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by Miles Mathis

Doth the eagle know what is in the pit, Or wilt thou go ask the mole?
                       William Blake

First posted May 24, 2007

Now that I have shown the mathematical errors contained in both Special and General Relativity, I am in a position to comment on the theoretical work done on the cosmological beast now termed the "black hole". By titling this paper The Myth of the Black Hole, I am not suggesting that the idea of a black hole is a myth in toto. I am suggesting, and will show, that the current theory of black holes is in large part either false, experimentally unsupported, or scientifically suspect. I leave open the possibility that black holes do exist in some form, and that a part of the astronomical data has been read correctly. But the greater part of current speculation could be called wild, and a significant part is demonstrably illogical.

The central error in the current theory of black holes concerns the singularity that is supposed to inhabit the center of the hole. This singularity is a mathematical outcome of the math of General Relativity, but I have attacked this math from many different angles in an array of papers. Firstly, I have shown that the singularity is a zero, and that the zero cannot exist in any differential math. A mathematical field led to the prediction of the black hole, and mathematical fields have led to all the current speculation about the properties of black holes. In my paper on the foundations of the calculus, I showed that an axiomatic muddle that goes all the way back to Euclid is to blame for this misunderstanding, and I showed precisely how this muddle had affected and been extended by Descartes, Newton, and many others. Put in a nutshell, the point can exist in a real field only, but never in a mathematical field. The zero-dimension point cannot enter any equations defined by cardinal or counting numbers. This mistake infected calculus from the beginning and it still infects all of contemporary math and science, since all higher math is founded on differential equations. This mistake is at the root of both renormalization in QED and of the inconsistencies within General Relativity.

I have shown precisely where it impacts the math of GR, going back to Einstein’s original paper to do so. Einstein had tried to express GR in terms of the motion of mass points in four dimensional space, but since mass points cannot exist physically, his math imploded at the foundational level (just as the math of QED did soon after). Points cannot exist in equations, and "mass points" can't exist in either equations or the real world. The term "mass point" is an oxymoron, since "mass" requires extension and "point" requires none. GR requires a sort of renormalization, although no one has ever stated it that way. In GR, the renormalizing is hidden in the idea of "field strength", and the lack of precision is usually not even noticed. But the lack of complete field strength is caused by trying to force equations to contain entities they cannot logically contain: points.

This problem does in fact come up within the math of black holes, since this math starts coughing up zeroes and infinities that can be contained only by a sort of renormalizing. As with QED, an extraordinary amount of time and energy has been spent trying to make the equations that express the actions of black holes continue to make some sense. But I have shown that this is all wasted effort, since it is after-the-fact fudging of equations that were not sound to begin with. The axioms that grounded the math were contradictory. To solve this, you must correct the axioms. No amount of pushing and pulling can make faulty assumptions into correct assumptions. Only by basing your math on correct assumptions or axioms can you create equations that do not need any later fixing or renormalizing.

Once the point is jettisoned from higher math, all these esoteric problems simply evaporate. We begin by basing the calculus on the constant differential—a differential which is not and cannot be zero. We then redefine the math of General Relativity, and seek the motion of some given mass or volume. This mass or volume may be as small as we like, but it cannot be zero. This causes several fundamental and far-reaching changes to the math of GR, but the first of these changes is of course the loss of the singularity from all solutions. Equations that do not contain zeroes or points cannot give us solutions that are zeroes or points or instants or singularities. Therefore the central “fact” of the black hole must be given up. Whatever may be at the center of a black hole, it cannot be a singularity.

