G Anthony Kent

I have an offer from the chief editor of a scientific journal to publish my paper on rapidly spinning black holes and their transformation of spacetime BELOW the event horizon to a two dimensional disk that, nevertheless, extends gravitationally beyond the horizon to far beyond a galaxy perimeter. It is important to note, as I do in my paper, that objects cannot experience a 2-D and a 3-D gravitational field simultaneously. The 3-D field arises from the mass of the galactic bulge and all the material that has descended to the horizon where the relativistic time dilation effect will cause time to literally stop (from our perspective) and the in-fallen matter's gravitational field to become frozen. But, the 2-D field arises as an observable effect of the descent of this same matter/energy that, by its own experience of continuing time, progresses all the way to as near to the singularity as may be necessary to achieve the transformative spacetime phase change effect. So, objects that orbit the central black hole in the plane of the galactic disk feel the coaxial 2-D, hyperbolic (1/r) gravitational field and objects that orbit chaotically within the bulge feel an inverse square (1/r2) field. So, measurements of central supermassive black hole mass that depend on Kepler's (3-D) laws to analyze the behavior of bulge stars should be correct. But, analysis of orbits of stars farther out in the disk should use the 2-D form of Newton's law and Kepler, wherein orbital velocity, v = (G*Mbh/r*)½ where G* is the 2-D gravitational constant and r* is the unit vector of r, for dimensional integrity. Mbhs for any given central supermassive black hole found by the two different ways should agree. But, there is a problem. Orbital v is found to be "virtually" constant from galaxy to galaxy. And, if G* is constant then Mbh should be constant also. But this cannot be so. Then, Mbh and G* must be allowed to vary inversely. This is hard to support. G* can found theoretically or from the anomalous velocity dispersion equation, above. This approach uses Mbh found from the M-sigma relation. This is essentially a (3-D) Keplerian method that uses stellar orbital velocity that encloses chaotic orbits of bulge stars that are at one standard deviation from the center. Then, G* = r*v2/Mbh. But, this implies that v and Mbh must vary inversely. The only way that this could be is for G* to be so small that variations in v cannot be precisely measured and have actually been overlooked. The fact is, G* is seen to be many orders of magnitude smaller than G, when this calculation is performed. This means that if v is measured more carefully, it should be found to vary as predicted here. This value for v, usually called σ (sigma), must not include contributions from the orbits of stars that are mainly in the plane of the galactic disk, for these stars will be under the influence of the hyperbolic (1/r) supermassive black hole gravitational field. Such stars will display effects of G*, not G. G* is the quantity that we would be trying to isolate, so such a calculation would contain a circular component. So, all the M-sigma relation data must be redone. This will take some time. The universe is over 13.7 billion years old. So, what's another decade or so? Gary A Kent

The human propensity to see patterns in ironic occurences is constanly amazing. Your prime numbers games are nothing new. Such games have been played for centuries. What would be more interesting is to find a connection between the iterative Koch equation for the prime numbers and the iterative relations defining fractal chaos patterns having complex variables. One could write the Koch equation in the complex plane and then try to relate it to some sort of fractal chaos function. The result might be to point the way to cracking public encryption codes with arbitrary numbers of digits in their prime number bases. see the Koch equation in the attachment

In my previous post I explained how a 2-D gravitational field can exist in our 3-D universe. It must be associated with a black hole having an infinite spin rate as well as infinite density and infinite gravitational field strength. Within the bounds of Heisenberg uncertainty, these singularities must exist. There is no point in trying to explain them away using some kind of advanced unintelligible sophistry. I show that the hyperbolic 1/r inverse gravitational field can exist as a spin disk surrounding any black hole with said disk extending far beyond the event horizon toward infinite r (radius). This explains MOND and the anomalous velocity dispersion because hyperbolic 1/r gravity means that orbital velocity, v, around a galactic center containing a black hole, v = (GMbh)1/2. That is v becomes constant dependent only on Mbh and G. But, G may not be the same G that applies in 3 dimensions so, I call it G*. One can get G* from the M-sigma relation. But, the mass of the supermassive black holes must first be refigured on the basis of the hyperbolic field if many of the orbits of the bulge stars that were used to get Mbh were coincident with the galactic plane. If all or most of these orbits were chaotic and not aligned with the galactic plane, the BH mass determinations may be okay. The meaning of the hyperbolic gravitational field of black holes is that MOND is explained without recourse to Dark Matter or to modifications of Newtonian dynamics. Newton and Kepler must be understood in two dimensions, that is all. All of the observations that are said to support Dark Matter as a huge halo of WIMPs engulfing galaxies and galactic clusters also supports the hyperbolic gravitational field postulate, even the Bullet Cluster effect. It is a postulate. This means that there can be no argument against it. It must be taken at face value and carried to its logical extreme where it will be either reduced to absurdity or else found to be correct. When extrapolated to the entire universe, the hyperbolic field mimics Dark Energy too. If Alan Guth's inflaton particle originated in 2-D space and began to roll down its gravitational potential slope toward a lower energy 3-D state, the higher energy 2-D potential energy would be progressively transformed in a time dependent quantum-like transition to the new 3-D "ground state". This potential energy would show up as apparently increasing kinematic momentum of all stars and galaxies in the universe. The universe would appear to be expanding at an accelerating rate. This is an exciting idea because the whole universe is thus to be regarded as a quantum object. It may provide a route to a valid theory of quantum gravity. It may point to the means to prove the existence of the multiverse. Hugh Everett could be right.

