Posts Tagged ‘inverse square potential’

Galactic M-Sigma Relation and the Anomalous Stellar Velocity Dispersion

June 4, 2012

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? 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

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.

Dark Matter is an unnecessary ad hoc fix

January 11, 2012

The singularity at the center of a black hole must be unique and have testable consequences.

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 of his spacetime metric under GR. 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, it becomes consant. 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 influenced 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 which is revised every time no Dark Matter is found.

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 at the extreme curvature of a black-hole singularity is not well defined”. This covers Birkhoff’s too.

It does too matter how the internal mass is distributed if it is contained within a single point. Then, in fact, it is NOT distributed at all! This is the point of Kretschmann’s invariant and Schwartzschild’s GR analysis of the consequences of his metric. Ordinarily, the distribution would not matter. But, a singularity must be different. If this is not explicitly acknowledged in some way, then to say there is a singularity with such intense curvature of spacetime in its vicinity that the laws of physics must begin to break down is a meaningless fatuous gesture to humility. It is false humility if it has no ameliorating effect on professional arrogance. Please, do not just restate Birkhoff.

I contend there is a loophole here or a gross misinterpretation. The consensus interpretation of Birkhoff and of Schwartzschild/Kretschmann cannot both be true at the same time. There must be a measureable consequence of the presence of a singularity that is beyond imaginary untestable gedanken experiments. The test is the hyperbolic gravitational field. It results in a nonzero constant rotational velocity distribution effect in spiral galaxies, ellipticals, globulars and galactic clusters. This is easier to believe than Dark Matter.

The very same phenomena that are used to argue for Dark Matter can be used to argue for the hyperbolic field. So, it is testable. But, how do we choose between them? I think that Occam ’s razor is the principle of choice here. WIMPS and neutralinos and the other oddball particles that have been proposed require ad hoc additions to theories or their complete rewrites. The hyperbolic field is far simpler. All that is needed is acknowledgement that the black hole singularity is unique. No rewrite of GR. No undetectable new heavy particles that get given self-serving, revised, lower detection limits every time they are determined to be really undetectable.

There seems to be a tendency of cosmologists to think inside the box. They never really consider anything that is outside the consensus. So too do journal editors rely on  conventional wisdom. They would all have been supremely comfortable with the Pope’s decision to censor Galileo.

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

A potential energy diagram is perfectly possible for a hyperbolic black-hole gravitational field

January 11, 2012

The “normal” Newtonian potential energy diagram derived from the inverse square relation versus the hyperbolic black hole gravitational potential energy diagram derived from the inverse (1/kr) relation

First of all, note that a potential energy diagram is perfectly possible for a hyperbolic black-hole gravitational field. The only trouble is with convention. Normally, one takes potential energy U as U = 0 at r = infinity. But, U keeps increasing forever with increasing r in the case of the HBH field. It does not level off to an asymptotic value. So, we would need to adopt a different convention, with U = 0 at r = 1. Then, we would have to remember that all U computed for the HBH case will need to be multiplied by -1 in order to be consistent with conventional usage.

We could represent how the ultra-massive universe excited inflaton HBH gravitational field collapses or transitions to the conventional inverse square field, thereby donating its potential energy to what is increasingly an inverse square gravitational universe, accelerating its expansion in the latter 2/3 of its evolution. We might use weighting factors. We could use a linear weighting factor or maybe an inverse square exponential form or even an hyperbolic expression.

Using x = r in the above diagram, let us try weight for the inverse square derived contribution, S = % and weight for the hyperbolic contribution, H = 1.00 – S, 0.00 < S < 1.00, so that U = S( -1/r +1) + (1.00 – S)ln(r), then total U will transition smoothly from the HBH hyperbolic potential energy to the inverse square potential. Here, 1 is a translation amount to let the inverse square derived curve superpose upon the hyperbolicaly derived. The potential energy lost by the HBH phase of the universe is made up by a gain in kinetic energy of expansion in the inverse square phase.