And there are many other specific problems with the math of the black hole, problems that concern not the axioms but the functions and terms themselves. In the 1930’s Chandrasekhar used Einstein’s field equations to calculate densities and accelerations inside a collapsing superstar. John Mitchell had first suggested the idea of a dark star in 1783, but Chandrasekhar was the first to do the actual math. This math suggested the singularity at the center, as well as other characteristics that are still accepted as defining the black hole. Einstein himself contradicted Chandrasekhar’s conclusions (though this is just one more instance where Einstein was ignored). Einstein’s argument against Chandrasekhar was more theoretical than mathematical, but historically it doesn’t matter one way or the other—except as a sample of politics within physics. It doesn’t matter because the math of both Einstein and Chandrasekhar was wrong. Beyond the matter of using mass points in GR, Einstein made several other basic errors—errors that Chandrasekhar did not correct—errors that had not been corrected until I uncovered them. Primary among these errors is the term gamma, which I have shown is faulty. Because Einstein had mis-assigned several length and time variables in SR—giving them to the wrong coordinate systems or to no specific coordinate systems—his equations began misfiring early on. He skipped an entire coordinate system, achieving two degrees of relativity when he thought he had only achieved one. Because his x and t transforms were compromised, his velocity transform was also compromised. He carried this error into the mass transforms, which infected them as well. This problem then infected the tensor calculus and GR.

In some experimental situations gamma is nearly correct, which explains why it has stood for a century. But in a majority of situations, gamma fails, sometimes very badly. This explains the various anomalies and variations and so-called violations within Relativity. Gamma is a central term in GR as well as SR, and its falsification has far-reaching mathematical consequences. Especially in extreme situations like the black hole, a small change in this term can radically alter the field. This effect is doubled and trebled once you realize that gamma is also the main term in the mass increase equation. To calculate volumes or densities in a field you must calculate both length and mass, and the term comes into play in both. I have shown that it is wrong in both places.

Newtonian mechanics is used in black hole field equations as well, and my corrections to Newton also come into play. I have shown how the Schwarzschild radius uses a=v2/r, an equation that contains many disclarities, to say the least. Other proof that orbital mechanics is poorly understood may be found in my paper on that subject, where I show the historical conflation of tangential velocity and orbital velocity. This is just to show that Chandrasekhar’s “black hole” equations fail in a multitude of ways, and that a new solution requires starting over from scratch.* Even with my hatful of corrections it is probable that there are other mistakes no one has yet teased out of the field equations.

[Added, August 2012: I just became aware of Stephen J. Crothers' analysis of the original Schwarzchild and Droste solutions, and his harsh criticism of Hilbert's solution—and all solutions after that. I send you to his equations and explanations, which tear apart the tensor math as I have torn apart the algebraic and calculus underpinnings. In my opinion, it is best to jettison the tensors in their entirety from GR, for reasons I have detailed extensively elsewhere; but for those who still won't look at any math that isn't curved, Crothers has your medicine, in a fullish dose.]

For this reason, I believe all the subsequent work on black holes has been wasted time. Since Einstein’s field equations aren’t correct, Schwarzschild’s solution of 1917 is not correct, Israel’s non-rotating solution is not correct, Kerr’s rotating solution is not correct, the no-hair solution of Penrose, Wheeler, Hawking, Carter, and Robinson is not correct, and all the twiddling since is also not correct. All these guys would have better spent their time combing the field equations more closely, rather than running off pell-mell on a century-long wild goose chase. I say this with some heat, since the most feted of them, Hawking, has since claimed that this wild goose chase—and all the other similarly composed wild goose chases—have taken them to the very brink of physical omniscience. This hubristic claim is so revolting that it excuses any amount of heat from me, or any of their other critics.

