Inverse gravitational decline versus inverse square decline

Analyzing the implications of a black hole singularity with near infinitely tight curvature close to the center and what this means to the mathematical form of the gravitational field, one concludes that a postulated singularity requires that black hole gravity declines as 1/r, not as 1/r^2. This effective “infinitely” deep gravitational “point-mass” geometrically implies a hyperbolic gravitational field profile. So, the concept has some bizarre twists.

But, general relativity does not permit a 1/r gravitational field in 3-D + t spacetime. However it does allow a hyperbolic field in 2-D + t spacetime. By GR, gravitational force must decline as 1/r^(n-1), where n = spacial dimensionality. If n = 2, gravity declines as 1/r. So, it is also posited (postulated) that there exists a 2-D, sub-event horizon, hyper-spinning, centripetally induced, infinitely broad disk singularity in all central galactic SBHs. Having mass probably concentrated nearer to the singularity center but being of spacetime in nature, the entirety of the disk singularity is immune to the event horizon of the black hole. It can therefore extend outward to far beyond the galactic rim even to nearby galaxies within a cluster or supercluster.

This 2-D gravitational field is also quantum renormalizable. It is well known that items in a 3-D space can be projected perfectly onto a 2-D surface – the holographic principle. Might this be a simple route toward validatable, falsifiable quantum gravity? It is interesting to contemplate that a supermassive central BH with its coterie of inner bulge orbiting stars may be a quantum object obeying quantum law.

This postulated set of logical statements is immune to criticism. If otherwise logical, it cannot be argued against. It must be experimentally tested. Observation is the only choice to conclusively validate or falsify such an argument. See the definition of “postulate” given below.

Definition of a Postulate

• A Postulate is assumed to be a true statement, which does not require to be proved.

• Postulates are used to derive other logical statements to solve a

problem. If a problem is thereby solved, especially if proven by

other data, the postulate must also be true.

• Postulates are also likened to axioms.

In other words, postulates are to be accepted at face value “for the sake argument” for whatever they may be worth as if they were indisputable axioms. THEN, if a whole argument containing such postulates actually works, there may be much joy. If not, it is back to the drawing board.

Newton’s law of gravity and Kepler’s laws are all easily adjusted to accommodate the hyperbolic 1/r G-field in two dimensions plus time. Kepler’s 3rd law in 2-D is derived from 2-D Newton analogously to the 3-D derivation. It is NOT the same result as if orbiting 3-D objects were limited to an Euclidean plane.

The G-field diagram is hyperbolic when its equal gravitational force contour lines are drawn with spacing in such a way that a 1/r relation is followed to the origin where spacing approaches zero. If the contour lines are then plotted having a z axis, Flamm’s hyperboloid is the result. This is a spacetime diagram, not a gravitational potential diagram.

No inner galactic bulge stellar orbits need be fitted to raw Kepler. Kepler does not define these orbits. Kepler’s laws are used merely to analyze them. The orbits are what they are. Kepler’s 2nd law applies no matter what the form of the central force. The “adjusted” Kepler’s 3rd law follows exactly from Newton’s law of gravity with reduced dimensionality according to GR. It is “adjusted” Kepler that should be used to compute central galactic supermassive black hole mass. See the Gary Kent post on WordPress.com.

There is nothing more to prove. What there is still to be done is to compare with observation.

Mathematically, the constant velocity distribution observed in spiral galaxies is explicitly derived. This means that the M Sigma relation is explained because peripheral stellar v = (GM/r*)^½. Also, Milgrom’s MOND constant, “a[o]”, is derived, where a[o] = GM/r*r[∞] = v^2/r*r[∞]. This implies that the universe must have a finite or maximum r because a[o] is an observed finite non-zero quantity. And, M, the black hole mass, may include the masses of many tens of thousands or more of very large stellar mass black holes that are thought to be embedded in every galaxy. The unit vector of r, r*, is used to maintain dimensional integrity.

No modification of Newton’s law is required. But, Newton must be regarded in the context of a 2-D hyperbolically curved spacetime. So, gravity for black holes declines as 1/r and is not an inverse square relation.

All the other effects that have been observed that have been traced to Dark Matter are also explained in this way. These include the anomalous velocity dispersion in spiral galaxies and in clusters, the weak gravitational lensing, the Sunyaev-Zel’dovich, the Sachs-Wolfe and the Bullet Cluster effects.

The hyperbolic G-field parsimoniously explains these phenomena without appeal to any unfalsifiable hypotheses of exotic dark matter. Weakly interacting massive particles and other alien perpetrators of Dark Matter effects have been researched avidly for a very long time. They must be regarded now as unfalsifiable hypotheses because it has become clear that there is no way to prove or disprove their existence or it would have been done by now.

The hyperbolic SBH singular ultra-spin disk G-field might have mass, perhaps like Alan Guth’s inflaton field in the false vacuum. Its mass, but not its hyperbolic gravitational spacetime configuration, could be confined to below the event horizon. The horizon itself could be greatly distorted – including any surrounding plasma or photon sphere. So, a photon passing through the expansive hyper-spin singular spacetime disk would experience therein an enhanced gravitational field, just as if it had passed through a Dark Matter “halo”.

