No Trouble With Tribbles

[G Anthony Kent]

No Trouble with Tribbles

 

 

There is no trouble with Birkhoff’s Theorem which says: All gravity fields (including BHs’) act like normal Newtonian fields because all gravity fields drop out of GR naturally and so must be “asymptotically flat”, that is, they must vanish at large distances, i.e. they must follow an inverse square law.

 

BUT, Birkhoff is based on the particulars of the massive bodies that are treated, like stars; such particulars as the metric are used as premises. The theorem says any spherically symmetric field must be asymptotically flat because any mass already behaves as if all its mass was concentrated at the center. It already behaves like a point mass. So, Birkhoff should rule out the hyperbolic (1/kr) supermassive Black-Hole singular galactic gravitational field.

 

Yet, none of the BH scenarios that are theoretically covered can be considered real. All real BHs are perturbed beyond recognition by their immense quantities of environmental matter and energy, including enormous external gravity fields. Such fields emanate from huge galactic disks or from other whole galaxies with their own embedded supermassive BHs.

 

Real conditions should invalidate the theorem.

 

One critical consideration is that black-holes are NOT mere point masses. They have been shown by Kretschmann and Schwartzchild to be physically real as infinitely dense point particles (within Heisenberg limits) with an infinitely deep gravitational potential well. They are NOT like a planet or a star. This is not properly reflected in the metrics with their singularities necessarily excluded, and is not adequately treated by Birkhoff, or else it represents an exception. This observation may indicate a flaw or shortcoming in the way that general relativity is interpreted for spacetimes in the vicinity of black-holes, particularly near the singularity at r = 0.

 

Birkhoff used the Schwartzchild Metric. But, he could not rightly use the existence of an infinitely deep gravitational well or an infinitely dense point particle because these singular infinities cannot be handled normally. “The physics at a singularity is not well defined.”

 

It is far easier to accept the possibility of a flaw or exception than to accept the idea of some sort of unfalsifiable Dark Matter comprised of, say, undetectable WIMPs (weakly interacting massive particles). By their very nature WIMPs are supposed to be so “weakly interacting” that they cannot even show up in particle accelerator experiments. The WIMP hypothesis is formulated to be as unfalsifiable as any of the other Dark Matter proposals. As such, it does not merit the label “science”. It is more like science fiction.

 

Unfalsifiable? Yes! Nobody has detected a WIMP or any other form of Dark Matter. The experiments that have been done result in excuses only. Then, the detection limits that detectors can achieve or particle accelerators can gain are moved (like moving the goal posts during the game). So by now, detection limits or achievable energies are beyond any known detector or device. This means unfalsifiable.

 

So, an hyperbolic (F = GMm/kr) supermassive BH galactic gravity field is possible after all: k = constant = 1m (S.I.), for dimensional integrity. Einstein referred to his equations as being hyperbolic/elliptical in nature. That is, hyperbolic geometry is not outside the realm of GR.

 

Kretschmann’s invariance and Schwartzchild’s analysis mean that the singularity at the core of a BH is physically real. From our external frame of reference, the exact location of a BH singularity cannot be found because of the Heisenberg limit. So, from our external perspective, a BH core density and central gravity strength cannot be directly “measured” to be “infinite”. But, mathematically, it is so. And, elementary analytic geometry says that an infinitely deep graphical gravity potential growing from an hugely heavy infinitely dense point mass MUST be asymptotic in nature (NOT asymptotically flat). By symmetry, the other arm of the graphical curve must be asymptotic too, the definition of a hyperbola.

 

If you can collaborate on a paper, let us prove that an hyperbolic spacetime geometry around a realistic supermassive black-hole can be genuine and that the postulated hyperbolic (1/kr) field can, indeed, account for all effects currently ascribed to so-called “Dark Matter”. As a partner, of course, I shall do a yeoman’s share of work, including the scut-work of referencing & literature search. I am in an ideal position to do this!

 

[email protected]

 

"It is far easier and demonstrates much less intelligence to shoot down an idea than to show how to make it work."