Closely tied to this mathematical mistake is the mistaken assumption that matter that enters a black hole escapes from our universe. This is the way it is stated in almost all popular books, articles, and encyclopedic entries (including Wikipedia). But there is and can be no evidence that this is true. This statement is simply a rather bald effort to create mystery. Science fiction is now seen as much more sexy than science, and the theory of black holes crossed long ago into science fiction. Most of the theory is no better than story-telling, and this idea of matter escaping into another realm is the first chapter of a long book of tall tales. It is doubtful that we know anything at all about real black holes, but if any of our astronomical data has been read correctly, the one thing we know is that all matter, including light, cannot escape from the gravitational well of the hole. If this is true, it only means that we cannot see or measure any of this matter or light; but it does not mean that the matter and light have exited our universe. When I turn out the lights at night, I can no longer see the TV, but that does not mean that the TV has exited the universe, or even the room. All it means is that the light rays can no longer reach me. The same is true with the black hole. The matter and light are still in the gravitational well, just as the TV is still in the room. Or this is the logical assumption. If we want to make any other less logical assumption, we should have some reason for making it. We cannot make a sexier assumption just because it is sexier.

It would appear that this make-believe story is based mainly on the word "hole". A sloppy title has misled not only a generation of credulous and lazy readers, it has misled a generation of poorly educated physicists. This is doubly surprising, in that the bad math that created the singularity should have disallowed any talk of actual holes—whether leading to other dimensions or universes or not. This is because a singularity is the definitional opposite of a hole. Even if the black hole contained a singularity, there would be no reason to propose it as a link to a parallel or alternate universe. A singularity is a singularity, not a hole. By definition, a hole must have volume, and by definition a singularity has no volume. You cannot logically enter a parallel universe through a singularity. If anything, a singularity would be precisely that thing that prevented you from entering a parallel universe, supposing one existed. Nothing closes a hole like a singularity.

But readers and theorists and popularizers all tend to ignore logic like that. They follow the hole and not the singularity, preferring to base their story-reading upon an imprecise title instead of a false mathematics. They prefer to believe that a black hole is really a hole, despite the fact that the name was chosen for PR purposes and not for scientific ones. The astronomers** who coined the term and popularized it had no intention of solidifying a misconception, but somehow the misconception has taken on a life of its own, and now even the astronomers are misconceiving the theory. It is as if the name had more power than the theory; and this is not entirely surprising, since the theory never had a lot of content from the beginning. The cart quickly out-sprinted the horse, and the readers’ desire for a colorful yarn led the theorist to supply the most interesting story, rather than the most likely.

Any rational analysis of the black hole must show that the collapsing star that creates it simply becomes denser. At a given density, no light or matter can escape. Density may increase in an infinite progression, but this progression cannot reach infinity. This statement is logical, and relies only upon the definition of "infinite." An infinite progression is infinite precisely because it is endless. If it ended it would not be infinite. A progression of density that ended in a singularity would not be infinite. It would be a finite progression that ended in a singularity. Likewise, a black hole may get smaller as it gets denser, but size and density both can continue to change forever, never mathematically nor physically reaching zero nor infinity. Zero and infinity cannot be reached, by definition. Infinite density cannot be reached, by definition, and in precisely the same way zero size cannot be reached. As a matter of logic and physics and mathematics, a thing can get smaller or denser or both, but it cannot be infinitely small or infinitely dense. You cannot reach the end of infinity, since infinity is endless. The combination of words “infinitely small” is like the combination “blackly white” or “sharply dull.” It is an error and nothing more.

What this means is that even if black holes are created roughly in the way we think they are, they cannot contain singularities and cannot be pathways to other dimensions, universes, or fields. They cannot even be said to contain mysterious subfields that somehow diverge from the known universe. Once the math and axioms are corrected, the only thing that can be said for sure is that the inside of a black hole is a field of very high accelerations, accelerations that must equal or exceed the speed of light. This in itself is strange enough, given the postulates of Relativity. In fact, this is the major contradiction that is never addressed: the black hole equations come directly out of GR—a theory grounded in the postulate that nothing can equal or exceed the speed of light—and yet the centripetal acceleration of the black hole must equal or exceed the speed of light in order to negate it or overcome it. And if this is the case, then all matter falling into a black hole would immediately achieve infinite mass. It is not clear how bits of infinite mass can be collected into a finite volume or increase in density, much less disappear into a singularity. It is not the universe as we know it that breaks down inside a black hole, it is the math and postulates of Relativity that break down. In other words, the assumptions that led to the math that led to the theory of the black hole do not work inside the created field. That is not a mystery, it is a failed progression of logic. It is not a paradox, it is a meltdown.