The open cell foam, network or spiderweb structure of the large scale universe is also explained by the extensiveness of the hyperbolic field and its form as a 2-D saddle shape “hyperboloid of one sheet” embedded in 3-D space. Galaxies and galactic clusters will be expected to align so that the hyperbolic surfaces of their 2-D fields tend to coincide. So, even the initial structure of the nascent universe would be influenced by supermassive BHs therein which could have formed very quickly at that time.

They might have been there from t = 0 + an instant, for all we know. After all, if the inflaton particle was like an unstable subatomic particle, it may have decayed into smaller particles including many SBHs. Some have said that the inflaton particle must have decayed all at once. Under these extreme initial conditions, what experimentally validated physical law or fundamental principle is quoted thereby? So, it decays all at once. To what?

In short, the hyperbolic 1/r SBH galactic G-Field explains all the phenomena that have ever been traced to Dark Matter. The hyperbolic G-field IS Dark Matter. Its potential energy profile is generally higher than the profile of an equivalent inverse square G-field. Since m = E/c^2, it accounts for the unseen and unseeable missing mass of Dark Matter. The HBHG field is mathematically derived rigorously and satisfies the mathematical requirements of all observations.

I have written a paper on gravitational decline with distance, but I need a reviewer to help check my mathematics. kentgen1@aol.com

]]>v = (GM)^1/2, a constant

and acceleration

a = GM/r

which implies

F = GMm/kr where

k = 1m (S.I.) , a constant for dimensional integrity.

Then potential energy is proportional to the integral of 1/r or

P.E. = q ln(r) where q is a proportionality constant, q = GMm

The natural logarithm of r, ln(r), continues to increase steadily as r increases so that P.E. becomes infinite at infinite r. Nowhere in the universe is exempt from the gravitational attraction of any black-hole anywhere.

]]>This is exactly the same as that predicted by hypotheses of “Dark Matter”. The hyperbolic black-hole gravitational field IS Dark Matter.

]]>When the BB occurred, its quantum renormalizable gravitational field stemming from its enormous mass began to collapse or “transition” from its hyper-excited state to its present unrenormalizable inverse square Newtonian gravitational field. In so doing, it released its potential energy in the form of the expansion of spacetime and the kinematic stimulation of massive cosmological bodies in their inevitable tendency to move farther apart.

This renormalizable gravitational field is one that follows an “inverse” relation, not an “inverse square” relation. That is, it declines as 1/r instead of as 1/r^2. It is hyperbolic in nature, not parabolic or exponential. Because this gravitational field falls off so much more slowly, it affects massive bodies far beyond, say, the peripheries of galaxies where stellar velocity distributions have been found to be anomalously high. It extends so far, in fact, that nearby galaxies in local clusters and even in super-clusters are affected. This hyperbolic gravitational field extends from supermassive black holes in the cores of virtually all galaxies. In the case of globular clusters, these black holes must be almost “naked” because they are “dark”.

Dark Matter composed of dark black holes is an old idea. But, that black holes may exercise their gravitational influence by means of a hyperbolic field is new. It explains the anomalous stellar velocity distributions in spiral galaxies perfectly. And so, the hyperbolic gravitational field is a candidate for an explanation of Dark Matter.

It does not eliminate Dark Matter. It explains it. The hyperbolic field is an excited renormalizable gravitational field in quantum spacetime. In the case of black holes, it is due to the in-fall of matter into it. Some of the energy from this in-falling mass raises the excitation energy of the gravitational field. Perhaps this is why this field can spread its influence so much farther than can a “ground state” inverse square field.

General Relativity can describe a hyperbolic gravitational field if the metric (the definition of spacetime) that is derived therefrom comes about due to correct assumptions and boundary conditions. Restrictive relativistic theorems like “Birkhoff’s Theorem” will not be in force then and they do not apply anyway. This is because they presume that the massive bodies that they describe, including black holes, are “unperturbed” and perfectly “spherically symmetric”. In fact, there are no such real black holes anywhere in existence. They are good ideas only for approximate calculations, not for “precision cosmology”.

So, Dark Matter in the form of the hyperbolic black hole galactic gravitational field is still required to account for all the matter in the universe. It is still needed to fill in the blanks of the Friedmann equations under the FLRW metric which is used as a “standard model” of the universe by cosmologists.

But, remember

“Cosmologists are always wrong, but never in doubt.”

Lev Landau

I’m afraid I’m no good at math, so I’m no collaborator. But I would be very interested to see if you can manage a theory that explains all of our observations without invoking dark matter.

]]>But Birkhoff’s Theorem is quoted almost religiously to dogmatically claim that the interior distribution of matter inside a massive body cannot affect the shape of the exterior gravitational field. But, singularities are different. They permit no “distribution” of matter. They are point masses. This must have very serious testable effects. There must be a loophole in Birkhoff. There must be misinterpretation. There is precedent for said misinterpretation.

