Spinning supermassive black holes part 1

Be sure to see the follow up paper
http://garyakent.blogspot.com/2014/03/galactic-gravitational-fields.html 

Spinning supermassive black holes: Dark Matter and MOND

G  A  Kentgen[1]

Abstract 

We rely on the formal definition of a logical postulate. A spatially two dimensional (2-D) hyperbolic (1/r) gravitational field (HGF) is postulated for all black holes (BHs), particularly for galactic central supermassive BHs (SMBHs). Virtual singularities are assumed to exist at the centers of these BHs and the nature of spacetime under the event horizon (EH) is deduced consistent with general relativity theory. Spacetime is posited to undergo centripetally induced phase change below the EH where it morphs to a “singular”, nominally flat or 2-D, gravitational “spin induced” thin platter with virtually infinite radius. The BH mass remains below the event horizon. This spacetime entity’s formal “spin” is coaxial with the overall orbital spin of the galaxy and of the whole central SMBH. The SMBH 2-D G* field declines as 1/r. (Distance is r, from the BH center.) Since this is a hyperbolic field, ordinary parabolic inverse ‑ square (1/r²) declining G may not be in effect for stars in the galactic bulge of spiral galaxies. G* affects all stars in the galaxy all the way to the galactic periphery and far beyond. Responsible for the anomalous velocity dispersion (AVD) observed in galaxies and galactic clusters, it actually becomes an asymptotic constant within a galaxy with uniform velocity dispersion v_o, dependent only upon black hole mass, M_bh, much like the M-sigma relation. In 2-D, a constant orbital velocity dispersion, v_o, implies asymptotically constant orbital radial acceleration, a_o, like MOND. The HGF complex surface shape is a hyperboloid of one sheet (at very large r). Such significantly more intense long range fields embedded in 3-D spacetime allow the hyperbolic fields of nearby galaxy central SMBHs to interact preferentially to form a complex network or 3-D array. Contrary to the current consensus and helping to evolve large scale structure, this also would allow creation of a facsimile of “Dark Matter” (DM) halos around galaxies and clusters so that it would account for all the phenomena attributed to DM, including the AVD. The posited 2-D hyper-excited SMBH gravitational field, if logically extended to the primordial ultramassive BH, to the inflaton point particle, accounts for Dark Energy as well.

Summary

This postulated (tentatively defined) premise[2] proffers DM, DE and elements of MOND without recourse to unwelcome modifications of general relativity, to superstring theory, to supersymmetry or to so-called M-theory. The postulated hyperbolic or inverse (1/r) gravity has major consequences for black holes. So envisioned here, with references, is the “why and how” 2-D gravity can actually exist[3], according to general relativity.

In a very rapidly spinning black hole where mass-energy has collapsed to a virtual singularity beneath the event horizon, the spin rate implies orbital rotation of the black hole material that requires it to move faster than c. Only the z dimension has shrunk à 0, according to the Postulate. Because all mass-energy entering a galactic central black hole must initially have had a nonzero, effectively co ‑ moving orbital velocity component, all such material must orbit at faster and faster speeds which advance toward infinity as the orbital radius shrinks[4] and a singularity is approached. Net orbital and rotational radii must shrink due to chaotic interactions and/or simple gravitational effects. “Orbital friction”, mutual tidal effects and even relativistic properties may also play a large role. 

Such in-falling material must orbit in concert. That is, initially, it all must orbit in roughly the same direction so that as its orbital radius shrinks the BH can be thought of as if it was rotating faster and faster as a whole. As it shrinks due to “orbital friction”, chaotic interaction and relativistic effects, it acts more and more like a single rotating body. Then, rotating faster than c below the event horizon, the nature of spacetime itself must transform. It undergoes a centripetal “phase change” [5],  [6].

This is truly plausible. What else can “plausibility” mean when we say GR and the known laws of physics break down at a singularity, quantum gravity notwithstanding?  But, the Postulate implies that we can nonetheless deduce what is going on at or near the singularity. Thus, GR is rendered “UV complete” because the 2-D (and 4-D) gravitational fields are quantum renormalizable states [7]^{, [8], [9]} and applicable to the highest energy regimes (see below) at the most extreme values of independent and dependent variables. So, the 2-D G* metric below the event horizon and even far above it is Minkowski asymptotically flat -- a pure spacetime entity in its own right[10].

According to our Postulate, collapse in the z dimension means that mass-energy-spacetime becomes unified in a new way. They form, not a ring singularity á la Kerr, but an indefinitely broad disk singularity that is singular only in the z direction and nominally infinite in the x, y directions due to centripetal and deeply hyper-extended pure relativistic frame dragging effects[11]. So, as part of the Postulate for which disbelief is to be temporarily suspended, this mass-energy-spacetime entity still exerts a far reaching 2-D gravitational field because the spacetime component of its nature can ignore the event horizon while retaining all its mass below it[12].

Gaining “virtually[13] infinite” implied mass density and nominally infinite gravitational field strength, G* nearer the singularity gives a black hole gravitational field strength diagram that changes proportionally with 1/r. Moving to the left in the diagram (Fig. 1), it must tend to infinity becoming a not quite flat curve, that is, it must approach the ordinate as an effective asymptote as r declines and other measures of mass-energy density go to extremes. That is, gravitational field strength progresses toward infinity as the diagram’s vertical axis is approached from the right, or with decreasing r, becoming almost perfectly constant. This behavior has quantum implications, 2-D gravity being renormalizable.

By the Postulate, G* indeed becomes “virtually infinite” and also asymptotically flat as the gravitational radius of curvature, r, approaches zero. Simultaneously, this 2-D G* field strength diagram must be flat at the horizontal axis too, only very slowly approaching a virtually constant value (a_o) as r increases without bound. The exact average value of the eventual approach to zero, as an asymptote, is MOND’s a_o.

This is the symmetry of a hyperbola, a gravitational field that declines as 1/r. Einstein calculated a gravitational constant, G (he called it K)[14], for the inverse square (1/r²) gravity field but he did not bother with “G*” for the 1/r case. This is left for a future paper.

This new form of G is among the factors that may cause an underestimate of the matter/energy present in galaxies and in the universe as a whole because this hyperbolic (1/r) supermassive black hole gravitational (HSBHG) field is never used. Unadapted for inverse (1/r) gravity, Kepler’s laws and Newton’s law of gravitation are always used as is, but sometimes corrected for relativistic effects. This leads to the premature conclusion that there must somewhere be more matter‑energy in the system in order to reconcile results with exhaustive total matter-energy audits, say, for a galaxy. This “more” is called Dark Matter (DM).

