About Ric = 0

On Wednesday, 5th December 2007, I sent a copy of my paper On Certain Conceptual Anomalies in Einstein's Theory of Relativity to all those scientists who were requested to write a submission in relation to the funding of the Australian International Gravitational Observatory (AIGO). The aforesaid paper provides proofs that Ric = 0 does not fully describe Einstein's gravitational field, assuming that Ric = 0 is permissible, that the Theory of Relativity forbids point-masses, that Ric = 0 violates Einstein's 'Principle of Equivalence' and is therefore completely inadmissible, that what is routinely called ''Schwarzschild's solution'' is not Schwarzschild's solution, that the quantity 'r' appearing in the "Schwarzschild solution" is in fact the radius of curvature by virtue of its formal geometric relationship to the Gaussian curvature, and that Einstein's pseudo-tensor is a meaningless concoction of mathematical symbols.

On Thursday, 6th December 2007, I received email response from one Professor Andrzej Krasinski of the Nicolaus Copernicus Astronomical Center, Warsaw, Poland, (and who is a member of the 'International Committee on General Relativity and Gravitation', and has published under Cambridge Univesity Press), and from one Professor Mike Ford of the University of Technology, Sydney Australia. According to Krasinski, "Your mails will be automatically qualified as spam. Discussing with someone who refuses to understand is pointless." and Ford merely said "Take me off this mailing list." Go here for their emails.

First, notwithstanding being invited to make a submission in relation to the AIGO, and given at least 3 months to do so, Krasinski and Ford did not make a submission, but simply ignored the invitation to make a submission. Yet it is I who "refuses to understand" according to Krasinski. Furthermore, Krasinski and Ford did not offer any invitation to discussion in relation to the aforesaid paper and did not offer any arguments as to why I am allegedly wrong. Once again Krasinski simply ignored arguments that invalidate the precious theories of black holes and big bangs. Now I wonder just what it is in the aforesaid paper that I do not understand. In the case of the Theory of Relativity forbidding point-masses the argument is exceedingly simple. According to Special Relativity, infinite densities are forbidden because that would require an infinite energy, or equivalently that a material object can travel at the speed of light in vacuo. This is easily seen as follows. Consider a cuboid rest-mass m₀ of sides length x. Let it move with constant rectilinear velocity v in the x-direction. Its mass is given by

and its volume is given by

So the density D of the moving mass is

D = mo x3 (1 - v2/c2)

which is infinite when v = c: but this is forbidden by Special Relativity since no material object can travel at the speed of light in vacuo. So infinite densities are forbidden by Special Relativity. Now the so-called "point-mass" has a finite mass and a zero volume, so that it is infinitely dense, which is what the singularity of the alleged black hole is supposed to be. Thus, if General Relativity permits point-masses it does so in violation of Special Relativity. Yet General Relativity is supposed to be a generalisation of Special Relativity to non-uniform motion. It cannot therefore violate Special Relativity. So if General Relativity is to be consistent with Special Relativity, it cannot permit point-masses, howsoever they are alleged to be formed, despite what Prof. Mr. Krasinski et al. might otherwise and vagariously claim.

Now what is it that I don't understand about R_μν = 0, by my claiming that it violates Einstein's 'Principle of Equivalence'? According to Einstein, in a freely falling inertial frame the laws of Special Relativity must apply. Now in Special Relativity there are masses, arbitrarily large but not infinitely large, so the freely falling frame must be able to admit the presence of masses; otherwise Special Relativity cannot be recovered. But R_μν = 0 forbids, by definition, the presence of mass and energy in the alleged gravitational field in which the said frame is falling. Thus, the laws of Special Relativity cannot be recovered in the spacetime of R_μν = 0. Contra hype! So R_μν = 0 violates Einstein's 'Principle of Equivalence' and so the spacetime of R_μν = 0 is not a dynamical system at all, only a system of kinematics, i.e. it is only a generalisation of Minkowski spacetime into a pseudo-Riemannian metric manifold: a pure geometry. Minkowski spacetime is not Special Relativity. Special Relativity only takes place in Minkowski spacetime. R_μν = 0 does not describe Einstein's gravitational field at all - it does not generalise Special Relativity. Another way of looking at this is to consider the Standard Model derivation of the alleged gravitational field for R_μν = 0. It begins with the usual spherical-polar coordinate line-element for Minkowski spacetime (using c = 1), given by