[Cyberia]

Some thoughts: Why should the material inside a black hole be a point mass? We have no evidence that fundamental particles like electrons and quark can break down. And we have neutrons in neutron stars with an escape velocity of 2/3 c. A point mass suggests a one dimensional dot, so it would not spin, and we know that all black holes do spin.

 

Surely "infinite gravity" is a contradiction in terms in that in the first moments of the big bang, the "singularity" was said to be so dense that gravity (and the other forces) did not exist.

 

What I find strange is that collisions aside, the SMBH of a galaxy is about in the center. In the case of our galaxy, a 181 million SMBH in a galaxy of some 600 billion solar masses, so fairly insignificant by comparison. Why should it not wander around the galaxy?

[G Anthony Kent]

Some thoughts: Why should the material inside a black hole be a point mass? We have no evidence that fundamental particles like electrons and quark can break down. And we have neutrons in neutron stars with an escape velocity of 2/3 c. A point mass suggests a one dimensional dot, so it would not spin, and we know that all black holes do spin.

 

Well, Karl Schwartzschild’s analysis of general relativity results is the Schwartzschild metric which has solutions that proceed toward infinity at the event horizon and at the center of a black hole. He defines the event horizon by the singularity that he finds at the “Schwartzchild radius”. Kretschmann’s invariant results from an analysis that shows that the singularity at the event horizon is an artifact of poor coordinate choice, but the singularity at the center is real and not an artifact of coordinate selection. It is independent of coordinates and is therefore called an “invariant”. He showed that by “real” he means that for all intents and purposes the singularity, as a singularity, is probably infinitely deep and infinitely dense. But, we cannot actually measure it this way because Heisenberg’s Uncertainty will intervene.

 

Perhaps a one dimensional dot cannot spin, but the space immediately around it can. This space is filled with the intense (approaching infinite) gravitational field which can spin. It can spin relativistically too, i.e. “frame dragging”. The North and South poles on Earth are said to not spin (formally), but the rest of the Earth certainly does. The fact that an infinite depth in the gravitational field profile necessitates an asymptote is lost on many. That symmetry demands another asymptote as one proceeds toward larger radius is not recognized because Birkhoff’s Theorem and its congeners seem to prohibit it. Gravity must be “asymptotically flat”. That is, a hyperbolic gravity field is impossible.

 

But these are the results of interpretations of GR. It is not necessary to interpret it this way. If one desires, the right assumptions and boundary conditions can be selected that will permit a hyperbolic gravitational field. Perhaps one must drop a dimension and treat a black hole as a 2-D entity. This will certainly permit a 1/kr proportionality. Furthermore, such a hyperbolic gravitational field is renormalizable and the inverse square form is not. This unrenormalizability of inverse square gravity is what makes it impossible to merge gravity and quantum mechanics/dynamics. Inverse square gravity has a hyperbolic gravitational potential energy profile.

 

Quantum physicists love to renormalize infinities away. One way to do this arbitrarily for a black hole is to simply assign to the value of the gravitational strength at the center a finite quantity equal to 1 or whatever amount is appropriate for one’s purpose. This is the real mathematical meaning of “approaches infinity” anyway. It means larger than necessary to meet whatever stringent test one may apply. Setting quantities equal to 1 is S.O.P. and is often called invoking the “natural number system”. Trouble is, one cannot do this indiscriminately. The gravitational constant G and the speed of light cannot simultaneously be set equal to 1, for example. Infinite gravitational strength can be set equal to 1 or any arbitrary quantity but severe problems arise if one tries to set the hyperbolic inverse square potential energy profile at r = 0 equal to 1.

 

The only point I really want to emphasize is that it would be worth it to admit the hyperbolic 1/kr supermassive black hole galactic gravitational field as a postulate. It leads to the conclusion that all the phenomena ascribed to Dark Matter can better be explained by this expedient.

 

Surely "infinite gravity" is a contradiction in terms in that in the first moments of the big bang, the "singularity" was said to be so dense that gravity (and the other forces) did not exist.