Vera Rubin, an astronomer at the Carnegie Institution of Washington, Department of Terrestrial Magnetism discovered the galactic anomalous velocity dispersion (AVD) in 1960 to 1980, using a new much more sensitive spectrograph to measure the velocity curve of edge ‑ on spiral galaxies far more precisely than ever before. Coworker Kent Ford and she declared at a 1975 meeting of the American Astronomical Society their observation that most stars in virtually all spiral galaxies orbit across nearly the whole width of the disk at about the same speed^[15] except nearest the center. One way to account for this AVD is to postulate that a “halo” of DM surrounds and suffuses galaxies, the halo having a radius far larger than any galaxy.

Profound are the consequences of a close match (shown below) between exemplar simple Milky Way mass, M_mw, central supermassive black hole mass-energy, M_bh and total mass of the Milky Way, M_mw + M_bh, found using the HSBHG field Postulate. Concurrent calculations with conventional methods that may even assume Dark Matter are vastly different.

The 1/r relation means that the gravitational field strength falls off far more slowly than the classical Newtonian field and accounts for not only the anomalous velocity dispersion itself, but also the M-sigma effect and all the other phenomena associated with Dark Matter. Appropriating all their supporting data, because of its association with black holes, this hyperbolic (1/r) supermassive black hole gravitational field IS Dark Matter. The implication is that maximally spinning black holes are the ultimate source of DM.

The 1/r gravitational potential energy profile follows a logarithmic relation. Moving from left to right in the usual diagram, P.E. rises as ln(r), while the 1/r² potential energy follows a hyperbolic profile, rising as 1/r. When they are placed on the same scale and plotted in natural or geometric units, they can be superposed so that the ordinate and the values at r  =  1 coincide. These superposed equivalent potential energies can be thought of as depicting the gravitational potential energy profiles of the present universe and also that of the primordial black hole (the inflaton particle).

Figure 1 depicts these gravitational potential energies. The P.E. of the primordial singularity is plotted together with that of the present day universe. The diagram portrays the hyper-gravitational field of the inflaton particle in Alan Guth's hypothesis of a highly excited inflaton field of the false vacuum. The inflaton field is here proposed to be just an enormously excited field state in the form of simple 2-D hyperbolic (1/r) gravitation.

According to Figure 1, beginning with the Big Bang (BB) at r = 0 (r = x = 0 in Fig.1), the time dependent quantum subsidence of the inflaton field donates its potential energy to the unexcited "ground state" as a function of t, it “collapses” into the 3-D gravitational field that we know. Of necessity then, there comes a time early on when both curves in the P.E. diagram simultaneously become equal to 1 (in natural or geometric units). They converge.

Thereafter, the disintegrating excited 2-D gravitational field begins to re-assert itself and again starts to fail from higher and higher energies, falling into the ground state field. This new divergence results in the fact that objects in the universe are once more observed to appear to move kinematically farther and farther apart and at an accelerating rate. It is said that we may as well regard expansion as a kinematic process because it avoids confusion and it may be meaningless to maintain that it is not.[16]

The Postulate

A postulate, a temporary proposed premise, must be provisionally accepted as is. The truth or falsity of a postulated premise must be deduced from experimental fact, not mere opinion. The components of the premise presented here cannot be argued away piecemeal because they constitute a single logical hypothesis that is falsifiable only as a whole. Only objections or reservations that are themselves falsifiable will be viewed as valid points of order requiring an answer.

The following points are part of the Postulate.

1.)       In a very rapidly spinning black hole where mass-energy has collapsed to a virtual singularity beneath the event horizon, the spin rate implies orbital rotation of the black hole material that requires it to orbit faster than c. In Cartesian coordinates, only the z dimension has shrunk, z à 0, according to the Postulate.

2.)       Because all mass-energy entering a galactic central black hole must initially have had a nonzero, effectively co ‑ moving orbital velocity component, all such material must orbit at faster and faster speeds which advance toward infinity as the orbital radius shrinks and a singularity is approached.

3.)      Then, rotating faster than c below the event horizon, the nature of spacetime itself must transform. It undergoes a centripetal “phase change

4.)        Collapse in the z dimension means that mass-energy-spacetime becomes unified in a new way. They form, not a ring singularity á la Kerr, but an indefinitely broad disk singularity that is singular only in the z direction and nominally infinite in the x, y directions due to centripetal and deeply hyper-extended pure relativistic frame dragging effects. So, as part of the Postulate for which disbelief is to be temporarily suspended, this mass-energy-spacetime entity still exerts a far reaching 2-D gravitational field because the spacetime component of its nature can ignore the event horizon while retaining all its mass below it.

5.)       Gaining virtually infinite implied mass density and nominally infinite gravitational field strength, G* nearer the singularity gives a black hole gravitational field strength diagram that changes proportionally with 1/r.

6.)      Gravitational field strength progresses toward infinity as the diagram’s vertical axis is approached from the right, or with decreasing r, becoming almost perfectly constant. This behavior has quantum implications, 2-D gravity being renormalizable. G* indeed becomes “infinite” and also asymptotically flat as the gravitational radius of curvature, r, approaches zero. Simultaneously, this 2-D G* field strength diagram must be flat at the horizontal axis too, only very slowly approaching a virtually constant asymptotic value (a_o) as r increases without bound.

7.)    This is the symmetry of a hyperbola, a gravitational field that declines as 1/r. Einstein calculated a gravitational constant, G (he called it K)[a], for the inverse square (1/r²) gravity field but he did not bother with “G*” for the 1/r case. A hyperbolic (1/r) gravitational field is permitted by GR if the spatial dimensionality is restricted to 2. For the gravitational field G, gravitational field strength, F  µ  1/r^(n-1) , declines with n = 2.

8.)      This new form of G is among the factors that may cause an underestimate of the matter/energy present in galaxies and in the universe as a whole because this hyperbolic (1/r) supermassive black hole gravitational (HSBHG) field is never used. Unadapted for inverse (1/r) gravity, Kepler’s laws and Newton’s law of gravitation are always used as is, but sometimes corrected for relativistic effects. This leads to the premature conclusion that there must somewhere be more matter‑energy in the system in order to reconcile results with exhaustive total matter-energy audits, say, for a galaxy. This “more” is called Dark Matter (DM).

