Posts Tagged ‘gravity’

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 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

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The Hyperbolic Black Hole Galactic and Universe Gravitational Field

February 4, 2012

The Hyperbolic Black Hole Galactic and Universe Gravitational Field

Figure 1   Proper Time versus Scale Factor a(t) or Hubble Distance, R and also versus Potential Energies, Expansion Velocity, and Acceleration with Dilated or Reduced “Root” Time or “Relativistic Time”

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 image #100 in this series

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Graph computed from equations given in Fig.2 for Hubble expansion of the universe in extensive units. With a proportional overlay of the associated potential energy state diagrams.

                Busy, Busy, Busy

This overly complicated ugly graph is really quite simple. The “underlying” set of curves, for which the legend applies, are composed as below and in

https://garyakent.wordpress.com

 First:

The “underlying” set of curves, for which the legend applies, are composed as follows:

1.) The straight black diagonal line represents expansion of the universe if it had occurred at the speed of light.

2.) The green curve represents the exponential expansion of the universe according to equation 3 against proper time, t, and the extensive variable R, below. It rises almost straight up at very very small values of t, then rises more slowly, nearly leveling off; then it rises again at a more sedate rate after about 2×10-14 u (time in geometric or natural units).

3.) The deep red curve refers to the slope of (2.), the velocity of expansion, i.e. the first derivative in units of U/u. It declines very steeply from an apparently infinitely high level very very early, passing through a minimum at extremely small t = 2×10-14u, whereupon it rises monotonically as shown. This is a unique and very fortuitous feature of this relation.

4.) The purple curve is supposed to represent the acceleration of (3.), i.e. the second derivative.

5.)  The sky blue curve represents “root time”, “reduced time” or “relativistic time”, t1, the (inverse e)th root of proper time, t where t1 = t(1/e)

An unexpected interaction between t1 and the rest of the exponential form produces a fortuitous minimum in expansion velocity, dropping to near zero when t is very small (which occurs much too close to the origin to show), which is a crucial feature of this graph (see the relevant plot in the image series, #97 at Gak on FotoThing.com).

This is due to the odd way that exp(B*t1) behaves at very small t. Otherwise there would be no time for the essential equilibration of temperature and density that is postulated by Alan Guth’s theory. Without such a peculiar minimum in the rate of expansion, because of how t1 works, there would be no initial exponential induction period as is assumed by Guth. If the parameter, e, is adjusted so that Hubble expansion decelerates overall, having negative or concave curvature, no such minimum in the expansion rate occurs at small values of t.

The curve (5.) is the key to the exponential equation and is the secret of why it works. Perhaps this reduced or root time may represent how proper time is vastly dilated especially as it rises from an ultra-massive physically real singularity at our initial proper time, when t = 0. For the inflaton particle is postulated by Guth to be a humongously strong gravitational point particle in the meta-time of a multiverse.

Now, this curvature parameter, e, can be adjusted to describe an open, flat or closed universe. It can be adjusted to show much less curvature or much more. So, the position and duration of the minimum that occurs very early for curve (3.) can be modulated.

But, changing parameters A or B will make (2.) completely miss passage through the point (1,1) on the graphical grid. This would not work at all because the universe with all its matter and energy must have mass/energy M == 1 µ at t = 1 u. So, here is another label for the abscissa.

When the vertical axis is interpreted as being the scale factor a(t), then the horizontal axis must be interpreted as having proper time t = 1u = 27.44 billion years (at least) because from the time of emission of light that became the CMB, to 13.72 billion years (until now, the Hubble time), our universe has expanded another 13.72 billion years (at least). If we insist that the horizontal axis t = 1u = 13.72 billion years, only the Hubble time itself, then the Hubble distance is only 13.72 billion light years or R = 1 U on the vertical axis.

Note that herein R = r, interchangeably. It does not matter how the axes are interpreted as long as one remains consistent.

The author worries that expansion velocity accelerates beyond the speed of light too soon. After over half of the universe lifetime from this point, by now we should have lost contact with the CMB. This can probably be fixed by choosing the parameter, e, so that the curvature of expansion in (2.) is rather a lot shallower. This should also move the point where its slope exceeds c by quite a bit to the right. Then, its effect could be viewed as being more benign.

According to Guth and the consensus of cosmologists and other astrophysicists who truly respect Inflation Theory, the universe was once a purely quantum entity. It still is. The very success of quantum theory is evidence that ours is a quantum universe. Why should the universe not follow a mathematically defined trajectory like this?

Second:

Overlain upon this graph is another graph with three more curves, a black, an orange red and a bright blue one.

orange red: the curve for potential energy (P.E.), from a graph of y = ln(x) representing the integral of 1/kr, where k = 1 m and x = r = t, representing either the Hubble time or the actual age of the universe (more or less).

Here, y is the vertical axis which is to be read as P.E. of the “inflaton” (or galactic) hyper-excited gravitational field or as whatever else may be the correct quantity, in natural units, depending upon which curve one is reading. Ideally, this ln(x) integral denotes the initially (near the origin) extremely high relative P.E. condition of the hyperbolic ultra-massive black hole gravity.

Hyperbolic gravity fields are allowed by GR if proper assumptions and boundary conditions are posited to find a metric, much like the Schwarzschild metric, for the space wherein is assumed to reside a singular black hole.

