About Ric = 0


Stephen J. Crothers

The Death of Julieka Dhu in Police Custody

The crux of the matter is this:

(1) According to Einstein his field equations couple his gravitational field to its material sources and his energy-momentum tensor describes the material sources. According to Einstein, everything that is not the gravitational field is matter.

(2) Einstein and his followers assert that when energy-momentum tensor = 0 there is present an arbitrary number of material sources (stars, galaxies, black holes, electric fields, magnetic fields, etc). An example is the so-called 'Schwarzschild solution'.

(3) Einstein and his followers assert that when energy-momentum tensor = 0 there are no material sources present whatsoever (de Sitter's empty universe).

(4) Claims (1) and (2) are contradictory. Claims (2) and (3) are contradictory. For (1) and (3) to be consistent, energy-momentum tensor = 0 means there are no material sources present, by mathematical construction. Indeed, that is the meaning gleaned from (1) and the very expression energy-momentum tensor = 0. Claim (2) is not consistent with (1) and (3). In view of (1) and (3), (2) is false. The black hole comes from (2). The black hole is therefore false.

Nothing more is required to invalidate the black hole. There is no need to investigate the mathematical complexities of metric spaces. There is no need for any mathematics. Observations can't be interpreted in terms of a 'theoretical' entity that has no basis in any theory. All claims for the detection of Relativity's black holes are false.

ENSTEIN'S 'FIELD EQUATIONS' SIMPLIFIED (especially for the non-mathematical reader)

I shall now elaborate on the above proof, without any complicated mathematics.

All other types of alleged black hole rely on the 'Schwarzschild' black hole because they all subsume it, and are therefore also invalid.

First and foremost, Einstein's field equations,

"couple the gravitational field (contained in the curvature of spacetime) with its sources." (Foster and Nightingale 1995)

This means that there is a causal connexion between the material sources of Einstein's gravitational field and his spacetime curvature (spacetime geometry). Let's take a quick look at Einsteinís field equations without the so-called 'Cosmological Constant', thus,

Gμν = -κTμν

This is an intimidating and uninformative expression. We shall therefore reduce it to simple words, thus,

Einstein tensor = -κ(energy-momentum tensor)

So the Gμν thing on the left side of Einstein's field equations is called the Einstein tensor and the Tμν thing on the right side is called the energy-momentum tensor. The symbol κ is just a coupling constant. This identifies the parts of Einstein's field equations, but doesn't convey any meaning, so we will write them more informatively as,

spacetime geometry = -κ(material sources)

According to Einstein if there are no material sources present (material sources = 0) his field equations reduce to,

Ric = Rμν = 0

So the Rμν thing on the left side of this set of equations is called the Ricci tensor (Ric for short). This identifies the parts of Einstein's field equations but doesn't convey any meaning, so we will write them more informatively as,

spacetime geometry = 0

There are no material sources present in this expression. Einstein (1916) said that this expression describes "the field equations of gravitation in the absence of matter". That's true, because material sources = 0 by mathematical construction. Ric = 0 contains no matter. Einstein first mathematically removes all material sources from his field equations by setting material sources = 0 (i.e. energy-momentum tensor = 0), and then, in his very next breath, linguistically reinstates the presence of a material source by saying that spacetime geometry = 0 describes his gravitational field "outside" a body such as a star. Let's see what Einstein said about Hilbert's solution for Ric = 0:

"M denotes the sun's mass, centrally symmetrically placed about the origin of co-ordinates; the solution (109a) is valid only outside of this mass, where all the Tμν vanish.Ē (Einstein 1967)

Einstein's argument for Ric = 0 is contradictory and therefore invalid.

We can easily reaffirm the absence of material sources in Ric = 0 (wherein material sources = 0) by considering Einstein's field equations "in the absence of matter" with inclusion of the so-called 'Cosmological Constant'. The Cosmological Constant is represented by the Greek letter Λ. In this case Einstein's field equations are,

Rμν = Λgμν

Once again we have before us an intimidating and uninformative mathematical expression. We shall again reduce this to simple words, thus,

Ricci tensor = Λ(metric tensor)

So the gμν thing on the right side is the metric tensor. The Cosmological Constant Λ is just a constant. This identifies the parts of Einstein's field equations, but doesn't convey any meaning, so we will write them more informatively as,

spacetime geometry = Λ(metric tensor)

Note that there are no material sources in this expression either because there is no energy-momentum tensor; in other words energy-momentum tensor = 0 because material sources = 0. The Λ(metric tensor) term is not a material source - it is not an energy-momentum tensor. The solution to this set of field equations is called de Sitter's empty universe. Now,

"the de Sitter line element corresponds to a model which must strictly be taken as completely empty." (Tolman 1987)

Eddington (2007) informs us that de Sitter's solution is,

"the solution for an entirely empty world."

d'Inverno (1992) remarks on de Sitter's universe that,

"This is not a model of relativistic cosmology because it is devoid of matter."

Weinberg (1972) also tells us that in the de Sitter model,

"there is no matter at all!"