9.)      The 1/r relation means that the gravitational field strength falls off far more slowly than the classical Newtonian field and accounts for not only the anomalous velocity dispersion itself, but also the M-sigma effect and all the other phenomena associated with Dark Matter. Appropriating all their supporting data, because of its association with black holes, this hyperbolic (1/r) supermassive black hole gravitational field IS Dark Matter. The implication is that maximally spinning black holes are the ultimate source of DM.

10.)    Part of the Postulate is that all cosmological measurements (made at great distances, reaching into the deep past) must conform to an extended version of the Heisenberg uncertainty principle, which we will call the cosmological Heisenberg uncertainty principle (CHUP). At the telescopic measuring instrument, the energies involved are just as small as the energies of, say, subatomic particle decay measured in accelerators or of absorption lines in spectrometers.

11.)       The shape of the BH must have flattened, mainly by centripetal acceleration, neither to a mere point mass nor to a simple ring singularity á la Kerr^[a], but to an axially contracted vastly distorted entity that subtends an infinitely broad planar subset (parcel) of excited spacetime that is only two dimensional (2-D). It must have become an indefinitely huge, nominally flat, 2‑D spacetime disk embedded in 3-D spacetime. It became a singularity in only one dimension but, its spacetime nature being immune to the limitation of the event horizon, it must still have retained all its mass and had a gravitational field that could reach out indefinitely far. Its mass alone remained subject to the event horizon but, its hyperbolic (1/r) gravitational field was not so limited.    

Relying on the formal definition of a postulate, it requires the existence of no exotic Dark Matter subatomic particles. No superstring or supersymmetry theory is needed to explain Dark Matter or even Dark Energy, for that matter. But, a quantum universe is implied with a Hugh Everett style Many Worlds[17] interpretation. Thus, part of the Postulate is that all cosmological measurements (made at great distances, reaching into the deep past) must conform to an extended version of the Heisenberg uncertainty principle, which we will call the cosmological Heisenberg uncertainty principle (CHUP)[18].

Then, the hyperbolic gravitational field postulate can not only explain all the phenomena of Dark Matter, by extension, it explains all the phenomena of Dark Energy also. The hyperbolic gravitational field IS Dark Energy too. It is a version of quintessence.

Therefore, it is “postulated” that supermassive black holes (SMBHs) must develop incredibly high spin rates that would far exceed the equivalent of c below the event horizon^[19],[20]. Exceeding c might merely reverse the priority or order of energy level hierarchies in the parcel and this will reverse[a] the flow of time within it[21] from an exterior frame. This may be just a kinematic side effect of the BB itself as would be the tendency toward increasing entropy[22].

Confined beneath an event horizon, SMBHs must have condensed and evolved out of galactic stellar components. These must have orbited “in concert” within the innermost shell or stratum of the bulge of spiral galaxies. Due to orbital rotation centripetally culminating with this ultra high spin rate, it “nears” infinity below the event horizon. As the singularity is approached, matter in such BHs never has a chance to infinitely compactify in all dimensions. A BH could not actually continue to shrink to form a nominally 0-D singular Planck point with infinite density. (Planck density, ρ_p  ≈  ∞, “as high as may be needed to explain effects”; the real meaning of infinity.) This implies uncertainty and is another reason to include the CHUP in the Postulate[23].

Instead, it must have flattened, mainly by centripetal acceleration, neither to a mere point mass nor to a simple ring singularity á la Kerr^[24], but to an axially contracted vastly distorted entity that subtends an infinitely broad planar subset (parcel) of excited spacetime that is only two dimensional (2-D). It must have become an indefinitely huge, nominally flat, 2‑D spacetime disk embedded in 3-D spacetime.

It became a singularity in only one dimension but, its spacetime nature being immune to the limitation of the event horizon[25], it must still have retained all its mass and had gravitation that could reach out indefinitely far. Its mass alone remained subject to the event horizon but, its hyperbolic (1/r) gravitational field was not so limited.

We shall call this “The Postulate”, see equations (1 & 4) below.

The Dark Matter Effect

This postulated premise guides us to a certain extraordinary connection that we deduce (eqs. 1 to 4), between the anomalous velocity distribution (AVD) of stars in spiral galaxies and the mass of galactic central supermassive black holes (SMBHs). Constant “radial” velocities or limiting orbital velocities, v_o of outer stars, those well beyond the bulge, should be seen to correlate accurately with the mass of the central supermassive black hole in such galaxies, at least once these velocities are measured more precisely[b].

With hyperbolic G*  =  2.128 x 10^-32 m³kg^-1s^-2  so small (see below), whereas classical G[26]^,[27]  =  6.67384 x 10^-11 m³kg^-1s^-2 [c], there is so much variation in measurements of constant velocity dispersion, v_o, even for the same galaxy, that there is ample room for this predicted future observation[28]. Thus, the Postulate implies that the AVD characteristic velocity is not truly constant, galaxy to galaxy[29], but varies as in eq. (1) given below[30], but it may approach a virtually constant value near an hyperbolic asymptote. 

The anomalous velocity dispersion is thus proposed to be like the so-called M ‑ sigma^[d] relation^{[31], [32]} which is also impacted by the Postulate.

On the other hand, it is said that there should be no such long range AVD correlation if gravitational force for galaxies with central supermassive black holes and big bulges is an inverse square force^[33]. Of course, the answer to this is: “It isn’t.” It must be a hyperbolic 1/r force, which unites M-sigma and AVD[34].

As far as M-sigma is concerned, no entirely plausible detailed mechanism has been proposed that explains exactly how the initial formation of a galaxy and its supermassive black hole could translate to or correlate with “σ” or v_σ of stars in the bulge. Even bulges with mini spiral structures have a small core having characteristic chaotically orbiting stars^[35], allowing for the required statistical treatment. It is the chaotic nature of inner bulge star orbits that gives rise to statistical M-sigma. Such chaos near the central bulge is innate and natural.[36]

With millions of stars in the bulge, this M-sigma relation has its root in the virtually continuous Gaussian distribution of v in this region and its natural connection to statistical σ, the standard deviation[37].

Why is it not self evident that the larger the central SMBH has grown, the larger the bulge may be and thus the larger the galaxy and so the wider would be σ and the larger would be v_σ ? If one concedes that we are dealing with a Gaussian or similar velocity distribution in the bulge, a natural connection between M_bh and velocity dispersion “σ” or v_σ is automatically implied. The question is: what is the nature of this connection?

Furthermore, as far as the anomalous velocity dispersion is concerned, no exotic Dark Matter (DM) is needed to explain the implied very long range action of this postulated, extensive, singular, spacetime spin-disk phenomenon. Yet, a “DM effect” is still implied by the Postulate, but in a radically unconventional way.

