General Relativity: In Acknowledgement Of
Professor Gerardus ‘t Hooft, Nobel Laureate

Stephen J. Crothers, Queensland, Australia


4th August 2014


I extend my thanks to Professor Gerardus ‘t Hooft, Nobel Laureate in Physics, for making more widely known my work on black hole theory, big bang cosmology, and Einstein’s General Theory of Relativity, by means of his personal website, and for providing me thereby with the opportunity to address the subject matter - supported by extensive references to primary sources for further information - in relation to his many comments, by means of this dedicated paper. The extensive mathematical appendices herein are not prerequisite to understanding the text.

Table of contents

I Introduction II Black Holes and big bangs in contrast III A black hole universe
IV Gravitational collapse V Black hole escape velocity VI The radius of a black hole
VII Metric 'extensions' VIII Black hole universes IX Big bang universes
X Gravitational waves and conservation laws XI Functional analysis Appendix A: Gaussian curvature
Appendix B: Riemannian curvature Appendix C: Appendix D: Isotropic coordinates
Appendix E: The Kretschmann scalar Appendix F: Geodesic completeness  

I got my own way, to go,
And now I want,
To take your minds;
I, believe, if you could see,
The blood between the lines,
I, believe, that you could be,
A better kind;
Please lead the way so the unborn can play,
On some greener hill;
Laugh as the flames eat their burning remains,
Fools die laughing still.

‘Fools’, Deep Purple, Fireball, 1971, (Gillan, I., Glover, R., Lord, J., Blackmore, R., Paice, I.)

I. Introduction

Gerardus ‘t Hooft is a Dutch professor of physics at the University of Utrecht in the Netherlands. He is a winner of the Nobel Prize for physics. He is currently, and for some years has been, the Editor in Chief of the journal Foundations of Physics. He has kindly brought attention to my writings on black holes, big bang cosmology, and General Relativity, on his personal website. I’m honoured that Professor ‘t Hooft has taken the time and trouble to inform people of my research proving the falsity of black hole theory, big bang cosmology, and Einstein’s General Theory of Relativity. Although he comments on the works of five particular scientists, he has allocated perhaps the most of his comments to me.

Mr. ‘t Hooft [1] refers cryptically to the five scientists as Mr. L, Mr. C, Mr. DC, Mr. E, and Mr. AL, although it is a well known secret that Mr. L is Dr. Chung Lo of the Applied and Pure Research Institute, Mr. C is me, Mr. DC is Dimi Chakalov (independent researcher)2 , Mr. E is Professor Myron W. Evans of the Alpha Institute for Advanced Study, and Mr. AL is Professor Angelo Loinger of the Dipartimento di Fisica, Universitá di Milano, Italy; for those Readers who were not aware of the well known secret. Mr. ‘t Hooft provided a link on his webpage to an interesting paper by Professor Loinger, but none, unfortunately, to me or the other scientists. I therefore elaborate herein on the many comments Mr. ‘t Hooft has made on his webpage concerning me and my scientific work.

I shall begin by comparing the generic defining characteristics of all alleged black hole universes to all alleged big bang universes as they require no mathematics to fully understand.

II. Black holes and big bangs in contrast

There are four different types of black hole universes advanced by the astrophysical scientists; (a) non-rotating charge neutral, (b) non-rotating charged, (c) rotating charge neutral, (d) rotating charged. Black hole masses or ‘sizes’, are not types, just masses or sizes of the foregoing types. There are three purported types of big bang universes and they are characterised by their constant k- curvatures; (a) k = -1, negative spacetime curvature and spatially infinite, (b) k = 0, flat spacetime and spatially infinite, (c) k = 1, positive spacetime curvature and spatially finite. Compare now the generic defining characteristics of all black hole universes with those of all big bang universes [2, 3, 4, 5].

All black hole universes:

(1) are spatially infinite
(2) are eternal
(3) contain only one mass
(4) are not expanding (i.e. are static or stationary)
(5) are either asymptotically flat or asymptotically curved.

All big bang universes:

(1) are either spatially finite (1 case; k = 1) or
spatially infinite (2 different cases; k = -1, k = 0)
(2) are of finite age (~13.8 billion years)
(3) contain radiation and many masses
(4) are expanding (i.e. are non-static)
(5) are not asymptotically anything.

Note also that no black hole universe even possesses a big bang universe k-curvature.