Note that there is no appearance of matter or energy in this expression, i.e. Special Relativity is not included. Then by generalising this expression subject to Einstein's equations R_μν = 0, the so-called "Schwarzschild solution" is obtained. But R_μν = 0 is due to the energy-momentum tensor being set to zero, which means that there is no mass or energy in the alleged gravitational field outside the supposed source of that field. So Minkowski spacetime, containing no matter and no energy, is transformed into a pseudo-Riemannian spacetime in which there are no masses and no energy. Thus, there is no transformation of the dynamics of Special Relativity and so Special Relativity cannot be recovered in a "freely falling" inertial frame in the spacetime of R_μν = 0. Thus, R_μν = 0 violates Einstein's 'Principle of Equivalence'. Further details can be obtained in my aforementioned paper.

What about my claim that "Schwarzschild's solution" is not Schwarzschild's solution? That is easily settled, by reading Schwarzschild's paper. Here is the so-called "Schwarzschild solution" (c = G = 1):

wherein r can go down to zero, one way or another, and m is the alleged mass of the source of the field. But here is Schwarzschild's real solution:

wherein Schwarzschild stated that the constant α is to be physically interpreted as a function of the mass of the source of the alleged gravitational field (based on the initial assumption however, that R_μν = 0 actually describes an Einstein gravitational field). He did not deduce the value of α because it cannot be done without introducing ad hoc arguments. In particular, Schwarzschild did not set α = 2m, and did not call it a radius. That was done ad hoc by the Standard Model relativists in order to fudge a Newtonian relationship, in the erroneous belief that since they have deduced a line-element for an Einstein gravitational field, there must be a source of that field, and so they insert the Newtonian potential to get it, ad hoc, not realising that they have merely inserted it and have inserted it as a centre of mass, so that it is not even in their alleged field (their line-element is undefined at r = 2m, and a centre of mass is not a physical object). Their introduced value 2m the Standard Modellers call the "Schwarzschild radius" of their black hole (.e. the "radius" of their alleged "event horizon"), and further claim that r can go down to zero, some way or another, in their line-element, down to an infinitely dense point-mass which they call the singularity of their black hole (where their line-element is again undefined), in violation of Special Relativity into the bargain, failing to realise that their point-mass actually occurs at their r = 2m, due to their ignorance of the mathematical fact that r is not even the geodesic radial distance from the centre of spherical symmetry of their line-element, but is only a "radius of curvature" by virtue of its formal geometric relationship to the Gaussian curvature (amplified in the next paragraph herein). Then, to get this black hole "singularity", they introduce the utterly nonsensical Kruskal-Szekeres "coordinates" to effectively drive r down to zero in their line-element. Reintroducing the usual values for c (speed of light in vacuo) and G (the Newtonian gravitational constant), the alleged "Schwarzschild radius" is 2Gm/c², which describes the radius of the hypothetical Michell-Laplace Dark Body, a purely Newtonian concept, for which the escape velocity is the speed of light in vacuo. The Standard Modellers make great fanfare of their claim that their alleged solution for the "gravitational field" for R_μν = 0 obtains the radius of the Michell-Laplace Dark Body (their "Schwarzschild radius" for their black hole's "event horizon"). That is not surprising, since they actually inserted it, ad hoc, into their line-element in the first place - it is not and never was a deduction, because it is a fudge to get Newton. Furthermore, the Standard Modellers claim that although the Michell-Laplace Dark Body has an escape velocity, they also claim that in the case of their black hole nothing at all (including light) can even leave their alleged "event horizon", let alone escape. Thus, since nothing can leave or escape their black hole, it has no escape velocity. Nothwithstanding the foregoing, the constant α cannot be physically interpreted as a function of the mass of the source of the gravitational field because R_μν = 0 violates Einstein's 'Principle of Equivalence' and so does not describe Einstein's gravitational field to begin with. It is merely a mathematical constant, the value of which moderates the purely geometric characteristics of a pseudo-Riemannian metric manifold - a pure geometry - and when α is zero Minkowski spacetime is recovered - also a pure geometry. So how, Mr. Krasinski et al., have I not "understood"? One only needs to read Schwarzschild to see what is what. No, I have reported accurately. Most Standard Model relativists haven't even read Schwarzschild, and when given access to Schwarzschild's paper evidently choose not to read it, just like Galileo's detractors who refused to even look at the heavens through his telescope, clinging instead to their fantasies.

