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Statement by Stepher Crothers

Dear Prof. Evans,

 It is an interesting coincidence that you have made these remarks about Ricci = 0. Only this morning I completed a draft of a paper I am working on, dealing with conceptual anomalies in the Standard interpretation of General Relativity. Here is a rough extract of the part in which I discuss Ricci = 0:-

 I believe that Ricci = 0 violates Einstein’s Principle of Equivalence, and so writing Ricci = 0 is erroneous in the first place. It is, I feel, inconsistent with the physical foundations of GR as adduced by Einstein. The motive to writing Ricci = 0 is due again to conceptual anomaly. First, as I have argued previously, Ricci = 0 does not generalise Special Relativity but only Minkowski space. That is, Ricci = 0 generalises the pseudo-Efcleethean geometry of Minkowski space into a pseudo-Riemannian geometry. Since Ricci = 0 imposes the centre of mass configuration on the perceived source of the field, the source of the field is not in the field (the line-element is undefined at the centre of mass). Since Ricci = 0 excludes by definition all masses and energy, the resulting curvature of spacetime has only kinematic properties. One cannot say that a material object follows a timelike geodesic in the field of Ricci = 0 because one cannot introduce any material object into that field. One cannot say that light follows a null geodesic in the field of Ricci = 0 because one cannot introduce energy into the field of Ricci = 0, and photons carry energy (if not also mass). One can only say that points travelling at the speed c of light in vacuo, in the spacetime of Ricci = 0, follow a null geodesic and one can only say that other points that move with a speed less than c follow timelike geodesics and that no points can move along a spacelike path. Time dilation and space contraction are kinematic effects of Minkowski space, which is a geometry in which points cannot move with a speed greater than c, by definition. The physical nature of light does not play a part in Minkowski geometry. The dynamics of Special Relativity are assumed to take place in Minkowski space, just as Newton’s dynamics are assumed to take place in Efcleethean 3-Space. Thus, it is assumed that masses can simply be inserted into Minkowski space, just as masses are assumed to be able to be inserted into Efcleethean 3-space for Newton’s dynamics. (This is not the case in General Relativity, wherein mass, energy and spacetime interact, acting one upon the other.) Then with the assumption that masses can be inserted into Minkowski space, the dynamics of Special Relativity are developed, subject to the kinematic nature of Minkowski space with its limitation on the upper speed of a point therein, and with the assignation of a point moving with speed c to a photon. The dynamics of Special Relativity are the result of the kinematics of Minkowski space (i.e. the mere geometry thereof) imposed upon masses inserted into Minkowski space and attached to moving points so that the distinction between point and mass is lost by subsuming mass into a centre of mass (a mathematical point). On the Principle of Equivalence, according to Einstein,

 "Let now K be an inertial system. Masses which are sufficiently far from each other and from other bodies are then, with respect to K, free from acceleration. We shall also refer these masses to a system of co-ordinates K’,uniformly accelerated with respect to K. Relatively to K’ all the masses have equal and parallel accelerations; with respect to K’ they behave just as if a gravitational field were present and K’ were accelerated. Overlooking for the present the question as to the ‘cause’ of such a gravitational field, which will occupy us later, there is nothing to prevent our conceiving this gravitational field as real, that is, the conception that K’ is ‘at rest’ and a gravitational field is present we may consider as equivalent to the conception that only K is an ‘allowable’ system of co-ordinates and no gravitational field is present. The assumption of the complete physical equivalence of the systems of co-ordinates, K and K’, we call the ‘principle of equivalence’; this principle is evidently intimately connected with the law of the equality between the inert and the gravitational mass, and signifies an extension of the principle of relativity to co-ordinate systems which are in non-uniform motion relatively to each other. In fact, through this conception we arrive at the unity of the nature of inertia and gravitation."

 (The Meaning of Relativity, chapter: The General Theory of Relativity, p. 56)

 Also, according to Einstein,

 "Stated more exactly, there are finite regions, where, with respect to a suitably chosen space of reference, material particles move freely without acceleration, and in which the laws of special relativity, which have been developed above, hold with remarkable accuracy."

 (The Meaning of Relativity, chapter: The General Theory of Relativity, p. 56.)

 However, Ricci = 0 does not generalise Special Relativity, only the geometry of Minkowski space. The source of the field, as a centre of mass, is not in the field of Ricci = 0. No masses or energy can be arbitrarily inserted into the spacetime of Ricci = 0. Thus, Ricci = 0 violates Einstein’s Principle of Equivalence. Furthermore, as pointed put in my papers, one cannot assign the value of the constant appearing in the Schwarzschild line-element to the Newtonian potential in the infinitely far field because Schwarzschild space is asymptotically Minkowski space, not asymptotically Special Relativity and not asymptotically Newtonian dynamics. And in Newton’s theory, the potential is defined as the work per unit mass, on a mass that can, in principle, be inserted into the gravitational field of another mass. One cannot insert any masses, by definition, into the field of Ricci = 0. The infinitely far field of Ricci = 0 does not become Newtonian – it becomes Minkowski space only. Newton’s law of gravitation is based a priori on the interaction of two masses; Einstein’s theory of gravitation is not. The claim that the constant in the Schwarzschild solution can be associated with the infinitely far field Newtonian potential was never made by Schwarzschild, because he clearly knew this cannot be done. He only stated in his 1st paper on the subject that the constant was to be physically interpreted as some function of the mass. That function cannot be ascertained from the line-element for Ricci = 0. The value of the constant was determined by Schwarzschild in his 2nd paper, on the sphere of homogeneous incompressible fluid. In that paper it is obtained that the constant is determined from the interior line-element, where the energy-momentum tensor is not zero, not from the field for Ricci = 0, and with it the fact that there are two non-Newtonian masses, the active and the passive mass respectively, both from the interior line-element.

 With a line-element for Ricci = 0 alone, one can only rightly say that the geometry is modified from that of Minkowski space, by the presence of a non-zero constant. When that constant is zero, Minkowski space is recovered, and with that recovery of Minkowski space, one can again arbitrarily insert masses and energies and develop the dynamics of Special Relativity. It does not follow, that with the setting of the constant to zero, that the pseudo-Riemannian metric manifold of Ricci = 0 collapses into Special Relativity. Special Relativity is merely an augmentation to Minkowski space by the arbitrary insertion of mass and energy into Minkowski space with the constrained kinematic features of Minkowski space applied to those masses and energies. The collapse of Ricci = 0 into Minkowski space takes with it only a geometry and hence only a system of kinematics, not a system of dynamics.

 Still, the writing of Ricci = 0 outside the source is erroneous, even though in the footsteps of Einstein, who claimed Ricci = 0 for a mass island. Schwarzschild only did as I have done - taken Einstein at his word. However, in writing Ricci = 0, Einstein has violated his own theory, by violating his Principle of Equivalence.

 This does not invalidate the detailed analysis by Schwarzschild, Abrams, Brillouin, and myself, since that work is based upon the implication, if Ricci = 0 outside the source of the field then certain things follow (but no black holes are possible). The validity of Ricci = 0 is entirely another question. Now, since Ricci = 0 violates Einstein’s Principle of Equivalence, it is erroneous. This invalidates the black hole from an even deeper level.

Yours faithfully,

Steve Crothers.


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