Because by extreme contraction of matter and spacetime to “virtually” a plane 2-D surface comprising a local nominally flat singularity with “virtually” infinite density, the gravitational force near the center is so intense that it’s mathematical behavior must be Minkowski or relativistically asymptotically flat there and at the other extreme of r[38]. Its gravitational strength profile must be asymptotically flat near the abscissa and ordinate, with spacial curvature, r = x in Fig.1, tending to zero at the extreme left, as an asymptote. And, v_o tends to a small virtual constant at the other extreme, at low curvature or large distance, as r → ∞.

This symmetry not only means that G* is hyperbolic (declining as 1/r) in nature but, Birkhoff's theorem^[39] cannot apply due to the fundamental gross limitations of his assumptions. If accurate, the gravitational strength diagram implies that general relativity should allow a gravitational field that declines as 1/r. But, this works well for our purpose here only in 2-D spacetime^[40], GR not permitting an inverse (1/r) gravitational field in three spacial dimensions.

Yet, we quote Milgrom^[41]:

“But, exactly which system attribute makes the difference? Galactic systems have masses, sizes, and angular momenta that are many orders of magnitude larger than those in the solar system. The large distances involved are a natural culprit. Indeed, there were attempts to modify the distance dependence of gravity: the gravitational force is still taken as proportional to the two masses involved but the decline at large distances is not as strong as in the r ^-2 law. Such a modification cannot, however, explain away dark matter. If the modified law is to produce asymptotically flat rotation curves of disc galaxies, as observed, it automatically predicts the wrong form of the mass velocity relation: it gives M proportional to V² instead of M proportional to V^α, with α = 4, as required by the observed Tully-Fisher relation (Milgrom 1983).”

Some believe the simplistic models that are being marketed in the academic bazaar. Tulley-Fisher, because it is a statistical correlation with observations, empirically takes into account the fact that galaxies probably contain many millions of large black holes. Hence the V^α, with α = 4 (or 5), dependence. Milgrom’s quote refers to an hyperbolic (1/r) 2‑D gravity which does not take into account embedded black holes explicitly and automatically.

Apparently, in order to take into account the embedded black holes in any galaxy, M_bh must be proportional to (V²)²:

1.)     v_σ² =  (G*M_bh/r*)² = (V^α)² versus  Eq. (3) below. Obviously, this impacts the form of the T-F relation as a function of v_σ .

Then 

2.)    a_σ²  =  G*M_bh/r*r  with near hyperbolic asymptotic value proportional, if not equal, to Milgrom’s a_o     

Example Calculations, AVD

The outer stellar limiting orbital velocity may be, for instance (for the Milky Way, MW),

AVD  v_o^mw = 230 km/s [42]^{, [43]},  M_bh^mw  =  4.1x 10⁶ M_ʘ [44] , M_ʘ  =  1.9891×10³⁰ kg [45]

If G is not correct for the 2-D case, we use a new version of G for the hyperbolic field. We may call it G*. (Note that v_σ^mw in the M-sigma relation for the MW is about 103 km/s[46]. It is interesting that these numbers are the same order of magnitude and may agree perfectly if v_o was better determined or if v_σ was properly redefined.)

The galactic Hyperbolic Supermassive Black Hole Gravitational Field (HSBHG field) conforms to an extensive 2-D “disk” singularity that has dimensionally contracted along the “z” coordinate. A peculiar result of this premise is that for stars somewhat nearer to the galactic periphery, but beyond the bulge, AVD stellar velocity dispersion[e] must be

(1)        v_o^avd =  (G*M_bh/r*)^½    

 (1a)      F*  = G*M_bhm_star/r*r    and      a_o  =  v_o²/r*

from Newton’s law of gravitation adapted to a 2-D parcel of spacetime (the main HSBHG-field Postulate). The unit vector, r* serves to preserve dimensional integrity. There have been many studies concerning 2-D spacetime^[47]. The idea is not new.

Rearranging eq. (1)

(2)     G*  =  v_o²r*/M_bh 

 

And, for the Milky Way:(2b)   G* =

(230,000 m/s)²(1 m) / ((1.25 x 10¹² M_ʘ)(1.9891 x 10³⁰ kg/M_ʘ))

               =  5.29 x 10¹⁰ m³s^-2 / 2.486 x 10⁴² kg

               =  2.128 x 10^-32 m³kg^-1s^-2 

Whereas classical G  =  6.67384 x 10^-11 m³kg^-1s^-2 as given above.

So, the conversion factor     

                 κ  =  G*/G  =  3.1884 x 10^-22 m^-1  and

κ^-1 = G/G* = 3.1364 x 10²¹ m

may be used to convert G into G* and to correct M_bhs and M_galaxys got previously. For all future calculations using the Postulate, a better value for G* might be obtained using successive approximation or else data for many M_bhs and many values of galactic v_os. Einstein did not bother to calculate G*, though he did calculate^[48] G, which he called K. 

An iterative process of successive approximations should probably be used to find G* because using a value for central supermassive M_bh that is determined using conventional methods to calculate G* is inconsistent with the implications of the Postulate. But if a better value for G* is found, it would only increase the disparity between M_bhs calculated by ordinary Kepler’s laws and those calculated using the Postulate.

This theoretical AVD galactic v_o ≈ v_σ. There have been some empirical equations developed that describe the M - sigma relation and Milgrom has proposed what is essentially an empirical relation that describes the AVD wherein he tacks onto Newton’s law an ad hoc constant term involving a_o.

For some empirical M - sigma equations, see ref. (74):

http://garyakent.wordpress.com/2013/07/18/normal-distribution-applied-to-milky-way-galaxy-m-sigma-relation-and-bulge-star-data/   

But, v_o may be modified by the presence of thousands or even millions of large BHs embedded in the galactic disk as well as in the bulge, increasing the effective central M_bh. Many of their spin-disks may rotate parallel with that of the central SMBH and they would have virtually infinite, slowly declining as 1/r, 2-D gravitational fields of their own with nominally constant additive a_o and virtually constant additive v_o. They may thus enhance the 2-D field of the central SMBH.

 

(3)     v_σ =  (G*M_bh/r*)^½            ,            a_σ  =  G*M_bh/r*r     and

                    F* =  G*M_bhm_star/ r*r

 

All stars in the central bulge may be affected by F* because we invoke CHUP to say that all the chaotic stellar orbits in the bulge must be able to experience the 2-D field. If they did not orbit chaotically, but did so as if they orbited together in the same galactic plane, we could measure the position and relative momentum of the BH to less than a factor of some small multiple of ħ/2 (whatever that might be from a cosmological perspective taking the limitations of our instruments into account). An impossibly precise triangulation would result thereby.