The Standard Model relativists have never rightly identified the variable 'r' occuring in the "Schwarzschild solution". It is in fact the radius of curvature by virtue of its formal geometric relationship to the Guassian curvature. It does not determine distance from the centre of spherical symmetry. The geodesic radial distance from the centre of spherical symmetry to the geodesic spherical surface of some given value of 'r' is given by the integral of the square root of the negative of the second term on the right side of the "Schwarzschild solution" (i.e. by the integral of (1 - 2m/r)^-1/2dr; and in the case of Schwarzschild's actual solution by the integral of (1 - α/R)^-1/2dR, where R = R(r) as determined by Schwarzschild) and so it is not the same as 'r' in general. Now I provided another simple proof of this fact in my aforementioned paper, by application of elementary differential geometry. In this case, for the spherically symmetric surface r²(dθ² + sin²θdφ²), as any book on differential geometry will confirm, the Gaussian curvature K is given by

wherein R_αβγδ is the Riemann tensor of the 1st kind, and g = g_θθg_φφ (because the meric tensor is diagonal). Simple calculations give:

for the "Schwarzschild solution". Thus, 'r' in the "Schwarzschild solution" is the inverse square root of the Gaussian curvature, i.e. the radius of curvature, as I have always claimed. Not one Standard Model relativist has ever realised that 'r' in the "Schwarzschild solution" is the radius of curvature. So once again, who is it that that has not "understood", Mr. Karinski et al.?

What now of my claims concerning the falsity of Einstein's conceptions of the conservation and localisation of gravitational energy and of gravitational waves, Prof. Mr. Karinski et al.? Einstein's field equations are given by,

According to the claims of the proponents of the Standard Model, if the energy-momentum tensor T_μν is zero, then the equations R_μν = 0 result (since the Ricci curvature R becomes zero also). However, since R_μν = 0 is inadmissible, because it violates Einstein's 'Principle of Equivalence', the energy-momentum tensor can never be zero for Einstein's gravitational field. Therefore, one can rewrite Einstein's field equations as

wherein the G_μν/κ are the components of a gravitational energy tensor. Thus, when T_μν = 0, G_μν = 0, i.e. they vanish identically - there is no gravitational field. This is an inescapable consequence of the inadmissibility of R_μν = 0. Consequently, the total energy is always zero and there is no localisation of gravitational energy and hence no Einstein gravitational waves. Once again, Prof. Mr. Karinski et al., just what is it that I have refused to understand? It is impossible to know, since you have not told me what it is that I have refused to understand.