If recent astronomical data is truly visible proof of black holes, and is not to be read in other ways, then the first order of business of theorists is to resolve the contradiction above. So far this contradiction has been buried by other contradictions. By a sort of sleight of hand the reader’s attention has been focused elsewhere, on sexier but less fundamental problems. The astronomer leads his tour group through wormholes and has them ride on tachyons and bump through virtual particle pairs and arrive—via spooky quantum leaps and non-linear i-trajectories—at 11-dimensional boson-massed fields in parallel universes, all in avoidance of this central contradiction. Is GR correct or not? Does the limiting aspect of c determine Relativity or does it not? If not, how is it possible to base the mathematics of the black hole on the mathematics of GR? If the postulates of GR are false inside the black hole, and the math of GR breaks down, too, then what sense does it make to claim that the theory of the black hole derives from Einstein’s equations? It may be possible to save GR and the black hole, but not by so extravagantly ignoring the axioms and outcomes of both.

Which brings me to my final critique of the myth of the black hole. From the beginning, research has been topsy-turvy regarding the black hole. The black hole is among the first cosmological entities—and is certainly the most famous entity—predicted before it was seen. Since that time, this progression has become standard practice, and we are now seeking many entities, cosmological and quantum, that simply fill theoretical holes. But until fairly recently, physics was the attempt to explain things that we had seen or discovered. In other words, we discovered them first and then tried to explain them. Starting with the black hole, we explained things and then tried to discover them.

It is clear that this method must skew research. When you have a nail, everything starts to look like a hammer. When you have a famous theory sitting in your observatory, everything in the telescope starts to look like confirmation. Every new emission becomes a telltale sign of an event horizon, every dark patch is a yawning mouth, every mysterious attraction or motion is a hole in the universe. The black hole is now being used to fill every gap and answer every question. We may have one at the center of the earth, to explain terrestrial gravity. Baby black holes are proposed as the source of sunspots, of the Tunguska Event, of the Bermuda Triangle. Next it will be of hair loss and erectile disfunction.

This is the predictable effect of theory that has outstripped observation. It is the predictable effect of glorifying theories and theorists beyond their merit. It is the predictable effect of the popularization of science, and its now total Bowdlerization. It is the predictable effect of trying to impress an audience, rather than trying to remain within the confines of your knowledge. The audience of popular science is like a tar baby, and physicists and astronomers have got caught in its sticky vulgarity. The entire field has become tarred by its own marketing success. Science has ricocheted off the slanted expectations of the milieu, and is now on a strange truth-exiting trajectory. Science has become infected by non-science, and the virus has already achieved a near-total destruction.

Science spends a good part of its off-time attacking religion and paranormal claims and other pseudo-science, ignoring the fact that both the foundation and top-end of contemporary science is itself pseudo-science. I have shown precisely how this is so in many subfields, including Relativity, QED, string theory, calculus, and other higher maths, but it is very transparently true in the theory of black holes. Some of these other fields contain large sections of usable information, but very little in the field of black holes is confirmed or confirmable. It is non-confirmable not because we can’t see beyond the event horizon, but because it is impossible to confirm illogic. No amount of data can either confirm or disprove a contradictory sentence. Like much of contemporary physics, black hole theory is an irrational muddle. Famous books by respected leading physicists like Stephen Hawking are mostly nonsense. They are not just metaphysical, in the positivist sense; they are irrational. They are full of sentences that have no possible physical meaning. These books are just story-telling with a physical flavor. Hawking gives them a mathematical and esoteric frosting, sprinkling his asides with equations that few can penetrate, and his readers think he has said something profound. But he has not said something profound. He has only made himself look smart. But those who know what all the words and symbols mean in the sentences and equations recognize that he is a fake.