According to general relativity, in a 3-D universe with time, the gravitational field of all compact objects behaves as if the objects are point masses and the field strength declines as 1/r2. In a 3-D universe, therefore, it is impossible to support a hyperbolic 1/r gravitational field. But, black holes are different. Why bother with the whole concept of black holes if they are not different? Collapse of matter into a black hole must not only create a singularity (within the limits imposed by the Heisenberg uncertainty principle) but, the spin rate of the black hole must also increase without bound as r decreases to values below the event horizon. Attempts to explain away these singularities on the basis of a non-existent quantum gravity scheme are vacuous extrapolations of tentative hypotheses that amount to pure conjecture. Black hole singularities exist. Einstein through Schwarzchild and others say so. Who claims to be more brilliant than these fellows? I appeal to authority here only because it seems to be the only thing that impresses some. If you want to claim that singularities are mere artifacts of an inadequate theory, show me the Math. Black holes are different. When matter and energy collapse under an infinitely strong gravitational field to a point mass that is as tiny as may be necessary to explain its properties (the true meaning of infinity), the result is a phase change. Spacetime phase changes are S.O.P. in the repertoir of theoretical cosmologists, like Alan Guth. Let us adhere to the hydrodynamics metaphor used by Einstein in his development of GR. Flat spacetime is a massless superfluid. Helium IV is a superfluid but, it is not massless. To extend the metaphor, it is not hard to imagine that spacetime could undergo a phase change. In a black hole this change involves a reduction in dimensionality. This is about the only change available to it. Analysis of the equations of GR shows that gravity G = 1/r(n-1) where n = number of spatial dimensions. In a 3-D universe, G declines as 1/r2. In a 2-D universe, G declines as 1/r. So, black holes must use the gravitational energy of in-falling matter to raise its gravitational potential, the gravitational energy level, to the 2-D "state". We are starting to talk quantum language now. The shape of the gravitational field strength diagram, as it is a 2-D entity embedded in a 3-D space, is a nominally flat disk or platter with a potentially infinite radius. I call it a "spin disk" because it arises from the infinite rotational and orbital spin rate of matter that is in-falling toward the singularity. As a spacetime entity, it ignores the event horizon and proceeds outward to beyond the edge of the galaxy wherein resides any central supermassive black hole. Here is one of the non-intuitive consequences of GR. It is known that matter in-falling toward the event horizon must experience time dilation. From our external perspective, we would perceive time for this matter as having slowed and even stopped at the event horizon. From any point outside the event horizon, time really does stop there. But from its own perspective time does not stop and such matter does indeed drop through the event horizon where it may take part in whatever processes it might (time reversed or not). There is simultaneously an inverse square gravitational field set up by this matter and an inverse gravitatinal field set up by matter that has infallen to the singularity. There is no violation of conservation laws here because no object can feel these separate effects simultaneously. If an object orbits the galactic center in the plane of the galactic disk, it feels the inverse 1/r field. If it orbits on a trajectory not aligned with the galactic plane, if it orbits chaotically, it feels the inverse square 1/r2 field. This has consequences for the analysis of the orbital motion of close-in Milky Way bulge stars like S2 for the determination of the MW's supermassive black hole mass according to Kepler's laws. Kepler is valid for the 2-D case as well as for the 3-D case. MORE later

Dark Energy, Dark Matter and Supermassive Black Holes – Is There a Connection? I am fascinated by black holes. When I read that Kretchmann’s Invariant meant that the central singularity of a black hole must be real, I thought this really means that BHs are unique. Because of this fact alone, I thought it verges on insanity to hold that BHs are ordinary and should be regarded like any other massive object in the universe. Let us suppose that there are such things as BH singularities or as close to being singularities as may be needed to produce the effect of “virtually” infinite density, virtually infinitely tight spacetime radius of curvature and virtually infinitely strong gravitational fields. “Infinite” means as great as may ever be necessary to account for whatever effects need accounting. Quantities just “tend” toward infinity. Infinite also means that if one attempts to graph an infinite quantity, one must truncate the plot and draw “asymptotes” that mean that the curve “approaches” infinity. Or perhaps it approaches a surrogate value very close to the value achieved at infinity, and would do so more and more closely if only there was a long enough piece of paper. An asymptotic curve never gets to where it is going. I thought that if this is true, there must be something wrong with the way the gravitational field of BHs is commonly described. One must draw a gravitational field strength diagram of an ordinary object as a parabola, nearer the center, having a maximum (or a minimum, if a gravitational potential well is intended) passing through the center of the object (as if it was a point mass). With a gravitational singularity, one cannot draw the curve so that it passes through the center of the object. It has no maximum (or minimum). This is absolutely not a “parabolic” or inverse square gravitational field strength diagram. This is something entirely different. I believe that the universe adheres to mathematical laws including the pure laws of geometry. Never mind about spacetime being curved and non-Euclidean. No interpretive theorem can upset strict geometry. This is a gravitational field diagram, so relativistic spacial curvature is already implied and taken into account. Besides geometry, I believe in the power of symmetry. If there is an asymptote nearest to the vertical axis, there must also be an asymptote nearest the horizontal axis. Wait! I know what this is. It is a hyperbola! Black holes must generate a hyperbolic gravitational field. So, I looked at Newton’s law of gravity, F = - GMm/r2, and wondered what would happen if F = - GMbhm/rr* where r* is the unit vector of r, for dimensional integrity. I call this an “adjusted” Newton’s law for black holes. It is hyperbolic. Using this, I followed analogous steps as with “raw” Newton’s law and I obtained v = (GMbh/r*)½ which is an amazing result. This says that the stellar orbital velocity around a spiral galactic center should not depend on r and, in fact, depends only on G and the mass of the galactic SBH at distances beyond the bulge. For a given galaxy with its central SBH, v is constant at larger r, say, well beyond 95% of the galactic stellar mass. I call it vo. This is the anomalous stellar velocity distribution, which is not really a distribution at all since all stellar orbital velocities are constant beyond the spiral galactic bulge. I also noticed that, for a series of galaxies having different SBHs with different masses, vo2 = GMbh This is the M-sigma relation wherein a simple monotonically increasing relation exists between central SBH mass and the orbital velocities of outer stars, where no correlation is supposed to exist when there is an inverse square gravitational force. (This is because inverse square gravity falls off so rapidly.) This explanation of the M-sigma effect might be complicated by the fact that galaxies contain many tens of thousands of large stellar black holes too. These would add significantly to an “effective” Mbh in ways that would be hard to measure or predict. But critics of the HSBH G-field postulate say that according to GR, gravitational force falls off as 1/r(n-1) where n = the number of spacial dimensions. So, a hyperbolic gravitational field can exist only in a 2-D region of spacetime. I said “No problem.” All real black holes spin very fast, are non-spherical, greatly perturbed and cannot be expected to obey Birkhoff’s theorem. This says that perfectly spherical, stationary, unperturbed, black hole gravitational fields must be “asymptotically flat”. This means that, among other things, the field diagram must have no asymptotes, which then means that it must follow an inverse square relation. But, central supermassive black holes must spin even faster than most stellar black holes because so many stars have constantly in-fallen concertedly toward the BH singularity, increasing its angular momentum. Eventually, this produces “infinite” spin rate at the singularity (not at the event horizon, however). Yes, the Kerr metric sort of stops or pauses (or at least the interpretations do) at the event horizon where a theoretically measurable finite non-zero angular momentum must transiently exist for this material. If it falls into the singularity, once there it should form a mere “ring” singularity. But, I deduce a possible new twist to the Kerr solution. After all, what about all the comments that have been made to the effect that we do not really know what happens when spacetime curvature becomes so tight? Everybody says of black hole singularities: “As r → 0, GR → ??? Might Kerr or his interpreters have got it slightly skewed? Perhaps SBH EMS (energy/matter/spacetime) collapses not to a ring singularity, but to a virtually infinite spin rate SBH (below the event horizon) with a very very wide 2-D disk singularity. This disk singularity is unique enough to be associated with the amazing properties of a real black hole. It befits the legend that we make of it. It becomes a very special 2-D EMS singularity that is a very broad, exceedingly thin flat disk. It is like a big whirling German pancake that spills beyond the griddle onto the floor and out the door. It is a strictly 2-D spacetime singular entity that can support a hyperbolic gravitational field. Maybe this is really what the Kerr metric actually implies, but rigid consensus mongers and obsessive seekers after conventional wisdom consistently misinterpret his analysis of GR. How broad is “broad”? I began to think large. Perhaps the mass or matter in the spacetime disk singularity stays under the event horizon (which must also be greatly distorted), but the gravitational spacetime coordinate component, which is immune to the event horizon, spreads out to include the periphery of the galaxy and beyond, even to the environs of other galaxies in a cluster or supercluster. It casts its hyperbolic 1/r gravitational field influence outward to infinity or to whatever passes for infinity in our universe. Maybe this is what it would really mean to be a 2-D infinite spin-rate disk singularity. The hyperbolic 1/r black hole gravitational field has a potential energy profile that is generally higher than the equivalent inverse square gravitational field. The difference might constitute Dark Matter because M = E/c2. The 2-D 1/r gravitational spin-disk influence would display thickness in the galactic stellar assembly because the stars are the observers that would measure the spin-disk’s location. They are limited by the Heisenberg uncertainty principle, so they see the 2-D flat plane as a volume having thickness. This flat plane is not totally flat, however. Being hyperbolic in nature, it would have an overall shape like an hyperboloid of one sheet. Looking like an hourglass or a saddle shape, It would not really be flat well beyond the galactic periphery. Its surface could therefore align more easily with other hyperbolic field surfaces generated by other SBHs in other galaxies. This alignment tendency may account for the large scale structure effect observed in sky surveys. I reasoned that the enhanced 1/r gravitational field effect accounts for not only the M-sigma relation and the anomalous velocity dispersion, but the weak gravitational lensing, the Sunyaev-Zel’Dovich, the Sachs-Wolfe and yes, even the large scale structure effects. These are precisely the phenomena that are cited in support of Exotic Dark Matter. But DM need not be exotic. It may result simply from how we compute the masses of cosmological objects containing large black holes. Maybe this can teach us something we did not know about black holes and cosmologists can quite pretending that they already know everything. Cosmologists at Cambridge, Cal Tech and the Perimeter Institute will not like this idea. Dozens of graduate students depend on Dark Matter research funding. I wouldn't dream of upsetting anybody's special interests. But, Nobel laureat Lev Landau said “Cosmologists are always wrong, but never in doubt.” But, maybe Dark Matter could still be real. A little extra midnight oil let me see that if extended to include the whole universe from the time of the BB, the hyperbolic field becomes Dark Energy, as well as Dark Matter, by virtue of its generally higher gravitational potential energy compared to the equivalent inverse square field that must pervade the universe. This is only because M = E/c2. The extra gravitational potential energy has mass and when it transitions to a 3-D + time inverse square field, its potential energy becomes real energy in the form of the accelerating expansion of space and the apparent kinematic "repulsion" of cosmological objects. As a matter of fact, Alan Guth says that the universe was once a quantum object. So, I say that it probably still is. Thus, the 2-D hyperbolic field may have been the highly excited “inflaton field” which decayed into the 3-D + time universe that we see today. This quantum transition would have been time dependent. It may still be going on. The higher potential energy of the hyperbolic field may currently be being realized as acceleration of expansion and the apparent “repulsion” between objects on a cosmic scale. The quantum superposition of yesterday fades into the relativistic reality of today. I did not yell “Eureka!” But, Dark Matter and Dark Energy may be essentially the same. It all hangs together quite nicely.

