Galactic M-Sigma Relation and the Anomalous Stellar Velocity Dispersion

Galactic M-Sigma Relation and the Anomalous Stellar Velocity Dispersion

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?

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.

Postulate

Definition of a Postulate

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

More about a Postulate

• 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

The Hyperbolic Hyper-Massive Black-Hole Universe

The Hyperbolic Hyper-Massive

Black-Hole Universe

Hawking did not buy his own pronouncements regarding the disappearance of information into black holes. Instead, he and some others invented a whole new theory of black-hole thermodynamics. So in a sense, the black-hole event horizon is a real surface. It is sometimes called a "quasi-surface". 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 quantum gravity (QG). 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. I appeal for collaborators to help find such a metric.

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 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 (to us) expression of a statistical process within the multiverse, we experience only the total sum, the superposed probability density form from which emerges has 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.

Dark Matter is an unnecessary ad hoc fix

Dark Matter is an unnecessary ad hoc fix to fill in the blanks in the Friedmann model under the FLRW metric. Galactic supermassive black-holes exist as true physical singularities according to the Kretschmann invariant and Schwartzchild's analysis. Therefore, as point masses, they must possess a hyperbolic (1/kr) gravitational field, NOT a field that falls off as 1/r². Now, k = constant = 1m, S.I., for dimensional integrity. It is not true that GR cannot tolerate hyperbolic spacetime geometries. "The universe is hyperbolic." said Albert Einstein in his classic paper of 1915.

An hyperbolic field will give constant orbital acceleration to orbiting bodies as far from the center of a black-hole as we might like to measure. This means that bodies near the periphery of a galaxy should seem to move at constant velocity because rotational acceleration does not drop to near zero there as with a 1/r² inverse square law. This constant velocity distribution effect has actually been measured and has given rise to the notion of Dark Matter.

Gravitation does not fall nearest to zero between galaxies in a cluster either. So they too can bend light and affect redshifts in ways that mimic Dark Matter. The rotation of galaxies in clusters is also influenced by the black-holes that they contain with their 1/kr gravitational potential profiles. The not quite counterbalanced redshift effects in the Sunyaev-Zeldovich phenomenon are influence by the hyperbolic galactic and galactic cluster gravitational fields that exist as light falls out of such clusters and super-clusters into a large void and as it climbs out of it again after the universe has expanded by another billion light years or more.

Scientists are mapping, not Dark Matter, but the huge extent of the network of hyperbolic galactic and super-galactic gravitational fields that behave like Dark Matter because of the mathematical properties of the hyperbolic gravitational field are similar to That expected for Dark Matter.

Primordial massive and supermassive black-holes with their 1/kr galactic gravitational fields can also mimic the “halos” of dark Matter that are postulated to have existed just after the big bang and before the emission of the cosmic microwave background. There is nothing that Dark Matter explains that cannot be accounted for just as well or better by the hyperbolic black hole gravitational field.

The hyperbolic 1/kr supermassive black-hole galactic gravitational field explains “the Dark Matter Effect” without Dark Matter and it is more parsimonious and is a falsifiable hypothesis, unlike Dark Matter

The conditions for validity of Birkhoff’s Theorem are not met for real black-holes. Therefore, Birkhoff’s Theorem does not apply. It sometimes may be used as a first approximation, but it cannot be depended upon as a rigid rule for precise calculations. “The physics near the extreme curvature of a black-hole singularity is not well defined”. This covers Birkhoff’s too.

By the way, any entity that possesses mass by virtue of its motion will be influenced by the gravitational fields that it encounters. It is not so much that a gas mimicking Dark Matter may be very much colder than other gases that such an entity might encounter, but whether such a gas may be much denser. But absolute zero is absolute. Only ground state vibration modes are allowed for gases at absolute zero, translational motion does not exist at K = 0 because it implies a temperature, T > 0 K. So, gases must not exist either. They must be solid crystals. Also, such ground state vibrational modes are only for multi-atom molecules. Intergalactic gas is almost non-existent, is not denser and is not a factor, so it cannot mimic Dark Matter.

No Trouble with Tribbles

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 unperturbed 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 rapid rotation and 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. The direct superposition of such axially coincident and rotationally concurrent mass concentrations with their enormous gravitational fields may well augment the black-hole field in such a way as to force it into compliance with the hyperbolic field "law" for black-holes. Relativistic frame dragging alone could effect this process.

Real conditions should invalidate the theorem.

Also, 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. Cosmologists say that the laws of physics break down at the intense spacetime curvatures present near the singularity of a black-hole. What else might this means except that even Birkhoff's Theorem cannot be depended upon. These observations may indicate a flaw or shortcoming in the way that Birkhoff's theorem and general relativity are 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.

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."