The reason why de Sitter's empty universe contains no matter is because in the field equations for which it is the solution, material sources = 0; in other words, precisely because energy-momentum tensor = 0.

We can now reaffirm why Einstein's alleged field equations Ric = 0 for "outside" a body such as a star are false. In the case of spacetime geometry = 0 the energy-momentum tensor = 0 because material sources = 0 there, yet Einstein still claimed that a massive source such as a star is present in Ric = 0, and according to his followers this star can 'collapse' to form a black hole. However, in the case of de Sitter's empty universe, energy-momentum tensor = 0 because material sources = 0 there too. Thus, material sources are alleged to be both present and absent when energy-momentum tensor = 0. That's impossible! It's a contradiction. Since material sources = 0 in both the equations Ric = 0 and Ric = Λ(metric tensor) the energy-momentum tensor = 0 in both equations, so there are no material sources in either set of equations. Thus, Ric = 0 does not describe a star or a black hole. It contains no matter by mathematical construction for the very same reason de Sitter's empty universe contains no matter by mathematical construction. But it is from Hilbert's solution (the so-called 'Schwarzschild solution') for Ric = 0 that the black hole was first spawned. So the black hole is a fantasy. It is not even predicted by General Relativity because Ric = 0 contains no matter by mathematical construction. Newton's theory certainly does not predict the black hole either because the theoretical Michell-Laplace dark body does not possess the alleged properties of the black hole.


d'Inverno, R., Introducing Einsteinís Relativity, Clarendon Press, Oxford, (1992)
Eddington, A. S., The Mathematical Theory of Relativity, Cambridge University Press, Cambridge, (2007)
Einstein, A., The Foundation of the General Theory of Relativity, Annalen der Physik, 49, (1916), The Principle of Relativity (A collection of original memoirs on the special and general theory of relativity), Dover Publications Inc., New York, (1952)
Einstein, A., The Meaning of Relativity, Science Paperbacks and Methuen & Co., (1967)
Foster, J. and Nightingale, J. D., A short course in General Relativity, Springer-Verlag Inc., New York, (1995)
Tolman, R. C., Relativity Thermodynamics and Cosmology, Dover Publications Inc., New York, (1987)
Weinberg, S., Gravitation and Cosmology: Principles and Applications of the General theory of Relativity, John Wiley & Sons, (1972)

SOME ENCOUNTERS (with notes)

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 inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section, 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. According to Sepcial Relativity its mass is given by

m = mo
(1 - v2/c2)

and its volume is given by

V = x3(1 - v2/c2).

So the density D of the moving mass is

D = m
= mo
x3 (1 - v2/c2)

which is arbitrarily large as v --> c. 'Infinite density' is forbidden by Special Relativity since no material object can travel at the speed of light in vacuo. The density is in fact undefined at v = c, because division by zero is undefined. 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 allegedly 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 Rμν = 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 Rμν = 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 Rμν = 0. Contra hype! So Rμν = 0 violates Einstein's 'Principle of Equivalence' and so the spacetime of Rμν = 0 is not a dynamical system at all; 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. Rμν = 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 Rμν = 0. It begins with the usual spherical-polar coordinate line-element for Minkowski spacetime (using c = 1), given by

ds2 = dt2 - dr2 - r2 (dθ2 + sin2θ dφ2).

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 Rμν = 0, the so-called "Schwarzschild solution" is obtained. But Rμν = 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 Rμν = 0. Thus, Rμν = 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):

ds2 = (1 - 2m/r)dt2 - (1 - 2m/r)-1dr2 - r2(dθ2 + sin2θdφ2),
0 ≤ r

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:

ds2 = (1 - α/R)dt2 - (1 - α/R)-1dR2 - R2(dθ2 + sin2θdφ2),

R = (r3 + α3)1/3,

0 < r

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 Rμν = 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 the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section (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 Rμν = 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 Rμν = 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 inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section. It does not of itself determine distance from the centre of spherical symmetry of the Schwarzschild manifold. 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

K = R1212

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

K = 1

for the "Schwarzschild solution". Thus, 'r' in the "Schwarzschild solution" is the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section as I have always claimed. Not one Standard Model relativist has ever realised that 'r' in the "Schwarzschild solution" is the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section. 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,

Gμν = Rμν - Ĺ gμνR = -κTμν.

According to the claims of the proponents of the Standard Model, if the energy-momentum tensor Tμν is zero, then the equations Rμν = 0 result (since the Ricci curvature R becomes zero also). However, since Rμν = 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, Einstein's field equations must take the form

+ Tμν = 0

wherein the Gμν/κ are the components of a gravitational energy tensor. Thus, when Tμν = 0, Gμν = 0, i.e. they vanish identically - there is no gravitational field. This is an inescapable consequence of the inadmissibility of Rμν = 0. Consequently, the total energy is always zero and there is no localisation of gravitational energy and hence no Einstein gravitational waves. Furthermore, the usual conservation of energy and momentum is violated thus placing the field equations in conflict with experiment on a deep level. 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:

"If you are right, our whole book is essentially useless and should be abandoned."

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 we will continue on our perhaps benighted way."

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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