Alan Guth's Inflaton & Mond

[G Anthony Kent]

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 = R₀e^rt, 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.

[G Anthony Kent]

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 a_N. The variant acceleration due to MOND expresses a_N 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 = a_N = 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 v_MOND 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, v_MOND = v_HBH. 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.

[G Anthony Kent]

 

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.

 

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.

[G Anthony Kent]

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

[G Anthony Kent]

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 = (GM_bh)^1/2. That is v becomes constant dependent only on M_bh 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 M_bh 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.

[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*M_bh/r*)^½ where G* is the 2-D gravitational constant and r* is the unit vector of r, for dimensional integrity.

 

M_bhs 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 M_bh should be constant also. But this cannot be so. Then, M_bh 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 M_bh 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/M_bh. But, this implies that v and M_bh 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

Edited March 27, 2013 by Gak