Comparison of the defining characteristics of all black hole universes with all big bang universes immediately reveals that they are contradictory and so they are mutually exclusive; they can’t co-exist. No proposed black hole universe can be superposed with any other type of black hole universe, with any big bang universe, or with itself. Similarly, no proposed type of big bang universe can be superposed with any other type of big bang universe, with any black hole universe, or with itself. All proponents of black holes are blissfully unaware of these simple contradictions and so they combine (i.e. superpose) their black hole universes with black hole universes and with big bang universes to conjure up black hole big bang hybrid universes ad arbitrium, and without ever specifying what black hole universes in what big bang universes they intend.

Furthermore, General Relativity is a nonlinear theory and so the Principle of Superposition is invalid therein. Let X be some alleged black hole universe and Y be some alleged big bang universe. Then the linear combination (i.e. superposition) X + Y is not a universe. Indeed, X and Y pertain to completely different sets of Einstein field equations and so they have absolutely nothing to do with one another whatsoever.

Despite the contradictory nature of the defining characteristics of black hole universes and big bang universes, and despite the fact that the Principle of Superposition is invalid in General Relativity, Mr. ‘t Hooft [1, 6] superposes and says that multiple black holes exist, along with other matter such as stars and galaxies, and all together in some (unspecified) big bang universe [7].

“We not only accept the existence of black holes, we also understand how they can actually form under various circumstances. Theory allows us to calculate the behavior of material particles, fields or other substances near or inside a black hole. What is more, astronomers have now identified numerous objects in the heavens that completely match the detailed descriptions theoreticians have derived. These objects cannot be interpreted as anything else but black holes. The ‘astronomical black holes’ exhibit no clash whatsoever with other physical laws. Indeed, they have become rich sources of knowledge about physical phenomena under extreme conditions. General Relativity itself can also now be examined up to great accuracies.” [6]

Mr. ‘t Hooft [7] begins his exposition of big bang creationism with the following words,

“General relativity plays an important role in cosmology. The simplest theory is that at a certain moment “t = 0”, the universe started off from a singularity, after which it began to expand.” and he concludes from the Friedman- Robertson-Walker metrics that,

“All solutions start with a ‘big bang’ at t = 0.” [7] All so-called black hole solutions for various respective sets of Einstein field equations are also said to pertain to stars and other masses, including the Sun and the Earth. For instance, according to Mr. ‘t Hooft [7],

“Einstein’s equation, (7.26), should be exactly valid. Therefore it is interesting to search for exact solutions. The simplest and most important one is empty space surrounding a static star or planet. There, one has

Tµv = 0.”

Consequently, all the generic defining characteristics listed above for black hole universes apply equally to stars and planets and such, and they too are supposed to subsist in some unspecified big bang universe. Black hole universes differ however to those of stars and planets described by the very same equations on a secondary level. For instance, all black holes have a so-called ‘event horizon’ within which is located an ‘infinitely dense singularity’ at which spacetime is ‘infinitely curved’; stars and planets have no event horizons or singularities. Mr. ‘t Hooft [1, 6, 7], as is usual for cosmologists, urges that singularities, which are actually just places in a mathematical expression where it is undefined, are physical entities. Mr. ‘t Hooft, along with the astrophysical scientists, reifies points in an equation where that equation is undefined.

Since Einstein’s gravitational field is spacetime curvature, it follows that the cosmologists, including Mr. ‘t Hooft, necessarily maintain that Einstein’s gravity is infinite at a black hole singularity. These infinities of density, spacetime curvature, and gravity are also said to be physically real. For instance, according to Hawking [8],

“The work that Roger Penrose and I did between 1965 and 1970 showed that, according to general relativity, there must be a singularity of infinite density, within the black hole.”

According to Carroll and Ostlie [9],

“A nonrotating black hole has a particularly simple structure. At the center is the singularity, a point of zero volume and infinite density where all of the black hole’s mass is located. Spacetime is infinitely curved at the singularity. . . . The black hole’s singularity is a real physical entity. It is not a mathematical artifact . . .”

According to Dodson and Poston [10],

“Once a body of matter, of any mass m, lies inside its Schwarzschild radius 2m it undergoes gravitational collapse . . . and the singularity becomes physical, not a limiting fiction.” According to Penrose [11],

“As r decreases, the space-time curvature mounts (in proportion to r-3), becoming theoretically infinite at r = 0.

And according to Mr. ‘t Hooft [1],

“C is ‘self taught’, so he had no math courses and so does not know what almost means here, in terms of carefully chosen limiting procedures.”