Here is one example from Hawking's most famous book, A Brief History of Time: "There are some solutions of the equations of GR in which it is possible for our astronaut to see a naked singularity: he may be able to avoid hitting the singularity and instead fall through a wormhole and come out in another region of the universe." Hawking does not lead us into or out of this statement. It comes up near the beginning of his chapter on black holes, and he does not clarify or support this statement at all. His next sentence is just a passing admission that "these solutions may be unstable." Of course he spends no time explaining how it is possible to see a naked singularity—which is a nothing; how an astronaut could avoid hitting this nothing (or how he would know if he had avoided nothing—it seems to me you could hit plenty of nothing on any possible trajectory); or why the wormhole would be hanging out near, but not in, the singularity. Like most of the rest of the book, this is just bald speculation; and it is not even good speculation, since it follows no logical or rational progression. There is no argument or explanation, just naked assertions.

Some have said that this book is difficult for laypeople, because Hawking argues quickly and assumes familiarity with the subject. But this is false. This assumes that the argument is fuller in some other place, but it isn’t. The theory has no fuller expression: it is just as airy and speculative in professional journals, and none of these gaps has ever been filled in (or ever will be). Neither Hawking nor anyone else can answer any rigorous critiques or serious questions, since these contemporary theories are just floating words—words that aren’t grounded in anything and usually don’t even refer to anything demonstrable or real or mathematical. They are ideas that are the children of an idle brain, begot of nothing but a very vain fantasy.

Here is another example from Hawking's famous book, an example with which I can snare both Hawking and Feynman. In it, Hawking avoids saying something meaningless only by saying something shockingly negligent. He is talking of the necessities within any unified theory and he begins with the use of Feynman's sum-over histories. He believes any future UFT will use these sum-over histories, although he gives no reason for this. Again, he just asserts it. But this is not the most negligent aspect of the example. He tells us that Feynman’s renormalization trick in sum-over histories is to sum the particle's histories in imaginary time rather than real time. This gets the correct answer and avoids the greater technical difficulties of trying to sum in real time. These greater technical difficulties are also considered a form of renormalization, but they require infinite renormalization, which is very sticky indeed. The problem is that neither Feynman nor Hawking seem to understand why their tricks work; in fact they admit as much. Renormalization of any kind is generally considered to be heuristic, which means the logic of the mathematical manipulations is not understood. But Feynman and Hawking seem to have a short memory. The reason the sums have to be done in imaginary time is that Minkowski assigned time to the imaginary axis to begin with. That is what the four vector field is in GR. So it is not that Feynman is using imaginary time; he is using real time which has been assigned to the imaginary axis. Minkowski assigned time to that axis for purely mathematical reasons: to make the field symmetrical. This is a mathematical convenience, not a physical necessity or fact. But once it is done, it cannot be undone. Which is to say that if you are using Einstein’s field equations as your basic mathematical background, you automatically inherit all the assumptions and assignments of that field.

Feynman and Hawking should know that, and it is possible that they do know it. But they seem to prefer to pretend that they don’t know it. They prefer to treat each manipulation as some mysterious trick, since that makes them seem like necromancers instead of physicists or mathematicians. A true physicist would downplay the mysterious aspects of his field, since he is trying to turn all mystery into science. But the contemporary physicist brags about the mystery, since it is the mystery that sells. Just look at the subtitle of Feynman’s most famous book: QED, the strange theory of light and matter. Feynman is not trying to make the strange into the unstrange, to turn the paranormal into the normal, or to turn nescience into science. He is trying to sell science as magic. He spends an inordinate amount of time trying to convince the reader that Nature does not make sense. And Hawking does the same thing. It is how he inherited Feynman’s mantle as the best-selling physicist. Hawking has spent the bulk of his career pursuing the details of the Big Bang and Black Hole precisely because they contained the most mystery. They are the sexiest fields.