I need a collaborator who would like to participate in initiating this paradigm shift.

]]>The consequences of the hyperbolic black hole gravitational field are momentous and would require a paradigm shift. If we could get this published, we could change the face of “precision cosmology”.

]]>Milgrom proposes a new model for gravity. He calls it modified Newtonian dynamics (MOND). But, MOND will require a rewrite of general relativity, one of the most validated theories in all science (only quantum mechanics is better verified). My comment leaves GR intact. It is simple, direct and jibes with the facts while being more parsimonious than MOND.

One does not observe the rotation of galaxies directly against the background of other galaxies. They rotate too slowly. One observes red-shifts from stars in different regions of each galaxy. Plotting rotational velocities got this way versus distance from the center of a galaxy, one should see a monotonic drop in velocity to near zero as one approaches large r. Instead, velocity reaches a constant nonzero plateau. This contradicts Newton’s Law of Gravity. Milgrom wants to add his tiny, residual acceleration constant to Newton’s Law. All I am saying is that it would be better to take into account the non-Newtonian hyperbolic black hole gravitational potential that simply must exist in almost all spiral galaxies and also in other types of galaxies that may harbor black holes. Galaxies that do not happen to show the MOND effect probably do not have supermassive black holes, or else their black holes have formed so recently that there has not been enough time for the effect to propagate all the way to and beyond the periphery.

Yes, Perlmutter and Riess both depended on the same Lambda/Cold-Dark-Matter model of the universe that uses the Friedmann equations as a basis. So, they really didn’t have to coordinate their results. But, they did. And, they used the model to predict the model, the ultimate retrodiction. The same thing is done when cosmologists use the model to interpret gravitational lensing effects, the SZ effect and other observations that they say give credibility to dark energy and dark matter.

I do not say there is any attempt at fraud here. In fact, I say that they all are clearly acting as honest scientists. But, the scientists who reported positive cold fusion results were all honest too. They did not realize that there were inherent flaws in the neutron detection devices that they employed to observe “fusion” in deuterium oxide electrolysis cells using palladium electrodes. Honest scientists fall for pseudoscience too. But, fudge is fudge and no-one is immune to wishful thinking. Perlmutter and Riess wished for a more exciting result and they got it.

]]>As far as dark matter is concerned, I report only what Milgrom says he discovered after carefully considering data from many many spiral galaxies. I am saying only that he ignores the fact that nearly all spiral galaxies, and most other types, have supermassive black holes embedded in them. This makes a huge difference. Black holes and the whole mass of the galactic disk will behave like a non-Newtonian entity having a gravitational potential that falls off as 1/r, not as 1/r^2. Comparing a graph of this hyperbolic versus a Newtonian parabolic potential one sees that there is a virtually constant difference at large r. This is the source of Milgrom’s residual centripetal acceleration constant that he says he sees in most of the galaxies he studied. It is constant also because “the periphery” of a galaxy is a self-selecting zone. Most of Milgrom’s pertinent data came for regions near a “periphery”.

I am not arguing with Milgrom’s raw findings. Far from it. I say he is probably right about the data itself. But, he needs to consider the implications of the existence of relativistic supermassive black holes.

]]>The difference between the relativistic black hole hyperbolic gravitational potential and the Newtonian parabolic one accounts for Milgrom’s proposed residual centripetal acceleration constant that he found for stars near the peripheries of spiral galaxies. So, the invention of unfalsifiable “Dark Matter” to acount for the MOND effect is as unnecessary as the belabored construction of the “Dark Energy” ediface to account for Perlmutter’s and Riess’s putative “acceleration”.

So, given the truly doubtful nature of Dark Energy and Dark Matter, what do we do about the “missing mass” necessary to account for the flatness apparent in the anisotropy shown by the cosmic microwave background radiation (CMB)? The easiest way is to postulate that the global universe is about 22 times as massive as our little telescopes can discern. The signal strength or amplitude pattern, the frequency nodes, statistical distributions and identifiable extra contributions (as from the SZ effect) to the CMB implies that our current inventory of matter and energy in the universe accounts for only about 4.5% of its total mass. So, 100%/4.5% = 22.2. That is, the mass of the universe must be around 22 times bigger than we can tell from our limited perspective here on Earth.

If the universe is that much bigger and more massive than conventional wisdom admits, it goes a long way toward accounting for the CMB characteristics, the other red-shift effects, the gravitational lensing effects and most SZ effects that are being used to give credibility to acceleration and Dark Energy. If the universe’s own global hyperbolic gravitational field effect, the transition from which must pervade the whole total universe (not just our local space) is acknowledged as a vestige from before the Big Bang inflationary era – even the sense of the Sunyaev-Zeldovich effect is explained parsimoniously.

In other words, Dark Energy and Dark Matter are subject to Occam’s Razor as mere whiskers on the chin of astrophysics.

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