 

Well, nobody really knows the nature of physical laws under these conditions. But Georgi Dvali and Lisa Randall have proposed the concept of “leaky gravity” that leaks or seeps out of our “local” expression of the universe into the metaverse or multiverse thereby accounting for gravity’s extraordinary weakness. If so, gravity can exist outside the singular point mass of the “inflaton” particle too and outside the center of a black hole. In fact, “excited” gravity may actually “be” the inflaton field, which is “like” gravity and may transform into it as it descends in energy levels to become, not a hyperbolic field, but the inverse square field that we are all familiar with.

 

Falling in potential energy, the residual hyperbolic gravity field that is still around after 13.72 billion years, donates its energy to the inverse square field that pervades the universe, kinematically increasing the size of the universe at an increasing rate (acceleration of the Hubble expansion). So, the hyperbolic field can account for Dark Energy as well as Dark Matter. One cannot get more parsimonious than this.

 

What I find strange is that collisions aside, the SMBH of a galaxy is about in the center. In the case of our galaxy, a 181 million SMBH in a galaxy of some 600 billion solar masses, so fairly insignificant by comparison. Why should it not wander around the galaxy?

 

. . . Because a big black hole was actually the seed nucleus around which our spiral galaxy began to coalesce in the first place. A supermassive black hole cannot spin eccentrically because it would have to emit gravity waves. It would soon lose enough energy to re-center itself. Plus, the field of the disk must be perfectly coaxial and concurrent with the field of the SMBH. So, it distorts the field of the SMBH such that Birkhoff’s Theorem and its siblings cannot apply. They are valid only for spherically symmetric gravity fields. The field of any real black hole must be so badly distorted that no perturbation theory can compensate. But, the field of the disk may add to the field of the SMBH so as to magnify its hyperbolic character. So, MOND is unnecessary to explain the anomalous orbital distribution of stars in spiral galaxies and elsewhere.

[G Anthony Kent]

If the author was a Mafia gangster, a really smooth big-time operator who had to hire a full time personal attorney to defend him from criminal charges, he would admonish his lawyer: “I do not need you to tell me so very simply just exactly what I can and cannot do. I want you to advise me precisely how to do what I want! Capisci?”

 

Now, the author concedes that GR, as it is most commonly interpreted, regards the hyperbolic black hole gravitational field as impossible. But as a system of sixteen complicated simultaneous nonlinear homogeneous partial differential equations, correct the author if he is wrong – having never actually done this, one must make some assumptions and define some boundary conditions in order to just begin to solve them. When this is done, one determines only the coefficients of these equations, many of which will be zero if one is lucky. The remainder will sum to at least one additional partial differential equation, whereupon more of the terms will cancel and drop out. This simplifying process is one of the main goals of many of the assumptions and boundary conditions and without which the equations might be unsolvable.

 

These final differential equations must still be solved and so, even more assumptions and boundary conditions must be assigned in order to do so. When this is done shrewdly, the equations can indeed be solved whereupon the results are equations that can be regarded as a physical rule that can be tested experimentally or observationally.

 

The author just cannot believe that there is no way to select assumptions and define boundary conditions in such a way as to permit the hyperbolic black hole gravitational field. This, especially when, in the case of the Friedmann equations for example, there happens to be a parameter designated ρ/ρcrit which determines whether the universe is spherical, flat or hyperbolic. The author knows that this is not quite relevant, so please do not focus on this stupid example and try to tear it to pieces. But, this is just to show that one can direct solutions of GR toward hyperbolic results by means of solutions having some mere adjustable parameters.

 

Einstein cannot have been so inflexible that he would have written a theory that could be rigidly used to prohibit reality. Such a prohibition would be a XXX atrocity wrought by a Grade A genius! Ha Ha! Let us not promulgate or propagate any such atrocities of our own.

 

Author’s challenge: a case of fine Spanish wine to the understanding personality who can tell him how to “force” GR to do want he wants!