Our use of instruments at an extreme of their operating capacity is not the culprit, nor is it a mere artifact or mathematical semantics. Heisenberg uncertainty is a real property of the universe, commanding the attention of the very stars.

But, the adapted AVD formulation (see below)

          (4b)   v_o =  (κGM_bh/r*)^½  AVD   ,     a_o  =  κGM_bh/r*r         

                                  F* =  κGM_bhm_star/r*r

and rearranging (4b), we get   

(4c)  M_bh  =  r*v_o²/κG =  r*v_o²/G*  especially for stars in the

          galactic plane.

If we replace G with G*, as above, because G* is such a small number, by this we compute a much larger central SMBH M_bh that may not at all be very similar to the M_bh got by conventional methods that use un-adapted standard Kepler’s laws (see below).

But, this picture comprises scenario #1 where all disk stars feel the (1/r) gravitational force, not just those that orbit in the galactic plane inside the bulge. An argument can be made for this less restrictive choice via the above mentioned enhanced form of Heisenberg uncertainty[49], CHUP.  A mechanism can be imagined whereby an inverse square force could also be extant, but cannot be felt by disk or bulge stars due to a “gravitational field exclusion principle” (see below). It would be felt far beyond a galaxy neither in the galactic plane nor aligned with the galactic poles.

In any case, under the Postulate, conventional M_bh determinations using standard inverse square gravity and unadapted Kepler’s laws in the M-sigma relation greatly underestimate the mass of central SMBHs.

And, use of the unadapted AVD relation must also underestimate masses of galaxies containing such BHs, particularly since galaxies must contain thousands, if not millions of very large BHs. But, audits of matter and energy cannot account for this diminished contribution which is simply due to this choice of 1/r² versus 1/r and G versus G*. So, to “balance the books” and reconcile with audits of mass-energy, we add a correction we call Dark Matter (DM)[50].

The Postulate aside, it would be invalid to use standard Kepler’s laws to estimate central supermassive BH mass for a galaxy using AVD data directly anyway. This is because the existence of AVD means that ordinary Kepler is not followed. Only hyperbolic (1/r) 2-D gravity might be used directly for the stars in the disk while standard Kepler/Newton might not be used even for the bulge stars.

Application of the Postulate to all bulge stars works better. Chaotically orbiting bulge stars have orbits that may all still respond to the 2-D field if some form of Heisenberg uncertainty requires that bulge stars may not orbit so precisely. The 2-D field is “smeared out”. This would mean that Heisenberg uncertainty is real, not a mathematical artifact nor a mere semantic device.

The M-sigma and the anomalous velocity dispersion effects are seen here to be natural outcomes of the hyperbolic field Postulate. Rather than refer to DM as a real phenomenon, perhaps, it would be better to refer to “the Dark Matter effect”. However, this effect is a real factor in the analysis of the nature of the universe. It deserves just as much, if not more, theoretical and research attention.

We can use the common form of Newton’s law of gravity which is easily adapted to the 2-D case by simple substitution. Then, the limiting orbital velocity attained by peripheral galactic disk stars

v_o^avd  =  v_σ   =  (G*M_bh/r*)^½    which is a main form of the Postulate.

(Remember r* = the unit vector of r, for dimensional integrity.)

And, using the anomalous velocity dispersion, v_o, or even the M ‑ sigma relation which shows a linear plot of M_bh versus v_σ (or v at the radius that encloses ±σ, the standard deviation of velocities of bulge stars chaotically orbiting a central supermassive black hole in a galaxy), we can determine a first approximation to G*. This is essentially what was done above.

We could use current empirical data for M ‑ sigma or for the AVD or both for a comparison. Right now, we shall use v_o for the limiting orbital velocity of stars beyond the bulge, approaching the periphery. It turns out that G*  ≈  2 x 10^‑32 m² kg^‑1 s^-2, for instance, in eq. (2b), when AVD data (given below) for the Milky Way is used.

For now, we will ignore the inconsistency in using a conventionally determined value for M_bh to get the 2-D gravitational value for G*. Successive approximation might be used to zero in on a more precise value. A better value for G* will make the disparity between conventional approaches and the Postulate even greater.

With G* this small, no wonder that the limiting stellar orbital velocity, v_o, for the AVD sometimes seems to be almost constant from galaxy to galaxy to some observers like Milgrom. The Postulate predicts that much more precise and accurate measurements of v_o will find that it is not nearly constant but closely follows the above equation with v_o  =  (G*M_bh/r*)^½  (eq. 4, below) which is a parametric constant within a galaxy and is dependent on M_bh from galaxy to galaxy. But a hyperbolic (1/r) decline means that v_o may approach the same asymptotic value, galaxy to galaxy. So, Milgrom is right, after all, since constant orbital v_o means asymptotically constant a_o especially when expressed in 2-D.

In order to take into account the numerous embedded BHs in all galaxies, perhaps  v_o  =  [(G*M_bh/r*)^½]²  =  G*M_bh/r*

We can measure outer star anomalous v_o to get M_bh with the above equation and then we can use v_σ  from the M - sigma relation for bulge stars in the same above equation to get another fix on M_bh. Unless embedded BHs in the disk get in the way, they should be quite the same if the Postulate is true. We would then see that classical Newton/Kepler vastly underestimates central supermassive M_bh and also the masses of galaxies which may contain thousands or even millions of large black holes.

The postulate yields a central supermassive Black Hole AVD M_bh that seems to equal the mass of the whole galaxy (example 1, below), including the central SMBH. It seems either that the mass of the whole galaxy adds to the hyperbolic SMBH G* field under the rubric of the Postulate because all the stars are caught in the flat 1/r G* field. Then, since we suppose M_bh^avd = M_bh^σ (for the Milky Way here), the M-sigma relation might reflect the mass of the whole galaxy too.

But, we have used v_σ = “σ” and ordinary classical G to calculate M_bh^σ . Comparing results for M_bhs got by each method is still a test. M_bh^avd could be much bigger if the millions of embedded large BHs are counted as contributors to the M_bh^avd G* field. And so, it might not have the same value as M_bh^σ. The difference might tell us something about how many BHs are extant in our galaxy.