Tom Erber of the Illinois Institute of Technology has published a paper on his "Few Loop Theorem" wherein he proposes that all digital random number generators eventually fall into one or another of a few possible endless loops. Has anyone ever heard of this? What are the consequences? How can one program a computer to recognize if a random number generator has fallen into a loop?

Hypothesis: Supermassive black holes must develop incredibly high spin rates, as would be measured by an observer under the event horizon. They condense initially out of massive supernovae and then from stellar components that orbit in concert within the innermost stratum of the central bulge of spiral galaxies. Due to this ultra high spin rate, nearing infinite rate below the event horizon, matter in such black holes never has a chance to compactify to a singular point with virtually infinite density. Instead, it flattens not to a mere point mass or to a simple ring singularity, but to subtend a "planar" subset of spacetime that is only 2 dimensional. It becomes a huge flat spacetime disk. But, this spacetime parcel still has mass. Oddly enough, this explains the strange correlation of the velocity distribution of stars in spiral galaxies wherein rotational v of outer stars correlates with the mass of the central supermassive black hole, the M-sigma relation. There should be no such correlation if gravitational force for black holes is an inverse square central force. This notion shows that no exotic dark matter is needed to explain this phenomenon. But, Dark Matter is still implied by this hypothesis. Because by extreme contraction under rapid spin of matter to “almost” a point singularity with “almost” infinite density (“almost” means “to below a Planck distance”), the gravitational force therein is so intense that it is asymptotic in behavior: it is hyperbolic (1/r) in nature and it changes the nature of spacetime. So, it emanates from an extensive centripetally induced 2-D “disk” singularity. A peculiar result is that for stars nearer the galactic periphery, stellar velocity v' = (GM/r*)½ , that is, it is a constant that is totally independent of r. (r* = the unit vector of r, for dimensional integrity.) Therefore, it is actually observed (the M-Sigma relation and the anomalous velocity distribution in galaxies and clusters) that this hyperbolic 1/r gravitational field of the supermassive black hole disk singularity can reach to the galactic periphery and far far beyond. And, because it is so much more extensive than an inverse square force, it has an effect on other nearby galaxies which also may have supermassive black holes with similar gravitational forces in play. This possibility explains all the phenomena associated with Dark Matter. It does not deny Dark Matter. It clarifies it. This massive flat relativistically plausible spacetime ultra-spin disk is a hyperboloid of one sheet. It has a saddle shape, its being embedded in a 3-D + time universe. The curvature of the hyperboloid does not become apparent until r becomes very large, far beyond a galaxy. So, to engage other galaxies with their own SBH hyperbolic fields, the "plane" or surface of this curved sheet will align with them more readily because it is not really planar or “flat”. This alignment is a way to account for the observed large scale network or spiderweb distribution of galaxies within clusters and superclusters. It also helps account for the primordial structure of the universe, as supermassive black holes may have been very common in the beginning. Newton’s law will accommodate a 1/r gravitational field only if spacetime is limited to 2 dimensions plus time, by general relativity. Kepler’s laws can be modified to accommodate a 1/r field because these laws assume Newton, and Newton’s law needs only to be rewritten for gravitational 1/r 2-D spacetime. This 2-D spacetime parcel possesses mass, like the highly excited inflaton field postulated by Alan Guth. Also, since it is spacetime in nature, it is immune to the event horizon of a black hole. Therefore, it can extend as far outward as is necessary to account for its effects - the real meaning of infinity. The potential energy profile of the hyperbolic 1/r supermassive black hole galactic gravitational field is generally higher than the profile of the equivalent inverse square profile. So, the difference between the P.E. of the hyperbolic field and the P.E. of the inverse square field is a real energy difference. Because M = E/c2, this difference represents mass – matter that is unseen and unseeable. That is, this is Dark Matter. I have written a paper along these lines. But, I need a reviewer to help check my mathematics.