That is, this curve represents the “state” of the universe’s initial hugely massive “inflaton” point particle and its associated “inflationary” hyper-excited renormalizable 1/kr gravitational field. Every field always has an associated particle, so there would have been an inflaton particle and it should have been an hyper-massive or “excited” quantum point particle already possessing that “renormalizable” higher energy hyperbolic gravitational field. How could it have been else in the Everett multiverse which Guth explicitly posits?

Now, there is no question that, if one accepts Inflation Theory then, one must accept the multiverse and meta-time.

The author compares the implied transition from an evolving higher energy gravitational field state toward a changing ground state to a time dependent quantum transition or to a Tanabe-Sugano diagram in transition metal ligand field theory.

B.)   black: the curve from y = -1/x, the lower energy P.E. state of the universe under the ground state of its normal gravitational field, which was proportional to 1/r2.

P.E. being the vertical axis now, as in (A.), it is equal to -1/x or -1/r or -1/t (because the scale has r = R = t = 1 since this is all in natural or geometric units). The ln(x) = ln(r) = ln(t) curve and the 1/x = 1/r = 1/t curves have been translated so that P.E. (A.) = P.E. (B.) when ordinate x = r = t approx. = 0.34 and then the whole graph was re-scaled and vertically re-positioned to fit other abscissa or P.E. versus t constraints.

Originally, this point was (1,0) or the intersection of both curves, where

ln(x) = 0 and

-1/x +1 = 0

got from a pair of equally scaled graphical curves adapted from figures in a textbook definition of the natural logarithm.

This was not an arbitrary adjustment. It was done so that the orange red ln(x) curve approaches the origin at the same time that the sum of the (A.) orange red, and (B.) black, curves equals the bright blue curve of (C.) as it passes through the point (1,1).

Then, the intersection of (A.) & (B.), at the bright green circle, which used to be at the point (1,0), had to be translated so that its position with respect to the underlying Hubble time axis corresponded to 0.344 u, or 9 billion years ago in Hubble time, when the universe is seen to have begun to “re-inflate”, consistent with observations of “acceleration”.  So, there is only one way that the overlay curves could have been re-scaled and repositioned to meet these constraints.

bright blue: the superposition or linear sum or mixture of the gravitational potential energy states represented by curves (A.) and (B.).

Now, as was said before, (C.) had to be made to pass through the point (1,1) on the underlying graphical grid. That is, the total P.E. had to equal all the matter and energy in the universe at t = 1, the present, including “Dark Matter” and “Dark Energy”, so that M == 1µ at t = 1u. So, total mass/energy M is yet another label for the abscissa.

The orange red curve in (A.) is identified with Dark Energy, and it is seen to keep on increasing into the future while the associated scale factor, which is to be read on the abscissa as R or a(t) in this case, also increases. Then, with these simultaneous increases, the P.E. density of the universe remains constant as is indeed postulated for Dark Energy.

At the instant of the BB, the hyperbolic inflationary gravitational inflaton field of the inflaton point particle had a potential energy curve that would have looked like the orange red curve in (A.) but, no matter/energy had yet had a chance to follow this P.E. curve at the instant that the BB occurred. And, afterward it might have been forced to follow the black curve in (B.). The matter/energy in the universe could not actually experience any of the individual states represented by (A.) or (B.) but could experience only the superposition of states in (C.), bright blue.

Still, it is all consistent. The orange red P.E. curve continues to increase as it should and the black P.E. curve becomes “nearly” constant as it must while the bright blue total P.E. curve increases as the energy density of the universe must remain constant, just as theory demands.

See? Simple. (Yuk! Yuk!)  By comparison, this should make filing an income tax-return seem like a piece of cake.

This is what time dependent quantum transitions mean. This is what the multiverse means. We can experience only the final resultant of the waveform vector sets for all terms in the total probability density wave sum. We experience only the superposition, not the separate states.

Yet, some cosmologists describe Inflation Theory minus the point particle concept and sans a multiverse with no meta-time and also without the implication that the inflaton field must be an excited renormalizable gravitational field with its associated hyper-massive particle. So, the quantum nature of Inflation seems foreign. One can pick and choose the ideas one likes only in regard to congressional legislation in a hidebound committee but, Inflation Theory will never become that kind of law.

If the hyperbolic black hole gravitational field can be validated and extended to the entire universe this way, then we would have hard evidence for a kind of a multiverse.

Figure 2   Rules Sheet from TK Solver Plus for the “e-Model” of Inflationary Expansion of the Universe

The graphical series in Fig.1 was computed using the equations presented in this rules sheet.

#98 A  Equations for the mathematical model of inflationary Hubble expansion of the universe according to extensive variables.

 98 A  universe accelerating hubble expansion guth inflation

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image 98A  in this image series

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This is the rules page from a UTS TK Solver Plus math program that was used to plot the exponential expansion curve shown in image 96. It depicts acceleration of Hubble expansion, the 1st and 2nd derivatives of this curve as well as a straight diagonal line showing a baseline of what expansion would look like if it occurred at the speed of light. Image 95, at FotoThing.com under Gak, shows a minimum in the 1st derivative curve, the expansion rate. The rate drops to near zero, indicating an extreme slowdown in expansion that constitutes a virtual pause at around 1^-14 to 3^-14 u or “universe time”, time in “natural units” or “geometric time”. This period lasts around 2 x 10^-14 u or 8,660 seconds (144 minutes) but is sensitive to somewhat arbitrary initial conditions like those chosen by Alan Guth in his first paper on inflation. This pause may have come earlier or later and lasted longer or shorter depending on these initial conditions. Such changes would have to be pairwise and in the correct sense or else the intersection with (1,1) will be lost.