is absent in the asymptotic Minkowski space. In other words, Schwarzschild spacetime is not asymtotically Special Relativity, only asymptotically Minkowski spacetime, and it is not asymtotically Newtonian either. Matter cannot suddenly appear asymptotically in the manifold of Ric = 0.

Steve Crothers
Posted: 1 May 2008 


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