Note that the M-sigma relation given here is not the same as that given in the literature. See ref. (74):

The literature versions of the M - sigma relation are purely empirical or phenomenological. The Postulate is theoretical.                                                     

For now, we shall use data:

M_mw  =  1.99 x 10⁴² kg^[51],[52]

since M_mw  =  Milky Way Mass  =  1.25 ×10¹² M_ʘ  ,  

and M_ʘ  =   1.9891×10³⁰ kg     the sun is a fairly small star

G*  =  2 x 10^‑32 m² kg^‑1 s^-2 for the approximate 2-D gravity adapted value of G (given elsewhere, deduced from Milky Way data).

For AVD, by the Postulate: 

 A good value of v_o  =  220,000 m/s for the Milky Way stellar AVD limiting orbital velocity given in ref. (74)

Example (1)            

M_bh^mw  =  v_o²r*/G*               r*  =  1 m      

            =  (2.2 x 10⁵m/s)²(1m)/ 2 x 10^‑32 m³ kg^‑1 s^-2   =  2.42 x 10⁴² kg 

      =  1.247 x 10¹² M_ʘ   

            ≈   M_mw   by the Postulate

which also happens to about equal the mass of the Milky Way and its central black hole by conventional calculations. The M - sigma computation by the Postulate gives a slightly smaller quantity (see Example 2, below).

We see that the mass of the whole galaxy might add to the mass of the central supermassive black hole – all of it in the hyperbolic sense. This may make the SMBH’s hyperbolic (1/r) gravitational field act as if all the mass of the whole galaxy was inside the BH adding to the (1/r) field, thus affecting v_o.

The separate fields would be additive, but this works for us only under the (1/r) rubric. If true, this result would reinforce the gravitational field exclusion principle because the disk stars would not be able to experience an inverse square HSBHG field in this case.

Or else, this really is M_bh^mw, the mass of the Milky Way’s central SMBH. Then the central BH would have had to consume hundreds of billions of solar mass stars since the birth of the Milky Way over 13 billion years ago, only on the order of a dozen per year, a not too implausible occurrence. At first, the rate of consumption would have been much greater, many dozens per month or even more. Such a rate would result in a prodigious release of energy: a quasar.

A result like this would then go much further in explaining Dark Matter. A better value for G* would improve it even more.

The inverse square gravitational field would be felt by entities outside the disk and bulge but not near the rotation plane of the disk. Yet, continuing the quantum metaphor, if the spacetime parcel below the EH is an excited state of the false vacuum (a plenum), such an excitation should be expected to split into two or more separate states under the inevitable perturbations it would experience. If changes in state imply changes in dimensionality, the change to a 2-D + t spacetime state might be accompanied by formation of a 4-D + t state (which may also be renormalizable).

There is some evidence of regions in the universe where gravitation follows a relation where the force of gravity has proportionality F   1/r^(n‑1) where the number of spacial dimensions is n > 3. A 4‑D + t state would have n = 4 and such a more rapidly declining gravitational field could be effective in such regions by alignment with properly oriented BHs in clusters and superclusters. On the other hand, there is evidence that, in other regions, gravity may decline slower than expected, as with n = 2, like the Postulate. See the article by Marcus Chown[53].

M-Sigma

In scenario 2, for stars in the bulge, which orbit chaotically around the BH in most spiral galaxies - some of which may be orbiting in the galactic plane where hyperbolic (1/r) gravity would certainly be in effect, a mixed equation might result.

(4)       v_σ  =  (G²M₁/r  +  κ²G²M_bh /r*)^½   ,  

         v_σ²  =  G²M₁/r  +  κ²G²M_bh /r*  with converted G

(5)        v_σ =  G²M_bh/r  +  κ²G²M_bh /r*  =  (G²/r  +  κ²G²/r*)M_bh 

(6)     M_bh  =  v_σ/(G²/r  +  κ²G²/r*)    the mixed result 

Alternately, where M₁ ≈ M_bh is the mass-energy of material that has in-fallen as far as the EH but its gravitational field is still effective even though its time dilation experience freezes it in place. Yet, its mass-energy still falls through the EH were it takes part in the formation of the 2-D flat gravitational field. All bulge stars would be affected by the inverse square field and all disk stars affected by the hyperbolic field. And, there could still be a normal (1/r²) galactic gravitational field for objects not located in the bulge nor in the plane of the disk. 

But now, to justify an alternate scenario, if the orbits of inner bulge stars were seen to be confined to the 2-D galactic plane, by watching several such stars in their various elliptical orbits at different values of r, or semi-major axes, we could determine BH relative momentum, M_bh and the BH position to less than would be the case if some enhanced version of the Heisenberg uncertainty limit[54] like CHUP were in effect[f]. So, in fact, the orbits are not restricted to the same plane. They are allowed to dwell in any plane, as they are indeed observed to do. This would be a probing quantum effect that would be consistent with the idea that the universe started out as a quantum entity and is still a quantum entity.

But, these other cases would all still be subject to the postulated, hyperbolic, 1/r, 2-D G* field (and perhaps also to a 1/r² gravitational field). This is just a bizarre suggestion meant to side-step the mutual exclusivity of the inverse versus the inverse square gravitational fields. Thus, this remark really suggests that all bulge stars are subject to the hyperbolic field.

The third scenario has bulge stars affected by (1/r²) gravity and standard Kepler’s laws. This gives eq. (4) while only disk AVD stars are analyzed by the Postulate. This would allow comparison of (1/r) and (1/r²) gravities in the same system.

The hyperbolic field originates below the event horizon of the central supermassive black hole. The inverse square field originates at the time-frozen event horizon from the very same matter-energy that has already in-fallen to the singularity. Time dilation at the event horizon should produce this non-intuitive result. So, to avoid risk of contradiction, we should say that no object can experience both types of field simultaneously. The contradictions include the violation of the conservation of matter and energy.

If relativistic descriptions of what happens when matter-energy falls into a black hole, with the passage of time falling to zero (from our “outside” perspective) at the event horizon is true, then all the matter-energy that had fallen in since the BH became a genuine BH is actually still falling in, right on up to the present time. Its gravitational force is still effective as a GMm/r² quantity. But, in its own frame of reference, this same material falls through the horizon to join the singularity where its mass-energy contributes once more to the 2-D flat, axially singular, spin-disk HSBHG field where F^* = G*M_bhm/rr* where superscripted F* refers to the HSBH G* field.

So, a BH might “have its cake and eat it too”. Its gravitational potential energy is manifest as kinetic energy as matter descends to the event horizon where it seems to come to a halt. But, simultaneously it passes through to the singularity where its kinetic and E  =  mc² mass‑energy helps grow the hyperbolic, spacetime G* field.