Multiverse and Metatime Maybe not on a par with Newton, but Hugh Everett may one day be seen as a thinker on a level with A. Einstein. Plus, John Archibald Wheeler’s suggestion concerning the quantum self-interference of probability density waves may be taken more seriously while Everett’s declaration of the “reality of probability” as a sort of substance gains credence. Self-interference can explain the virtual absence of antimatter (AM) in our universe. AM would be confined to our virtual twin, which must exist according to the logical extension of Alan Guth’s inflation hypothesis wherein a virtual particle came into existence from a hyper-excited false vacuum which itself came to exist precisely because of its ultra-high energy level. It would be seen as the deeper mechanism behind apparent “symmetry breaking” and unbalanced annihilation of fundamental sub-nuclear particles and antiparticles to give our universe with matter as the dominant form. There is no such thing as M-Theory. It is just an ideal. As such, it and super-string theory are unfalsifiable fantasies. But quantum principles must be seen to apply to the universe directly, due to the widespread acceptance of Guth's hypothesis. The implications are at least as profound as any M-Theory that could be imagined. The existence of an interference twin could also be helpful in explaining the hyperbolic (proportional to 1/r) black-hole gravity field as the resultant of a superposition of states. As the real expression of a statistical process within the multiverse, we experience only the total sum, the superposed probability density form from which emerges probability, P ---> 1. There are ways that such a superposition might affect the shape of a gravitational potential well. Gravity itself may be viewed as a probability vortex or wave in the Einstein Aether. There is much that has not been considered. The hyperbolic black-hole galactic gravitational field (HBGF) explains the anomalous stellar velocity distribution in galaxies and anomalous velocity distributions in galactic clusters and all the other phenomena that have been ascribed to so-called "Dark Matter". The hyperbolic hyper-massive black-hole universe gravitational field (HHBF) explains how the super-excited "inflaton" gravitational field has mostly collapsed and is now transitioning to a normal inverse square gravitational field, donating its potential energy. It thus fuels accelerating expansion, playing the role of "Dark Energy". See http://garyakent.wordpress.com .

The Universe is as it is because of Black-Holes: The Hyperbolic Hyper-Massive Black-Hole Universe The hyperbolic (declines as 1/r) black-hole galactic and universe gravitational field explains Dark Energy and Dark Matter. Stephen Hawking did not buy his own pronouncements regarding the disappearance of information into black holes. Instead, as a retraction, he and some others invented a whole new theory of black-hole thermodynamics. So, in a sense, they concluded, the black-hole event horizon is a real surface. It is sometimes called a "quasi-surface". However, the center of a black-hole is a physically real singularity. It is constrained only by the Heisenberg Uncertainty Principle. There is no such thing as a valid quantum gravity (how many papers are published in ArXiv on unicorns? By their standards, there should be dozens!) So, any appeal to QG to put the Kibosh on black-hole singularities is therefore bogus. See The Hyperbolic Hyper-Massive Black-Hole Universe and Galactic Gravitational Field (HHBF), which is a paper written for the blog http://garyakent.wordpress.com that describes the e-Model for inflationary expansion of the universe. The hyperbolic hyper-massive black-hole gravitational field is a phenomenological postulate, that is, it is a tentative premise that should be confirmed by experiment or observation and need not wait for theoretical justification. In the case of galaxies and galactic clusters, there is already enough observational support for the galactic hyperbolic super-massive black-hole gravitational field (HSBF). The point is emphasized that Birkhoff’s Theorem and other interpretive principles derived from general relativity cannot apply to any real black-holes. These rules presume that the massive bodies that are considered are always “unperturbed” and are perfectly “spherically symmetric”. No real black hole meets these criteria. The rules are good only for approximate calculation, not for “precision cosmology”. Besides, GR should not prohibit a gravitational field that declines as 1/r if a metric is found, similar to the Schwarzschild metric, using assumptions and boundary conditions wherein a singular black-hole is presumed at the outset. If such a gravitational field can be confirmed, the e-Model will serve as more evidence for the existence of our universe as part of a multiverse in meta-time. Hugh Everett may one day be seen as a thinker on a par with A. Einstein. And, John Archibald Wheeler’s suggestion concerning the quantum self-interference of probability density waves may be taken more seriously while Everett’s declaration of the “reality of probability” as a sort of substance gains credence. Self-interference can explain the virtual absence of antimatter (AM) in our universe. AM would be confined to our virtual twin, which must exist according to the logical extension of Alan Guth’s inflation hypothesis wherein a virtual particle came into existence from a hyper-excited false vacuum which itself came to exist precisely because of its ultra-high energy level. It would be seen as the deeper mechanism behind apparent “symmetry breaking” and unbalanced annihilation of fundamental sub-nuclear particles and antiparticles to give our universe with matter as the dominant form. The existence of an interference twin could also be helpful in explaining the hyperbolic field as the resultant of a superposition of states. As the real expression of a statistical process within the multiverse, we experience only the total sum, the superposed probability density form from which emerges probability, P ---> 1. There are ways that such a superposition might affect the shape of a gravitational potential well. Gravity itself may be viewed as a probability vortex or wave in the Einstein Aether. There is much that has not been considered.

Take a look: 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 become apparent after the 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)  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 a 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 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 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. One has to solve for a metric first, and then use even more assumptions and boundary conditions to solve for a useful equation that can be experimentally or observationally tested. Of course, using the conventional multiple layers of assumptions and boundary conditions, they 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/r1.999 , 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). This blackboard presentation is better to be accompanied by a full length lecture or explanatory text. Click on this link http://www.fotothing.com/photos/4a8/4a862b908b060b0f56a68ac89901c54e_fcc.jpg The Hyperbolic Super-Massive Black Hole Galactic Gravitational Field can be allowed by GR. One need only use the correct boundary conditions and assumptions. Here is a summary comparison of MOND and the HBH Field. The hyperbolic field is more parsimonious and is extendable to other situations like the whole universe’s gravitational field and Dark Energy.