 

In the meantime, all he is saying below is that there is sufficient reason to go ahead and allow the hyperbolic field as a postulate. The ideas below are not meant to be picked apart and eaten alive. But, they should still be digested. They are not logically necessary and sufficient, so demolishing them may be pointless. They are meant only to illustrate the notion that to allow the hyperbolic field as a postulate might make good practical sense.

 

The hyperbolic black hole gravitational field (HBHF) is said to be prohibited by common very reasonable interpretations of general relativity. But, the consequences of finding some loophole, some valid formulation of the HBHF are potentially momentous. They may even be capable of causing a revolutionary paradigm shift in the science of cosmology. Reasons that could motivate the search for some means to validate the HBHF are manifold.

 

1.) The HBHF field can explain the anomalous orbital velocity distribution of stars in galaxies.

 

2.) The HBHF can explain anomalous velocity distributions of galaxies in galactic clusters.

 

3.) HBHFs can explain the dependencies and magnitudes of the Sunyaev-Zeldovich effect. It can even explain the progressive changes in the SZ differential redshift offset effects that are seen when this phenomenon is observed to occur through voids closer and closer to Earth.

 

4.) The HBHF can explain the apparent offsets in the barycenters of colliding galactic clusters – the so-called “Bullet Cluster effect”.

 

5.) The peculiar galactic thermal distribution effects can be traced to the HBHF.

 

6.) The HBHF can more fully explain gravitational lensing phenomena.

 

7.) The HBHF can explain the inhomogeneity that is seen to have developed in the early universe, said inhomogeneity having been present since before the time of “recombination” of electrons with atomic nuclei. This inhomogeneity probably persisted as the hot plasma produced from the Big Bang “recombined” to produce redshifted emission of the cosmic microwave background radiation (CMB). Acoustic variation and long prior quantum perturbations are said to have been insufficient to fully account for the deviations that are now observed in the CMB.

 

8.) The HBHF process in 7.) can provide a confirmatory rationale for the Inflation theory of cosmogenesis. An Inflationary Big Bang, behaving like a hyper-massive, decaying, excited, quantum, fundamental point particle might have resulted in a large number of big primordial black holes as well as a lot of electromagnetic energy and many subatomic particles. This decay debris as these primordial black holes, with their super-extensive hyperbolic gravitational fields, would serve to induce an unusually broad gravitational “halo” effect similar to the one postulated for Dark Matter that is supposed to have been largely responsible for the inhomogeneity observed today in the CMB and in the actual observed distribution of galaxies.

 

9.) An extension of the HBHF hypothesis to the whole universe can provide a mechanism for a positive lambda in the LCDM Friedmann model of the universe. But, the label “lambda cold dark matter” might be replaced by the “lambda apparent cold dark matter” or LACDM model, since “cold dark matter” will then have been seen to be utterly superfluous.

 

 

One angle to deal with criticism along the lines of Birkhoff’s Theorem and its siblings might be to postulate that a black hole is wholly a quantum object so that its gravitational field is really a quantum field of a different form from the kind of gravity in GR. Perhaps Alan Guth’s “inflaton field” is related to gravity, but is not actually gravity, exactly. And, it could have a hyperbolic normalizable form because it originates, not in a Hugh Everett style meta-universe, but in an “infra-universe” or “sub-world” of fewer dimensions. So, it could then indeed be hyperbolic in its mathematical description. This “Many Worlds” interpretation of the nature of black holes and/or the Inflaton may include laws of physics that no longer pertain except in regard to black holes, especially since black holes involve physically real singularities. Inside black holes, the laws of physics not only break down, but may be delocalized outside the singularity and even outside the event horizon. And, yes, I know that I speak of black holes and the whole universe in the same breath.

 

After all, if the universe was once a quantum entity, then it still is. Macroscopic quantum effects should still be discernable in larger systems than in just tiny globs of Bose-Einstein condensates. Would super-massive black holes be large enough for you? Yuk Yuk!

 

The contention that some future theory of quantum gravity will erase the physically real singularities in black holes is a dream. The author thinks that theoretical physicists have been thrashing around for long enough. It is time to acknowledge that no such TOE or GUT will be forthcoming. No GUT has been proposed that uniquely and competently predicts anything new that has actually been verified, is falsifiable and actually unifies what it claims to unify.