From this point of view, the HSBHG field could be regarded as just an excited state gravitational field like Guth’s hyperexcited inflaton field. The central SMBH singularity is this excited fundamental field’s own unique point particle manifestation, which therefore must be real, by the field/particle duality theorem^[55].

That is, M₁  =  mass-energy of material below the photonsphere and ergosphere that has not yet fallen through the event horizon to become part of the singularity. This really means that M₁ accounts for virtually all of the mass-energy of the black hole, by appeal to the relativistic gravitational time dilation effect. We and the rest of the universe, including bulge stars, see matter-energy approaching the event horizon from the outside. We see that time there really does stop. We, on the outside, never see this material drop through the horizon. It all becomes frozen in place, and its active gravitational field freezes in place too[56].  Simultaneously, the cumulative mass-energy really does drop into the singularity. Then,

(7)  M₁  =  naked central supermassive Black Hole M_bh via M ‑ sigma with 1/r or 1/r² gravitation arising from mass-energy sequestered near the EH. Yet, AVD M_bh has the about the same value [g] but arises from that same matter-energy which has already become present in the singularity itself.

                (10)   M_bh^σ  =  M₁  =  M_bh^avd [57]  

One can see why it might be necessary to postulate the 1/r² versus 1/r gravitational field exclusion principle. This may be a mechanism how M_bh^σ could be found for all chaotically orbiting bulge stars while M_bh^avd could apply to disk stars via the postulate.

But, the real restraints put on the chaotic orbits of bulge stars by CHUP works too. The 2‑D gravitational field provides better results for both bulge stars and disk stars than conventional Newton/Kepler because it accounts for Dark Matter.

There is a “halo” of stars surrounding most spiral galaxies including the Milky Way. These stars do not orbit in the plane of the galactic disk. Rather, they are distributed roughly spherically around the center. These stars should orbit according to the (1/r²) force law.

For a sample calculation, it would be fortunate if it were not always necessary that all stars should be subject to the hyperbolic field. In fact, if they are not, central galactic supermassive M_bh got by standard calculations of M - sigma, say, and then M_bh values that are got by the anomalous velocity dispersion (adapted to the hyperbolic field) could be compared. This first approximation of M - sigma is essentially how we obtained G*, above (there is a more explicit example calculation below).

But, this procedure is probably incorrect.  

So, this highlights inconsistencies. First of all, the M_bhs got by concurrent calculations according to the alternative approaches do not even roughly compare. And then, we use a value for M_bh that is got by standard methods to calculate G*. This is bad enough. But then, we use this G* to calculate new values for M_bh via AVD or M - sigma (herein) using the Postulate. It’s not exactly circular, but it would be nice if an iterative method of successive approximation could be worked out that results in a value for G* that is more fundamentally sound.

The best way would be to use GR to calculate G*: in good time, dear reader.

Yet, here is an area that is suggested for much more research because the current data are much too imprecise to tell, substantially disagreeing among each other (see below).

Because of the presence of numerous other large massive black holes embedded in it[58], stars in the MW’s outer disk might not be usable to precisely determine M_bh and the position of the central supermassive black hole by the Postulate. Yet, appealing to cosmological Heisenberg again, these stars should themselves be unable to detect the exact position of the SMBH or the exact location of the hyperbolic, gravitational spin disk. So, this kind of uncertainty might contribute to thickening of the galactic disk.

So, one could determine M_bh by studying inner bulge stars according to eq. (4) and also by simultaneously applying eq. (4b), the AVD form of the Postulate, to stars well beyond the bulge, nearer the periphery. In other words, let’s say that one wishes to use the forms in eq. (4) and eq. (4b) determining both “σ” = v_σ and AVD v_o by redshift. Then one could use the 2-D adapted (1/r) Kepler’s laws that are got by a derivation parallel to Kepler’s own development, given in Serway & Beichner^[59] or by simple substitution, as well as classical Kepler/Newton. Then we can compare a result for G* with G and possibly M_bh versions, say, got in the same system.

Unfortunately we cannot do this. Only M_bh^σ and M_bh^avd both got by the Postulate are appropriate for comparison. This method might be also used to highlight the difference with standard calculations, however.

Explicitly, the implication is that there should be major differences between M_bhs got by purely conventional methods that do not use adapted Kepler/Newton and Ms got by methods that do. Conventional methods underestimate M_bh and by extension, the mass of galaxies[h]. So, mass-energy audits will inevitably find too little matter to account for observations. The magnitude of this quantitative underestimate is DM[60].

That is, if it was true that chaotic inner bulge stars feel only inverse ‑ square gravity and outer disk stars feel only hyperbolic inverse gravity, M_bh got by each should at least be very different (and they are). This result would be meaningful for the hyperbolic gravity Postulate. This is what we hoped to find for the Milky Way example (see below and see footnote k). The question is: which model works best.

Be sure to see the follow up paper



http://garyakent.blogspot.com/2014/03/galactic-gravitational-fields.html 




 

Bibliography

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[a]    The factor γ by which lengths contract and time dilates is known as the Lorentz factor and is given by γ = (1 − v²/c²)^½, where v is the speed of the object. If v exceeds c, the expression has an imaginary result. In Einstein’s language, “imaginary” does not mean “unreal”. If time is relatively reversed, energy relations must somehow be affected too because E = mc². If t becomes -t, c² becomes -c² and E becomes -E.

[b]     Some say that the AVD is constant not only across the width of a galaxy, except near its center, but that it is constant from galaxy to galaxy. Milgrom’s a_o implies this. MOND (ref. 17) was invented to explain the AVD. But, AVD is expressed as constant velocity dispersion, not constant radial acceleration.

 

[d]      Since there are millions of stars in the galactic bulge, we take chaotic stellar “radial” orbital velocity, v  = x, as approximately continuous variables. In the M ‑ sigma relation, this “σ” or v_σ is the stellar radial orbital velocity at near one standard deviation from the mean of the distribution. See the blog entry for specific details: http://garyakent.wordpress.com/2013/07/18/normal-distribution-applied-to-milky-way-galaxy-m-sigma-relation-and-bulge-star-data/

[e]   This anomalous velocity dispersion depends on M_bh, but some data may seem to imply that it might be constant from galaxy to galaxy. More accurate and precise data might be needed to confirm that AVD is not really constant between the galaxies. MOND says a_o is constant, at least asymptotically so, between galaxies. But G* is so small and r can be so large that it may require really precise measurements to tell that it is not. Yet, if the AVD results from a near zero asymptotic hyperbolic gravitational field, it may be impossible to discriminate it from a constant.