This may be relevant to the Higgs discussion: 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. 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. . . . 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. In order to bow to the subject of this thread: being renormalizable, the hyperbolic super-massive Black Hole galactic gravitational field and the universe hyperbolic gravitational field, if they can be allowed somehow under GR via proper selection of boundary conditions and other assumptions, offers a way to incorporate gravity into Quantum mechanics/dynamics rather easily and directly. Then, a mechanism might be found that gives better account of "mass" than the Higgs field and the Higgs boson. It might mean exactly what Einstein said it means: Gravity and mass curve spacetime and spacetime tells matter how to move. Each particle distorts the fabric of the cosmos by its very existence and such distortion makes spacetime act like a viscous fluid but without any energy dissipation. Like Helium 4, perhaps. Spacetime has been called an "Einstein Aether". Hyperbolic gravity may provide a means to incorporate this notion into quantum.

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/r1.999 , 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).

(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

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")

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. 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. . . . 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.

Many Worlds is an invention of Hugh Everett, a student of John Archibald Wheeler. Everett also invented a calculus of quantum mechanics that does away with probabilities. The Universal Wave Function is a term introduced by Hugh Everett in his Princeton PhD thesis The Theory of the Universal Wave Function, and forms a core concept in the relative state interpretation or many-worlds interpretation of quantum mechanics. It has also received more recent investigation from James Hartle and Stephen Hawking in which they derive a specific solution to the Wheeler-deWitt equation to explain the initial conditions of Big Bang cosmology. So, to say that MWI has no use unique to itself that goes beyond the Copenhagen Interpretation is simply untrue. Many physicists take MWI seriously, like Stephen Hawking. It may be unimaginable that events bifurcate in infinite progression from the time of the “first event” at the Big Bang, but much of quantum mechanics is unintuitive and basically unimaginable anyway. Universes that spin off from this bifurcation process themselves undergo a complex multiple bifurcation process in their own event trains. So, the multiverse may be not only infinite, it may be infinitely infinite. Some of these alternate universes may be detectable because gravity may be “leaky” so that their gravitational effects can impinge on us (Georgi Dvali, Lisa Randall). So, MWI is not necessarily unfalsifiable. Wheeler’s idea of the quantum waveform always having an interference partner is intriguing. If the universe that we observe has an interference twin, this may be the home of antimatter. There could be harmonics of this pair, each having an allotment of mass equal to our own universe. The total mass of this little multiverse may be sufficient to account for the “missing mass” of Dark Matter and Dark Energy. Our universe could have some spin in the environment of this multiverse and this spin may allow for accelerating Hubble expansion as it slows down upon getting larger. There may, in fact, be testable consequences. If Alan Guth’s “inflaton” ultra-massive point particle sprang into existence like a quantum mechanical virtual particle in an excited inflaton field because its high degree of excitation made its appearance more probable, even inevitable, then it (the universe) was indeed a quantum mechanical entity. If it was once a quantum entity, then it still is. This makes Wheeler’s notion plausible. Everett’s ideas cannot be dismissed so easily either. Once upon a time, the wave-particle duality was unacceptable to many rigid thinkers. Quantum mechanics gets weirder every day.

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

Some criticize because GR must result in gravitation with a 1/r2 dependence. They say GR cannot tolerate any other geometry. It is this kind of "inside the box" thinking that cannot bridge the gap between GR and quantum. The hyperbolic black-hole galactic gravitational field formally does indeed have a 1/r2 dependence, or else it would be dimensionally bankrupt. Okay, so this is contrived just to provide dimensional integrity. But, it in fact DOES SO! So, in effect, it really is a 1/r2 relation. It is just that r2 is repressed by the intense gravitational field inherent in a black-hole.