 

A 2-D origin of the universe is not inconceivable. And, 2-D components of a non-spherical 3-D gravitational field are not ruled out. One can imagine that 2-D cosmogenesis or galactic orbital motion around a black hole was conceived when such motion or even the entire universe began to unfold or deconvolve from a compactified form, perhaps like opening a child’s “pop-up” book.

 

This rationale would include Guth’s hypothesis of an energetic massive “inflaton” particle in a hyper-excited “inflaton field” that decomposed, decayed or deconvolved, thus forming our universe. It seems unlikely that the inflaton particle would decay directly into gazillions of photons and little fundamental particles directly, all at once. This is not the way short lived excited particles typically decay. It probably split first into thousands, then millions of large black holes and simultaneously and/or subsequently into a lot of electromagnetic energy as well as many small particles.

 

(to be continued in "The inflaton gravity-like field itself also surely would not have collapsed or changed all at once")

[G Anthony Kent]

No Trouble with Tribbles

 

Kretschmann’s invariance and Schwartzchild’s analysis mean that the singularity at the core of a BH is physically real. From our external frame of reference, the exact location of a BH singularity cannot be found because of the Heisenberg limit. So, from our external perspective, a BH core density and central gravity strength cannot be directly “measured” to be “infinite”. But, mathematically, it is so. And, elementary analytic geometry says that an infinitely deep graphical gravity potential growing from an hugely heavy infinitely dense point mass MUST be asymptotic in nature (NOT asymptotically flat). By symmetry, the other arm of the graphical curve must be asymptotic too, the definition of a hyperbola.

 

If you can collaborate on a paper, let us prove that an hyperbolic spacetime geometry around a realistic supermassive black-hole can be genuine and that the postulated hyperbolic (1/kr) field can, indeed, account for all effects currently ascribed to so-called “Dark Matter”. As a partner, of course, I shall do a yeoman’s share of work, including the scut-work of referencing & literature search. I am in an ideal position to do this!

 

[email protected]

 

"It is far easier and demonstrates much less intelligence to shoot down an idea than to show how to make it work."

 

(continued from "If I was a Mafia Big-Time Smooth Operator")

 

The inflaton gravity-like field itself also surely would not have collapsed or changed all at once. Its transition might have been a process that may still be going on. Then, the present epoch’s breakdown of the residual inflaton field may act like gravity in whatever proper kind of space it may need to give a hyperbolic asymptotic effect for the whole universe (so that it would have a higher potential energy than the inverse square field).

 

In an infinite array of 2-D slices (if necessary to allow the HBHF) the universe HBHF’s ongoing stacked or packed 2-D asymptotic cross sections might devolve or transform into the lower energy 3-D inverse square gravitational field. This process might then result in acceleration of the expansion of the universe and putative Dark Energy.

 

Or else, a black hole is a tunnel or portal to another universe (Everett’s “Many Worlds” interpretation of QM) with different physical laws spilling over into our world and which simply do not prohibit the HBHF.

 

Another avenue might be to say that a super-massive black hole galactic gravitational field can be hyperbolic by virtue of analytic pure geometry in a non-Euclidian space, by an appeal to Schwartzschild’s analysis which certainly includes a non-Euclidian metric and to Kretschmann’s invariance which does not depend on any coordinate system. Then, if proper assumptions are made and correct boundary conditions are set, GR cannot be seen to override these sets of principles, however fundamental GR itself may be. As well, under the circumstances that would allow F = GMm/kr, GR might not be seen to trump the symmetry argument that is used to extend the asymptotic hyperbolic field to the far right on the ordinate of a gravitational field strength diagram. (Such a diagram needs to be given some latitude because it is a plot in 2-D Euclidian space, LOL). Symmetry representation is one of the most powerful tools available to the quantum physicist.