Remember, it is legitimate to think of Hubble expansion of spacetime carrying the objects embedded within it as a kinematic growth process. One need not always regard it as a “stretching” of spacetime, though for some other purposes, this may help.

Obviously, the resulting composite curve will have a significant positive slope on the right, connoting Dark Energy. But, the curves explicitly describe Dark Matter. So there is a strong link between DM and DE.

There have been misstatements and misinterpretations of Birkhoff’s Theorem. For instance, it has been shown by Kristin Schleich and Donald M. Witt (“A simple proof of Birkhoff’s theorem for cosmological constant”, arXiv: 09084110v2, 27oc09) that Birkhoff does not demand staticity in spherically symmetric solutions to Einstein’s vacuum field equations. Static solutions have heretofore been thought of as being required. There may be other misstatements and misinterpretations that are not yet recognized.

For instance, Birkhoff’s Theorem must actually leave black hole singular gravitational fields as an exception to the commonly quoted rigid rule that only asymptotically flat (commonly assumed meaning: inverse square) gravitational fields are allowed. Otherwise there is no way to measure or unequivocally determine that the center of a black hole is a singularity since electric charge and gravity are the only items the influence of which can escape the interior of a black hole.  Then the theories of Schwarzschild and Kretschmann that say such singularities are physically real are largely meaningless as unfalsifiable hypotheses.

There simply must be a measurable consequence of a true singularity at the center of a black hole or else its existence cannot be postulated. That the mathematics seems so very precise is not good enough. There must be a way to experimentally verify or falsify the equations.

If the gravitational singularity at the center of a super-massive galactic black hole results in a hyperbolic gravitational field, there is a way. By measuring the velocity distribution of stars in the surrounding galactic disk, it can be determined if they move with a constant velocity, v = (GM)^½ at large r, as they must if they move in a hyperbolic gravitational field. As a matter of fact, the velocities of stars in spiral galaxies do indeed move with constant velocity at large r. This can be seen as proof of a singularity at the center of a spiral galaxy’s black hole.

We often state that “The laws of physics must break down at the incredibly tight curvature of spacetime near the singularity of a black hole”. What does this mean? One thing it could mean is that Birkhoff’s Theorem breaks down too. The metrics to which Birkhoff applies probably are not strictly valid near the singularities that they themselves predict so, the “asymptotically flat” dictum may not be strictly true either. Otherwise, our cautionary statement is meaningless.

Besides, the intense benefit brought by the postulate of a hyperbolic galactic super-massive black hole gravitational field is too great to be ignored. It explains the anomalous stellar velocity distributions in galaxies, anomalous velocity distributions in galactic clusters, galactic lensing phenomena, temperature distributions within galaxies, Bullet Cluster type apparent offsets in the barycenters of colliding galaxy clusters, etc. It does everything that Dark Matter is supposed to do! So, Dark Matter is an unnecessary complication that violates Occam’s Rule.

Cosmologists will not like this idea. The LCDM model would have to be drastically revised. The consensus would have to change. Since journal editors and referees endorse only papers that conform to the consensus (they would have been comfortable with the Pope’s decision to censor Galileo), no-one will publish a paper that challenges the commonly accepted interpretation of Birkhoff’s Theorem. And, if Birkhoff does break down in the vicinity of a gravitational singularity, how can it ever be proven? One would have to develop a whole new physics of the ultra strong curvature near black hole singularities. Unless it also had consequences outside the black hole, such a theory could not be falsified so, it could not be admitted as a part of science.

Catch 22 says “Anyone who wants to get out of combat duty isn’t really crazy.” Hence, pilots who request a mental fitness evaluation are sane, and therefore must fly in combat. We would be better off flying in combat duty than trying to fly the hyperbolic super-massive galactic gravitational field into some journal’s pages.

The amazing thing is that the HBHG field does have consequences outside the BH and the event horizon.