What Imbues the Higgs Boson with its Mass? Speculation or fringe theory is really what we are all about here on this forum, no? In some way or another, this is true. If we were all in the business of writing texts, we would be paid. OR, we would pay journals to publish our junk if we wanted to propound fully qualified articles or developed papers. I understand Brian Greene's, Alan Guth's and other astrophysicists’ descriptions perfectly well. But, I am not about to duplicate their formulations and recount their descriptions just to make a point. There is not enough space in the forum server for me to do this anyway. Take my whistling in the wind for whatever it may NOT be worth. My MAIN POINT, these days, is always that the hyperbolic (1/kr) black-hole singular galactic gravitational field is acknowledged to be for real and is being studiously ignored... Now, if that other big unfalsifiable massive particle we call the Higgs Boson is the particle that imbues all other particles with their mass, what imbues the Higgs Boson with its mass? Higgs theorists are pulling their "pud". The Higgs is an ad hoc addendum that is a poor band-aid for the kink it was supposed to fix. Just what was that, anyway? Oh yeah, no explanation of "mass" in the standard model. Higgs is not really part of the standard model (yet). If the Higgs is not found, they will simply add in another ad hoc splint. The standard model will not collapse. Eventually, they'll get it right, though, I'll bet. Funny, there is no explanation of the origin of gravity in GR either, only that it exists mathematically associated with mass. Why cannot we be satisfied with two sides to the same coin? Yin and Yang? If mass and gravity are two ways of looking at the same thing, is it not futile to try to merge them into one - when they are NOT one? OR, if they are already merged as best they can be? This implies quantum and GR are just "so" - two facets of the same reality. If we try to merge the two, we shall go blind. The GUT or TOE is a fantasy. What if I am right? Millions, perhaps billions more will be spent pursuing Harvey down his rabbit hole. We will get just a mouthful of mud Much less than mass, there is no implicit validated account of gravity in the standard model of particle physics either. If there is a Higgs boson and Higgs field, it should be possible to derive the existence of the full fledged macroscopic gravitational field from them by means of the "correspondence principle". Then we shall have quantum gravity. Nah! Too easy. On the other hand ... ****** But, as far as other unfalsifiable new hypothetical heavy bosons are concerned - try Alan Guth's "inflaton" particle: A hyper-massive excited particle in a humongously excited "inflaton field" that cannot be distinguished from gravity itself, except by its degree of excitation. Suddenly, it decays. It decays into daughter particles and these then decay. Some of this decay debris has a long half-life. And enormous mass. The rest decays into matter and energy as we know it. But, the long half life particles remain as ultra-massive black holes. These decay, not via Hawking radiation, but by virtue of their intense infinitely deep singular gravitational fields that cause them to erupt into this same universe (somewhere "else"). They spew out smaller black holes and matter/energy detritus like a Roman candle. The daughter black holes they generate this way should follow a "normal" or "Poisson" distribution, perhaps. Statistically, this might be verified. It would take time for these BHs to start gathering in more matter to form full fledged galaxies. Some additional BHs may then form by accretion in the expected way. Perhaps this process would result in very ancient super-massive BH masses indeed following a Poisson distribution. If I was a mathematical physicist, I am sure I could derive it. But, I am just a modeler. Now for Black-Hole existence: the singularity case of a mass with radius r = 0 is very different, however. If one asks that the solutions to the simultaneous nonlinear partial differential equations in GR be valid for all r, one runs into a true physical singularity, or gravitational singularity, at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant (which says) at r = 0 the curvature blows up (becomes infinite) indicating the presence of a singularity. At this point, the metric, and space-time itself, is no longer well-defined, but not undefined. For a long time it was thought that such a solution was non-physical. However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the GR theory and not just an exotic special case. Such solutions are now believed to exist and are termed black-holes. Because they certainly are gravitational singularities, they must have a unique gravitational potential field profile. By simple geometry, they must be distinguished by a hyperbolic (1/r) fall off in the gravitational field strength. This fact is currently being ignored. F = GMm/kr, k = 1m (S.I., for dimensional integrity) means black-hole gravity falls off hyperbolically, not parabolically as according to Newton. This F equation is fully Newtonian, however. It just focuses on black-holes as being unique, and, of course, they are. ****** Mordechai Milgrom is a reputable careful worker. His raw data are used to support the idea of Dark Matter, not MOND. Not used this way by him, though, he still teaches MOND. Where do we get Dark Matter from GR or from the standard theory of particle physics? Where? WIMPS are even more hypothetical and unfalsifiable. DM itself is just a patch used to fill in the blanks in Friedmann. If one can derive Newton from GR, then one can derive the hyperbolic 1/kr black-hole galactic gravitational field using the right assumptions. These would be interesting in themselves... Unfalsifiable hypotheses cannot be used to refute facts. TeVeS theory is such an hypothesis like quantum/GR hybrids all are. They have never predicted one single unique item and no such prediction has ever been verified. A theory that does not predict competently is not a theory and does not deserve the attention of mathematicians nor scientists. All math, all science, is metaphor. All language is ultimately just metaphor. It is impossible to fully capture reality with any kind of human language. This is what many people mean when they claim that scientists are insufferably arrogant and grossly naive. These critics go too far, though. Then they claim science is Myth. They create this Myth. Let us endeavor NOT to do so ourselves.

Origins, emergence and eschatology of the Universe: Dark Energy & Dark Matter Should we mean "the universe" or "the meta-verse" or "the multi-verse"? (Hugh Everett) Presumably, when the universe formed from an ensemble of some sort of "inflaton" point particles (Alan Guth) as a statistically inevitable child of an extremely excited field, possibly the gravitational field itself, its hyperbolic (proportional to 1/r) field began to collapse into a parabolic (1/r^2) one as soon as Inflation began. The universe did not arise out of “nothing”. Nothing is NO THING. Therefore, as an object or “thing” it does not exist. Its existence would be an oxymoron. This collapse continues to this day. But, the process is almost done. There cannot be an infinite amount of energy sequestered in the hyperbolic 1/r field that would be available to fuel acceleration of the Hubble expansion rate by such a transformation. Transition to a lower potential energy parabolic field must provide a distinctly limited supply of extra impetus. Surely, after 13.72 billion years, the (1/r) potential energy mainspring has almost run down by now. The remaining (1/r) potential energy is called Dark Energy. It accounts for the "missing mass " or "Dark Matter" and "Dark Energy" in audits of universe contents and provides a convenient, theoretically rigorous and parsimonious basis for "acceleration". Dark Energy could account for around 80% of the universe's total mass, but audits are not so accurate. Dark Matter accounts for another 15%. Still, The Mainspring may still has enough oomph to last for at least 140 billion years more! The total mass, including Dark Matter, of the universe is enough to "flatten" it while acceleration may stop, but Hubble expansion will not. The hyper-excited gravitational field sprang into existence simply because it could. It came to be in a tremendously excited state because very high excited states are much more probable than lower ones, because of the zero point cut-off. This is just like virtual particles come to exist and be annihilated all the time on the quantum level (this is confirmed by experiment). None of them become universes, though, because there is already one here. It's a sort of a Pauli exclusion principle. There has been some confusion about variable labels. So, let us switch definitions of "r". In the following, r is the rate of acceleration of expansion of the universe (or rotational acceleration around black-hole). If the acceleration of the expansion rate is called a, and its present value is called P, then a = P at any given time, including the present. The simplest equation for the expansion rate's effect on P would be an exponential decay expression, P = ho e^(-rt), where ho is an initial value for h, r is the rate of increase in this expansion and t is time. We can get an estimate of a value for ho from Alan Guth's formulation of the theory of simple inflation. The presentvalues of both the expansion rate, P1, and acceleration rate, r, is observable. We can set t = 1, for the present value of t. So, we can summarize all relevant observations with this simple equation or the associated exponential expansion equation, R = Ro ^e(rt),where R is the putative instantaneous "radius" or scale factor of the universe. The current value of the expansion rate is Ho, the Hubble "constant", so P1 = Ho. Back to our original definition of r (not R) as a radius or scale factor: Exponential decay equations exhibit what is called a "dormancy" period or late plateau region. In this part of the discussion, here, "r" refers to distance from a center of rotation. Sorry. I missed this inconsistency in previous posts. I need a nicer symbol for the exponential period, another name for r; maybe Cyrillic backward "R"? Maybe a lower case Cyrillic "r"? Using "r" as a radius or scale factor: the hyperbolic 1/r curve levels off to near zero and continues to subside gently almost linearly for an indefinite time. Plot a graph yourself on the back of an envelope! Use mass M = 1, the smaller mass, m, drops out for acceleration. And, assume G is any self consistent constant like G = 1. This is just for comparison purposes, so it matters not. The equation for orbital acceleration around a galactic center, say, levels off to a constant, even at infinity, for a hyperbolic 1/r black-hole galactic gravitational field potential diagram. (You have just DERIVED modified Newtonian Dynamics or MOND!) You must multiply r by the constant k = 1m (Systeme Internationale) for dimensional purity. NOW, let us MIX the "r" metaphors. The current state of the universe itself may be considered as being in this (1/r) – implying both of the ways we defined "r" - exponential decay late dormancy or plateau period. The conclusion here is that acceleration of expansion may continue for a long time while very slowly decreasing nearer to zero. The black-hole rotational acceleration connection implies that the universe may be rotating very very slowly right now. But, we cannot know. We would have to observe the universe from the outside, from the perspective of the meta-universe, to tell. From the standpoint of general relativity, we simply cannot tell from our perspective here and now. Yet, in other words, even with acknowledged acceleration of the Hubble expansion rate, there does not necessarily have to be a "Big Rip" wherein the fabric of the cosmos is irreparably torn apart as expansion proceeds beyond a certain point. By the way, "M Theory" doesn't exist. M Theory is just an "ideal". Brane Theory is not M Theory. Neither one has ever predicted anything that can be experimentally verified and neither one is falsifiable. Therefore, they cannot qualify as legitimate scientific propositions. Not one single unique result has ever come from either. Furthermore, they are both unnecessary. Shrewd development of general relativity and quantum are slowly causing both to merge. What's the hurry? Let true "M Theory" and "Brane theory" grow organically directly out of quantum and GR. Each step will be independently validated, then. No worry. Origins, emergence and eschatology are fertile fields for philosophers. This is why we scientists are sometimes called "Doctors of Philosophies", Ph.D. Philosophi Doctori. I took Latin for three years and I am still not sure of this. German and Russian too, but this is no help. What happened to my old Latin grammar texts? I got tired of converting superscripts and subscripts to their proper forms. So, I quit. You can figure it out.