 

 

With additional assumptions or slightly different boundary conditions, the Schwartzschild treatment and Kretschmann’s invariance will still work if the overall geometry of spacetime in the broader galactic zone around a black hole is not Euclidian such that this whole local space could be strongly hyperbolic. And, there may be a way around the necessity to consider gravity as always operating under an inverse square relation, especially if there are “perturbations” that are really more like very strong distortions (like a train wreck somewhat distorts the rail cars) so much so that perturbation principles cannot truly be used for a mathematical description of a real black hole. The spacetime geometry in the distortion zone of a galaxy or galactic cluster containing black holes may be so strongly warped and hyperbolic in nature that any type of field, hyperbolic or not, can exist, persist and never cease to desist. Yet the overall hyperbolic or “open” geometry of the universe may be counterbalanced by the mere existence of all the matter and energy that it contains so as to “behave” like it might be flat.

 

The hyperbolic gravitational field, being normalizable, could pertain to the Higgs field and the Higgs boson and perhaps replace them.

 

Therefore, the author thinks that there is something fishy about the way GR is used and Birkhoff’s Theorem and its siblings are cited in order to put the kibosh on the HBHF.

 

 

All this author is saying, once again, is that there is sufficient reason to go ahead and allow the HBHF as a postulate. The above notions are not meant to be picked apart by intellectual sharks, however kindly, gentle, well meaning, gifted and dedicated. These ideas are not logically necessary and sufficient, so their demolition may be pointless anyway. They are meant only to illustrate an idea. This is that to allow the HBHF as a postulate might make very good practical sense, eventually.

 

Let us do this in the same way that Louis DeBroglie promulgated the postulate that the Bohr planetary model of the atom that he defended simply did not and could not undergo an “ultraviolet catastrophe” as classical physicists insisted that it must. DeBroglie almost single handedly invented quantum mechanics, by means of his postulate. But, he had a little help from Albert Einstein, Irwin Schrödinger, Werner Heisenberg, P.A.M. Dirac and a few others. This is what is needed now. Some help. If you or someone you know can collaborate on a paper for the Astrophysical Journal or some other platform, please consider it.

 

Furthermore, allowing the HBHF may provide yet another link between quantum mechanics and general relativity. There are a number of links already and when we forge enough critical connections we will have a ready-made unified theory of quantum gravity without having made any special fuss. We need not invent seemingly unfalsifiable, incompetently unpredictive, almost infinitely numerous, unmitigated psychedelically novel and inordinately complex hypotheses. I hope I am wrong, but such as these look like a whole boatload of Aristotelian theories of baroque “epicycles” that might accomplish little new that is uniquely proven. Except to satisfy the anal retentive urges of some who may otherwise be very fine workers, what purpose is to be served?

 

Theoretical physicists have long been fascinated by Eastern philosophies. They say that many principles of modern physics, including relativity, are reflected by philosophical concepts therein that are millennia old. The philosophical point that they chose to ignore is the tenet of “Yin & Yang”. Why should we not be satisfied with “two sides to the same coin”? Quantum mechanics and general relativity are not truly in opposition in any way. They do not address the same issues. They may be mathematically incompatible because they were assembled by different people who used different symbolic conventions. But, why should one necessarily be able to express gravity on an exceedingly small quantum scale? Why should we be able to compute the properties of a galactic cluster from quantum principles? Why?

 

So what if computations show that gravity becomes infinitely strong when the distances between even very tiny almost massless objects becomes excruciatingly small? Maybe we can learn something about what may have held the “Inflaton” infinitely dense point particle together before the Big Bang. Mathematical physicists are uncomfortable dealing with “infinities” and try to banish them whenever they can. Might not this trend be carried too far? Quantum scientists do not seem to fear some infinities. Why should cosmologists?

 

 

next reply: Comparison diagrams of the Hyperbolic Super-Massive Black Hole Gravitional Field and of the Inverse Square Law

[G Anthony Kent]

 

No Trouble with Tribbles

 

So, an hyperbolic (F = GMm/kr) supermassive BH galactic gravity field is possible after all: k = constant = 1m (S.I.), for dimensional integrity. Einstein referred to his equations as being hyperbolic/elliptical in nature. That is, hyperbolic geometry is not outside the realm of GR.