I turn now to Einstein's pseudo-tensor. Evidently that I understand a mathematical proof to mean a proof is a misunderstanding. No matter what Mr. Krasinski et al. have to say, the fact remains that Einstein's pseudo-tensor is a meaningless concoction of mathematical symbols and cannot therefore be used for anything, because it implies the existence of a 1st-order, intrinsic differential invariant that depends only upon the components of the metric tensor and their 1st derivatives. But the mathematicians Ricci and Levi-Civita proved in 1900 that such invariants do not exist. That minor detail has not stopped the Standard Modellers from using it to describe the flow of energy and momentum and the localisation of gravitational energy, or from claiming that I don't or won't "understand". The detailed proof is given in my abovementioned paper and again here (by Levi-Civita himself). Consequently, Einstein's conceptions of the conservation and localisation of gravitational energy, and also of gravitational waves, are also false. Evidently Prof. Mr. Krasinski et al. do not understand, or refuse to accept, that a mathematical proof is a proof; a very definite proof.

The reader can obtain more detail in my aforementioned paper. He will assuredly see that I have not misunderstood or that I refuse to understand. On the contrary, it is Mr. Krasinski and the Standard Modellers, so irrationally attached to their moronic black holes and big bangs (and of course their cosy jobs), who have not understood and who have routinely demonstrated a refusal to look at the facts ("none are so blind as those who will not see"). Superstition, magic and ineptitude are much easier than science - that is why there are so many Standard Model relativists, and they proliferate like coat-hangers, lurking about furtively in the darkness of closets, ever ready to squirt their snake oil at the unwary and the gullible.

It should also be noted that the signatures of the alleged black hole are (1) an infinitely dense singularity (point-mass), and (2) an event horizon. Almost daily the astronomers and astrophysical relativists claim discovery of yet another black hole here or there. However, if you ask them for the coordinates of just one infinitely dense singularity (point-mass) or the coordinates of just one event horizon, for any of the many hundreds of black holes they allege to have found, you will get no set of coordinates, because nobody has ever found the tell-tale signatures of a black hole - no singularity and no event horizon. Nobody has ever found a black hole. The claims for black holes being found are patently false. In addition, there are no known solutions to Einstein's field equations for the interaction of two or more comparable bodies. It is not even known if his field equations admit of solutions for such configurations of matter since no existence theorem has ever been proven. Before one can talk about two or more comparable bodies interacting gravitationally according to General Relativity, one must first obtain an appropriate energy-momentum tensor, and then solve Einstein's field equations for it. This has never been done (nobody has even been able to define an appropriate energy-momentum tensor) and so General Relativity has to date been unable to account for the simple observational fact that two bodies, initially held fixed, will attract one another upon release (e.g. the experiments of Cavendish).

I received notification from an interested 3rd party of his exchange of email with Mr. Krasinski. The latter advised the former that he should reject my scientific arguments for the following reasons:

1. I refuse to understand;
2. I fiendishly infected Mr. Krasinski's computer with destructive viruses that are hidden away in the code for my webpages (this one too!), to extract revenge upon him (and presumably others);
3. I am a maniac.

That is the sum total of his "reasoning" - a "professor" of physics? Go here for the email exchange.

Here is an example of how physics is done at MIT. On 8th December 2007 I emailed Professor Edwin F. Taylor, at MIT. He is the coauthor of a book entitled "Exploring Black Holes". The first edition he wrote with Professor John A. Wheeler. The second edition is with Professor Edmund Bertschinger, also at MIT. In preparation of the second edition, Taylor invited comment. So I did so, on 8th December 2007, and sent him a copy of my abovementioned paper. On the 13th December 2007, Taylor emailed me, after he had discussions with Bertschinger. He and Bertschinger could offer no objections to the matters treated in my paper. He did say:

Nothwithstanding their appreciation of the consequences for their book and their physics, Taylor stated that neither he nor Bertschinger would enter into any discussion, and would publish their book anyway. Taylor said:

So Taylor and Bertschinger have deliberately ignored the facts for their own covenience and that of the black holers and big bangers at large. You can read the whole correspondence here. They refused to state their considered position on the matters raised in my abovementioned paper. Evidently physics is done at MIT by ignoring anything and everything that invalidates the (preposterous) claims of its professors.

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My email address: thenarmis@gmail.com.

Page established: 2nd December 2007.

Updated: 24th May 2009

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