THE NEW COSMOS and the HYPERBOLIC FIELD HYPOTHESIS for MOND (modified Newtonian Dynamics) Mathematical, graphical models of the Universe according to the Black-Hole and Hot Big Bang Theories using "extensive variables" and therefore giving new insights Albert Einstein easily derived a relativistic differential equation that was guaranteed to reproduce Newton's Law when it was integrated with certain simplifying assumptions. He could have written a differential equation that reproduced MOND, but he didn't. He just didn't. He had no reason to do so because the various MOND observations had not yet been made. But, considering the very definition of a black-hole, it must be accompanied by a very characteristic and very different gravitational field. Because it is a singularity, a single point-object with infinite density, it must possess a gravitational field that is determined by its single point, its singular property. Its gravitational field graphical potential plot in 1/r must therefore approach an asymptote (boundary line having a limiting value) at radius = 0 and, by symmetry, it must approach another (perpendicular) asymptote at radius = infinity. This is consistent with the definition of a hyperbolic 1/r field, not one that follows Newton's inverse square law, which is parabolic 1/r2 in nature. A hyperbola follows an inverse law, 1/r while a parabola follows an inverse square law, 1/r2. If black-holes posses hyperbolic 1/r gravitational fields, then there is no mystery in MOND. Look at this image of a whiteboard derivation of the hyperbolic black-hole (HBH) gravitational acceleration and velocity profile near a galaxy containing a supermassive black-hole at its center. On the whiteboard, I also have written a synopsis of the MOND development: See http://www.lonetree-pictures.net/MOND%20&%20HBH%20%20.htm For MOND, Newtonian gravitational acceleration is denoted aN. The variant acceleration due to MOND expresses aN in terms of an acceleration that is modified by the function μ(a/a0), which is equal to 1 when the radius from the center of the galaxy is small enough for the overall gravitational acceleration to be large relative to the MOND constant, a0. When the putative acceleration is small relative to a0, μ is equal to a0 such that the equations on the lower left are satisfied. This happens when radius r is large enough for a2/a0 = aN = GM/r, and the total force of gravitational attraction on a given body enters the MOND regime. These values for r are equal to and beyond the point where the velocity distribution of stars encircling the galaxy in its outer regions becomes constant. Remember M2 = the supermassive black-hole mass while M1 = mass of the stars in the disk inside the radius r, to a given star. This radius might enclose more than 95% of all the stars in the disk and so may as well be considered to enclose all of them. But, a graphical model would have to take a step by step percent enclosure into account in order to plot a theoretical velocity distribution. So, it will be a little while before I actually do this. According to Newton’s inverse square law, velocities should fall off rapidly toward zero in this outer zone. But, observations show that it does not. Instead, they fall off much more slowly and become constant according to the vMOND equation on the lower right. But, the hyperbolic 1/r field contribution to the overall galactic gravitational field (shown as the blue curve, y1) in the graph above, gives precisely the same result. Thus, vMOND = vHBH. If it is admitted that black-holes are different, that they have special properties, among them that they are gravitational singularities, then this is not too surprising. In fact, it should have been thought of before, and it most probably has (see comments by Michael Rowan-Robinson). But, it is now being overlooked. I did not invent the hyperbolic 1/r gravitational field. I hope this effort on my part may prevent this oversight from being propagated indefinitely. Note that the usual depiction of the velocity profile of a spiral galaxy shows velocities rising to a maximum as one moves toward the center whereupon they fall off virtually to zero as one gets very close to r = 0. My simpler velocity distribution profile is for just one or for a very few stars. The standard picture of a maximum and fall-off in velocity as r --> 0 occurs because stars get crowded as one approaches the center and their orbital paths become chaotic. As one moves inward, just as many stars move clockwise as move counterclockwise (and also often on more nearly perpendicular trajectories relative to the galactic plane) and the net velocity declines. The stellar distribution becomes more spheroidal too, resulting in the classic galactic “bulge”. When I formulate my model, I will have to take all this into account as well. This is going to be fun. It is interesting to imagine what a galaxy would look like if purely Newtonian F = ma = GMm/r2 ≠ GMm/r for large r. Remember, I multiply 1/r by 1/r1 where r1 = the unit vector associated with r or r. These quantities are all directed in the same way, so we don't really need to use vectors except in order to introduce this unit vector. Then, v∞ ≠ (GM)1/2 ≠ constant. If the parabolic 1/r2 case held, rotational velocity would fall off toward zero as r increases without bound. The stars would lag much further and further behind and the spiral arms would wrap around the center of the galaxy much more tightly, like the mainspring of an old windup clock. So, one can actually see the MOND effect by just looking: http://www.lonetree-pictures.net/MOND%20&%20HBH%20%20.htm see image of a typical galaxy. This means that there may well be no such thing as dark matter or twin matter or any such Baroque complication festooning the simple picture of the universe that we, as scientists, should be looking for. It is in the nature of human beings to overdo. Dark matter and MOND are in danger of becoming vastly overdone. So, now let me engage in a little bit of my own overdoing. If this version of MOND is correct, I predict that there will never be a measurement of a0 obtained in the laboratory in a supersensitive Cavendish experiment. That is, not unless we can produce a sufficiently long-lived black-hole in the lab. This prediction will not go over very well with a lot of people who depend upon strange weird science for funding, so it will not be readily accepted. In elliptical and globular galaxies wherein the MOND effect may be observed, HBH MOND will require that one or more supermassive black-holes shall be found, naked or dark black-holes at that. The intragalactic black-hole presence in galactic clusters and superclusters may not be enough to account for MOND in these objects either, so some dark naked black holes may well be found embedded within them but outside the galaxies too. Nobody is looking for them now, so it is not surprising that they have not yet been found. The hyperbolic field concept can be extended to the entire universe, too. If it can be verified, it may go a long way toward an accounting for the mistaken interpretations of acceleration (Nobel Prize or not) and dark energy in the universe. This would require that the primordial black-hole, the mother of all black-holes or MOAB, must have persisted in some form, probably diminished, for a long time after it started to decay into the universe that we see now. In other words, the highly excited inflaton particle may have taken some time to deconvolve, decompose and collapse so, remnants may even still persist today. Certainly one remnant would be some degree of persistence of the original hyperbolic 1/r MOAB gravitational field. Continued collapse of this field will provide the energy to fuel acceleration and may well be the identity of Dark Energy. Remember, if one plots a hyperbolic field potential profile and compares it to the equivalent parabolic 1/r2 potential profile, one sees that the energy is higher for the hyperbolic case. Transition from a high energy field to becoming part of a lower energy field will provide energy from this which may amount to be a vast potential energy reservoir. If strange weird science is needed to justify funding, this is it. The questions implied here will be worth answering.