Unfortunately, the details were all dependent on the foundations, and the foundations were all wrong. Which means that except for the money and fame, all his work was in vain. It was a glittering palace built on a sandpit.

The contemporary theoretical physicist is no longer a scientist, he is a very clever and very technical conman, one who knows how to make money in his field. Like a used car salesman, he is mainly a master of PR. He doesn’t know how the universe works or how logic works or how mathematics works or how mechanics works, but he knows the politics of his own field. He is the most central insider, the hub of a wheel of other wheeler-dealers. Physicists are certainly better at memorizing long lists of fake equations and big words—and using these equations and words to deflect any criticism—but beyond that their superiority over car salesmen is not so clear.

As a closer I will give you a clue to the solution of the central contradiction of the black hole. Those who have read my other papers will know that I have not dismissed SR or GR, neither the postulates nor the bulk of the math. I have corrected Relativity but I have not jettisoned it. I have fine-tuned both the math and the postulates, making Relativity more consistent and more secure. Therefore, this paper is not to be read as an oblique attack upon Einstein or Relativity. I believe that Relativity is true on both sides of the event horizon.

The central contradiction is solved by recognizing that Relativity is a theory of observation and measurement, not a theory of existence. That is, Relativity is operational, not existential. What this means, specifically, is that mass and velocity transforms are performed on data, not upon particles. A given mass may look like it is increasing, from a distance, but locally there is no mass increase. A given velocity may look like it is decreasing, from a distance, but locally there is no change. Relativity does not apply locally. Einstein himself stated this explicitly and implicitly in a number of places, as I have shown in other papers, but he was not always consistent in his interpretation. And subsequent interpretations have not kept this distinction. The standard model now interprets Relativity in existential ways, claiming that Relative changes are in fact Absolute changes. Meaning that mass and velocity change not only in measurement, but locally—for the particles themselves. But, I may point out, if they were absolute and existential changes, Einstein would not have called Relativity “Relativity”. A relative change requires a relationship—a distance, a viewpoint. Local measurements are self-measurements where there is no distance, no relationship, and no relativity. Locally, there are no transforms, no mass increases, and no velocity changes.***

This changes everything in regard to the theory of black holes. We can no longer ask how something looks or seems or is measured by an outside observer, since there is no possible data. The event horizon is a total operational wall. Transforms cannot take place across this horizon, and all equations and transforms and thought problems must take place between particles inside the hole. If this method is scrupulously followed, then logic can be maintained and contradictions avoided. The hole remains a strange place, but it is no longer a place where Relativity or logic or universal rules break down.

Given this, I think that most readers can create their own new story inside the hole, a story that is logical and interesting at the same time. I will offer my own telling of this story in a subsequent paper. For now, I have more pressing problems to solve. In my opinion, the conundrums of the black hole are mostly manufactured, or come out of bad math. Most likely the black hole is just a very dense star, and the only real question is when and why it later explodes. In this way it is a variation of the supernova question, differing only in that we are dealing with much greater masses, and dark masses at that. It seems probable that black holes, having a greater vacuum effect, lie dormant longer and collect during that dormancy far greater amounts of energy. This would make them candidates for quasars. Beyond that, the whole issue of black holes is not nearly as compelling as it has been made out to be. Once you peel away the possibility of time travel, wormholes, superluminal velocity, parallel universes, and so on, the black hole is just a glorified superstar. Much richer veins of thought exist in physics and astronomy, and it is time we returned to them.

To learn more about black holes, you may now consult my newer paper on black holes and quasars, showing a simpler and more logical reading of the data on both.

*The new theory of the gravastar also does not correct any of these errors, so it may be considered useless.
**Led by John Wheeler in 1969.
***Some physicists like Feynman have understood this (at times), as I show in my papers on him.

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