 

Kretschmann’s invariance and Schwartzchild’s analysis mean that the singularity at the core of a BH is physically real. From our external frame of reference, the exact location of a BH singularity cannot be found because of the Heisenberg limit. So, from our external perspective, a BH core density and central gravity strength cannot be directly “measured” to be “infinite”. But, mathematically, it is so. And, elementary analytic geometry says that an infinitely deep graphical gravity potential growing from an hugely heavy infinitely dense point mass MUST be asymptotic in nature (NOT asymptotically flat). By symmetry, the other arm of the graphical curve must be asymptotic too, the definition of a hyperbola.

 

If you can collaborate on a paper, let us prove that an hyperbolic spacetime geometry around a realistic supermassive black-hole can be genuine and that the postulated hyperbolic (1/kr) field can, indeed, account for all effects currently ascribed to so-called “Dark Matter”. As a partner, of course, I shall do a yeoman’s share of work, including the scut-work of referencing & literature search. I am in an ideal position to do this!

 

[email protected]

 

"It is far easier and demonstrates much less intelligence to shoot down an idea than to show how to make it work."

 

Gravitational Field Strength and Potential Energy Diagrams for the Inverse Square and Hyperbolic Cases

 

 

Figure 1

http://www.fotothing.com/photos/d98/d98a611bae38b8ed1ac943e8344717d1_7bc.jpg Figure 2 http://www.fotothing.com/photos/574/5740c48da23f95a8a869d5fc022221b3_8af.jpg

 

 

The Figure 1 caption remarks that interpretations of Birkhoff’s Theorem and its siblings may well be misinformed. One such common misinterpretation is outlined in detail by Kristin Schleich & Donald M. Witt, A simple proof of Birkhoff's theorem for cosmological constant, arXiv:0908.4110v2, wherein they prove that the common belief that Birkhoff's Theorem implies staticity is false for the case of a positive cosmological constant. So, it is not the Theorem itself that may be a problem, it is the ways in which it and GR is commonly interpreted which could be at fault. Let us use whatever type of interpretation that might be needed to allow the HBHF. Let us be creative. We are, after all, just “creative” cosmological accountants (LOL). We shall try not to cook the books, but we cannot guarantee it.

 

Figure 2 presents plots of the equations 1.) y = ln(x) and 2.) y = -1/x +1 according to common axes in such a way as to accurately represent the overall relative shapes of an inverse square gravitational potential energy diagram (2.) and a hyperbolic gravitational force P.E. diagram (1.). Note that P.E. keeps increasing without bound to the right in the case of the hyperbolic black hole P.E. trace which is actually a plot of ln(x). What this really might mean is that the cosmological influence of black holes might extend to infinity as a strong influence, or to whatever passes for infinity in our universe. So, the black hole gravitational effect may pervade the space well beyond a galaxy wherein it is contained, far beyond the space in a galactic cluster wherein BHs may be found and beyond even the envelope of galactic super-clusters or “walls” into large voids where the HBHF’s decline going deeper into the void could amplify the effect of such a void vis a vis the Sunyaev-Zeldovich effect.

 

In other words, the hyperbolic super-massive black hole gravitational effect might mimic a “halo” of Dark Matter that envelopes galaxies and galactic clusters. It could even deepen the difference between the gravitational fields present in large superclusters or galactic “walls” and the relative absence of said fields deep inside voids.

 

If the hyperbolic black hole galactic gravitational field can be generalized to the entire universe, its transformation or time dependent quantum-like transition to an inverse square gravitational field that may have begun with the Big Bang. And, it might be characterized as a process that is still ongoing. So, potential energy from a higher energy hyperbolic gravitational form, as in the red curve, might become available kinematically to objects under the influence of inverse square gravitational potential energy, consistent with the black curve.