This would also mean that a black hole cannot grow. But, certainly they do. I think the answer lies in the idea that one should compare apples to apples. One should evaluate equivalent referenced times in each segment of each scenario. Crossing over to different reference frames AND scenarios is like mixing metaphor. It's a NO NO. In other words, one must stay within a given scenario until one reaches an experimentally testable result, then one may compare results. While one may actually do what Einstein suggested and ride a photon or at least travel at the speed of light (if one had an infinite source of energy - which may be the whole universe itself) it may be "pulling one's pud" to imagine scenarios that are even theoretically impossible. Such would necessitate making false assumptions. False assumptions can be expected to achieve false results.

Presumably, when the universe formed from Alan Guth's inflaton, its hyperbolic (proportional to 1/rr1) gravitational potential field began to collapse into a parabolic (proportional to 1/r2) one (see posts elsewhere). This collapse or transition continues to this day. But surely, the process is almost done. There cannot be an infinite amount of gravitational energy sequestered in the hyperbolic 1/rr1 field that would be available to fuel acceleration of the expansion rate via such a transformation. That is, transition to a lower energy parabolic 1/r2 field must provide a distinctly limited supply of extra impetus. Surely, after 13.72 billion years, the mainspring has almost run down by now. If the expansion rate is called h, and its present value is called P, then h = P at any given time, t, including the present. The simplest equation for the expansion rate’s effect on P would be an exponential decay expression, P = h0e-rt, where h0 is an initial value for the expansion rate, h, r is the rate of increase in this expansion. P can be used for a region of empty spacetime, perhaps even a galaxy, cluster or supercluster (if appropriate). It can also be used to model the whole universe. Transition between an initial higher energy hyperbolic 1/r gravitational potential field and a Newtonian 1/r2 field accounts for Dark Energy. We can get an estimate of a value for h0 from Alan Guth’s formulation of his theory of simple inflation. The present values of both the expansion rate, P1, and acceleration rate, r, are observable. We can set t = 1, for the present value of t. So, we can summarize all relevant observations with this simple equation or the associated exponential expansion equation, R = R0ert, where R is the putative “radius” or scale factor of the universe. The current value (at t = 1) of the expansion rate is H0, the Hubble “constant”, so P1 = H0. Exponential decay equations exhibit what is called a “dormancy” or "senescence" period or final plateau region. The hyperbolic 1/rr1 curve levels off near zero and continues to subside gently almost linearly for an indefinite time. The current state of the universe may be consistent with this dormant period. The conclusion here is that acceleration may continue for a long time while slowly decreasing nearer to zero. In other words, even with acceleration of the expansion rate, there does not necessarily have to be a “Big Rip” wherein the fabric of the cosmos is irreparably torn apart as expansion proceeds beyond a certain point. The essential detail made about point masses and singularities engendering a hyperbolic 1/rr1 gravitational field is a mathematical necessity. Consider what a point mass as a singularity actually means. If it does not mean that they generate a hyperbolic gravitational field potential profile, then the words point mass (remember Guth) and singularity are meaningless. Karl Schwartzchild would not agree with this negation of his analysis of general relativity. Some say that general relativity predicts that black-hole singularities must possess a gravitational field that falls off as 1/r2 with no difference from other Newtonian entities. I don’t believe general relativity says this under the rule that a black-hole must contain or “be” a point-mass (Heisenberg bounded) singularity. If treated as a real singularity (see Schwartzchild metric, Wikipedia), black-holes must have hyperbolic 1/rr1 gravitational potentials. This is a geometrical necessity. Such a gravitational potential falls off as 1/r, or more accurately, as 1/rr1. The symbol r1 is the unit vector associated with r and r in order to make dimensional analysis valid in equations like F = GmM/rr1. Here, r1 is not a variable. As a unit vector, it is a constant. So, this is a hyperbolic equation in 1/r. In association with the galactic disk which has a coincident and coaxial gravitational field equivalent to a couple of hundred billion sols, the residual hyperbolic 1/rr1 field at large r coincides with Milgrom’s extra gravitational acceleration seen near the periphery of galaxies. The periphery is a self defining zone that is responsible for Milgrom’s leftover acceleration “constant” that he wants to tack onto Newton’s Law. These observations mean that Milgrom’s MOND is unnecessary. And, Dark Matter is superfluous too because all the phenomena associated with Dark Matter are explained equally well by the hyperbolic 1/rr1 field effect. The “MOND effect” itself is evidence for the hyperbolic 1/r field.

Collision between branes is called the ekpyrotic theory. Only if one adds in the supposition that these branes are not exactly flat can one reproduce observations. It's another ad hoc add-on that makes these kinds of speculations suspicious. Now, if branes can bounce too, it is another ad hoc. If one surmises that they bounce repeatedly, one gets a periodic universe. Another ad hoc. Then, it is highly questionable that any these are falsifiable hypotheses. See my recent post on the scientific method and quintessence (2dc11). It is safe to say that no experiment can disprove that any of these phenomena exist. This is mainly what "falsifiability" means. The scientific method prohibits consideration of unfalsifiable hypotheses. Plain and simple.