How does Mr. ‘t Hooft know if I have taken any mathematics courses or not? He doesn’t! He certainly never asked me about it. What evidence does he adduce for his charge? None! Mr. ‘t Hooft just invented this charge for his own convenience. And for what it’s worth, I have taken formal university courses in mathematics; not that it makes any difference to the scientific realities.

As for “carefully chosen limiting procedures”, Dodson and Poston have already told us that a black hole singularity is “not a limiting fiction”. Carroll and Ostlie have already told us that “The black hole’s singularity is a real physical entity. It is not a mathematical artifact”. Hawking and Penrose have already told us that “there must be a singularity of infinite density, within the black hole.” Penrose has already told us that spacetime curvature becomes “theoretically infinite at r = 0.”

It is not difficult to see when a limiting procedure is employed or not, and it is certainly not employed by the foregoing Authors, in their very own words. Such is the nature of the alleged black hole.

There are two types of black hole singularity reported by cosmologists and astronomers, according to whether or not their black hole is rotating. In the case of no rotation the singularity is a point; in the case of rotation the singularity is the circumference of a circle. Cosmologists and astronomers call them ‘physical singularities’; and so does Mr. ‘t Hooft [6]. These and other mathematical singularities of black hole equations are reified so as to contain the masses of black holes and to locate their event horizons. Black holes are said to range in size (by means of their masses) from micro to mini to intermediate to supermassive to ultra- supermassive, up to billions of solar masses.

Since singularities are actually only places in an equation where the equation is undefined, owing for example, to a division by zero, singularities are not real physical entities, contrary to the claims of the cosmologists and astronomers.

Similarly, astrophysical scientists assert that there was a big bang singularity, also possessing various associated physically real infinities. According to Hawking [12],

“At the big bang itself, the universe is thought to have had zero size, and to have been infinitely hot.”

That which has zero size has no volume and hence can’t contain mass or have a temperature. What is temperature? According to the physicists and the chemists it is the motion of atoms and molecules. The more energy imparted to the atoms and molecules the faster they move about and so the higher the temperature. In the case of a solid the atoms or molecules vibrate about their equilibrium positions in a lattice structure and this vibration increases with increased temperature. According to Pauling [13],

“As the temperature rises, the molecules become more and more agitated; each one bounds back and forth more and more vigorously in the little space left for it by its neighbours, and each one strikes its neighbours more and more strongly as it rebounds from them.”

Increased energy causes atoms or molecules of a solid to break down the long range order of its lattice structure to form a liquid or gas. Liquids have short range order, or long range disorder. Gases have a great molecular or atomic disorder. In the case of an ideal gas its temperature is proportional to the mean kinetic energy of its molecules [14, 15, 16],

wherein is the mean squared molecular speed, m the molecular mass, and k is Boltzman’s constant3.

Now that which has zero size has no space for atoms and molecules to exist in or for them to move about in. And just how fast must atoms and molecules be moving about to be infinitely hot? Zero size and infinitely hot - there is no such thing. Nonetheless, according to Misner, Thorne and Wheeler [17],

“One crucial assumption underlies the standard hot big-bang model: that the universe ‘began’ in a state of rapid expansion from a very nearly homogeneous, isotropic condition of infinite (or near infinite) density and pressure.”

Just how close to infinite must one get to be “near infinite”? There are no such things as infinite or “near infinite” density and pressure either, just as nothing can have infinite gravity.

Near infinities of various sorts are routinely entertained by cosmologists and astronomers. Here is another example; this time it’s Professor Lawrence Krauss [18] of Arizona State University, who says, “But is that, in fact, because of discovering that empty space has energy, it seems quite plausible that our universe may be just one universe in what could be almost an infinite number of universes and in every universe the laws of physics are different and they come into existence when the universe comes into existence.” Just how close to infinite is “almost an infinite number”? There is no such thing as “almost an infinite number” at all. Krauss [18] reaffirms Hawking’s zero size beginning of the big bang universes with the following, “There’s no real particles but it actually has properties but the point is that you can go much further and say there’s no space, no time, no universe and not even any fundamental laws and it could all spontaneously arise and it seems to me if you have no laws, no space, no time, no particles, no radiation, it is a pretty good approximation of nothing.” Thus, the Universe sprang into existence from absolutely nothing, by some big bang creationism, “at time t = 0” [7] and nothing, apparently, is “a good approximation of nothing” [18]. And not only is nothing a good approximation of nothing, Krauss [18] says, “But I would argue that nothing is a physical quantity. It’s the absence of something.” Krauss [19] reiterated the big bang universes creation ex nihilo dogma, thus,