 

Acceleration of Hubble expansion would becomes apparent after the Fig. 2 curves begin to substantially diverge (diagrammatically and not to scale) at about x = 1.5. The present time, t, or the present scale factor of the universe, a(t) could be represented to be located at maybe around x (or y) at 3.5, so that acceleration becomes apparent at maybe around 40% of the way toward t = 1, the present, or toward a(1) = 1, the present era scale factor. If we took the time, we could make these to scale so that actual predictions or depictions of real observations could be symbolized. Crudely diagrammatic or not, this scenario seems to be close to what may have been actually observed. Of course, it may be said that these curves were shrewdly constructed in an artificial manner that was deliberately meant to show this very thing. But, it was too easy to be an accident and this author is not smart enough to have contrived it.

 

Look at the enormous difference between the red curve and the black curve in Fig. 2 to the left of r = x = 1. This difference grows and becomes virtually infinite as one moves his attention further to the left, approaching the abscissa. Maybe this would provide a rational for initial inflation, which may then be said to have ended at r = x = 1, not after just a few seconds. Then, to the right of r = x = 1, the curves diverge again as the universe experiences acceleration or “reinflation”, gaining new vigor from the infusion of energy from the hyperbolic field’s residual gravity-like field.

 

Well intentioned, sincere, dedicated, competent and very intelligent people have tried to prove that the hyperbolic black hole gravitational field is impossible. They may sometimes use direct application of GR without prior recourse to any metric. But, one has to solve for a metric first, and then use even more assumptions and boundary conditions to solve this final partial differential equation for a useful expression that can be experimentally or observationally tested. Of course, using the conventional multiple layers of assumptions and boundary conditions, our indefatigable and diligent workers must logically arrive at the notion that the form of the gravitational field has to be an inverse square relation in any universe with 3 spatial dimensions.

 

But, how do they handle the fact that the black hole gravitational field must be physically real, infinitely deep and having a central infinitely dense point mass? No non-existent quasi-quantum gravity theory will “normalize” this singularity away. Somehow, this infinity must be included in any computation without “normalizing” it away after some fashion accidentally, implicitly or not. Which “normalization” might be in the form of the uncritical application of Einstein’s derivation of Newton’s law. This is insouciantly done as if contrast to said application was not the whole point underlying the concept of the hyperbolic field in the first place.

 

They must actually make the implicit assumption or silent premise that the singularity in a black hole doesn’t even exist in order to handle it mathematically at all. That is, they find a very plausible excuse to simply ignore it. Thereupon, one just naturally arrives at the idea that F is proportional to 1/r^(n-l), which boils down to 1/r² for 3 spatial dimensions.

 

It seems as if there is actually no way to explicitly acknowledge the physical reality of a black hole singularity in any way in such superficial treatments of GR. Unless such proofs explicitly treat the singularity as a physically real singularity, which is not merely another simple internal “distribution of matter”, they may end up proving nothing. Whatever odd geometry, queer boundary conditions or kooky assumptions may be necessary to admit the HBHF, they should be considered.

 

Let’s face it. Black holes are real and unique. We need to treat them mathematically this way, as if the central singularity is not a myth. Some observably exceptional properties must propagate far beyond their unobservable event horizons or else black holes are just ordinary objects in an increasingly dull universe. So, if this is the day of the GUT, cosmology is dead.

 

Finally, speaking of renormalizability, the hyperbolic black hole gravitational field may be renormalizable precisely because it is not represented by an inverse square relation. So, perhaps this would be a means to force gravity into the rigid klogs of quantum dynamics. The advantages and therefore the motivation to admit the hyperbolic gravitational field, even as it may be an unlikely postulate, may be much greater than anyone thinks.

 

We could keep Newton for laughs by joking that F = GMm/r^1.999 , there being no such thing as a real sphere in the locally distorted geometry of our universe, at least not near black holes. What does in fact happen to a gravitational field if it is not spherically symmetric as all the theorems presume? Might perturbation theory have to introduce an hyperbolic field component?

 

This F equation may be renormalizable too because it is not quite an inverse square relation. Ugh! This could be a form of MOND, modified Newtonian dynamics. Everyone knows that all modified gravitational field theories are intrinsically illegitimate (LOL).