[f]      When cosmological distances are involved, quantum principles may be restored.  

[g]     See example calculations below.

[h]   If v_o is larger than v_σ M_bh^avd may include the mass of the galaxy. In any case, M_bh is very much larger in the case of a 2-D hyperbolic gravitation G* in any calculation done via the Postulate.  

---

[1]    Mr. Kentgen obtained a Ph.D. equivalent M.S. degree from the Illinois Institute of Technology in 1985. His unfinished (due to a funding issue) dissertation critically involved mathematical modeling. He then became interested in heavy element nucleosynthesis and went on to develop an interest  in cosmology and black hole modeling. Since black holes arise from galactic supernovae as do heavy elements, the evolution of the Postulate presented here seems natural.

[2]    A postulate, a temporary proposed premise, must be provisionally accepted as is. The truth or falsity of a postulated premise must be deduced from experimental fact, not mere opinion. The components of the premise presented here cannot be argued away piecemeal because they constitute a single logical hypothesis that is falsifiable only as a whole. Only objections or reservations that are themselves falsifiable will be viewed as valid points of order requiring an answer.

[3]     2-D gravity is only obliquely evidenced via a reference. The 1/r^(n-1) decline with n = 2 must be substantiated more fully.

[4]     R. A. Serway, R. J. Beichner, Physics for Scientists and Engineers, 5^th ed., Saunders College Publishing, 2000, www.harcourtcollege.com,  Kepler’s 3^rd law, p432

[5]   Brian Greene, The Fabric of the Cosmos, Alfred A. Knopf, 2004

[6]  Ibid, page 254

[7]  Quantum black hole and the modified uncertainty principle  http://adsabs.harvard.edu/abs/2011PhLB..701..384M

[8]     L. L. Samojeden, G. M. Kremer, F. P. Devecchi, Accelerated expansion in bosonic and fermionic 2-D cosmologies with quantum effects,  http://iopscience.iop.org/0295-5075/87/1/10001,   2-D quantum renormalizable gravitational field

[9]     Jan Ambjørn, Kazuo Ghoroku,     2-D quantum gravity coupled to renormalizable matter fields   http://arxiv.org/pdf/hep-th/9312002.pdf,  a quantum renormalizable 2‑D gravitational field

 

[10]  http://encyclopedia2.thefreedictionary.com/Asymptotically+flat

[11]     Ignazio Ciufolini,  Erricos Pavlis,   Earth dragging space and time as it rotates     http://www.phy.duke.edu/~kolena/framedrag.html    contains an inference about enhanced frame dragging.

[12]  (Forgive the clumsy way the following is expressed.) This is a point that would not be covered by GR because GR has “broken down” at such singularities. But, only the z dimension is singular. Kerr does not consider that this might happen. It is part of the postulate and its validity cannot be argued away and cannot be refuted unless it is shown falsifiably that it is contrary to proven experimental facts. As part of the Postulate, it must be experimentally verified or falsified, not argued away as a matter of opinion. All other assumptions, especially the implicit ones concerning metrics and their underlying bases, cannot withstand such a postulate. Spacetime phase change is a new concept and its implications are novel too. If the Postulate is accepted at face value, at least tentatively, quantum gravity may already be here with little fuss and few extra complications. The real question is, can mathematical physics and cosmology stand it? Can scientists, even if only tentatively, abide ideas that might make a paradigm shift that could render their whole careers obsolete? The author’s grasp of gravitation theory is not an issue. At issue here is the institutional open mindedness of Physics. 

[13]  Even if there are valid versions of quantum gravity, one must allow that all gravitational singularities are bound by the Cosmological uncertainty principle, which is part of the Postulate and must be entertained unless it contradicts strong empirical data. The uncertainty is largely in measurements at the instrument. These must still apply to the object under observation because of the cosmological logical positivistic principle that “seeming is being”, a very existentialist attitude. The questions are what we mean by “seeming” and the definition of “being”. Our senses cannot escape the limitations of our instruments which only extend their reach. The universe is WYSIWYG, but “seeing” may not be so intuitively obvious.

[14]    Albert Einstein, The Meaning of Relativity, Princeton University Press, 1984,  page 89

[15]     Rubin V, Ford W K Jr., Thonnard N. Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R = 4kpc) to UGC 2885 (R = 122kpc), The Astrophysical Journal 238: 471, 1980

[16]     This new divergence results in the fact that objects in the universe are once more observed to appear to move kinematically farther and farther apart and at an accelerating rate. It is said that we may as well regard expansion as a kinematic process because it avoids confusion and it may be meaningless to maintain that it is not. 

 

Fulvio Melia    arXiv:1207.1332v2 [astro-ph.CO] 31 May 2013   Proper Size of the Visible Universe in FRW Metrics with Constant Spacetime Curvature 

 

[17]      Everett, Hugh, "Relative State Formulation of Quantum Mechanics". Reviews of Modern Physics 29, 1957 : 454–462.

[18]   The CHUP has to be respected and provisionally accepted at face value in order to maintain its formal status of a logical Postulate.

[19]    Carroll, Sean M (2004), Spacetime and Geometry: An Introduction to General Relativity, San Francisco: Addison-Wesley, http://spacetimeandgeometry.net/.

[20]     A. Zee, Time Reversal: can you tell the past from the future? Physicists can't -- not yet, anyway, Discover, October 01, 1992 http://discovermagazine.com/1992/oct/timereversal140#.UcHxZpyTuuI   physically possible, physical laws are time invariant.

[21]     Fine, Arthur, The Einstein-Podolsky-Rosen Argument in Quantum Theory, 2009,  http://plato.stanford.edu/entries/qt-epr/ ………………………

[22]      Benjamin Gal-Or, Cosmology, Physics and Philosophy. Springer Verlag, 1987

[23]     (Planck density, ρ_p  ≈  ∞, “as high as may be needed to explain effects”; the real meaning of infinity.) This is another reason to include the CHUP in the Postulate.

[24]     Kerr, R P   Gravitational field of a spinning mass as an example of algebraically special metrics, Physical Review Letters 11 (5)1963: 237–238

[25]   But, its spacetime nature is immune to the limitation of the event horizon

[26]        Lide, David R (Ed.)   Handbook of Chemistry and Physics, CRC Press, 78^th edition, 1997

[27]     Constants of Physics and Math http://www.ebyte.it/library/educards/constants/ConstantsOfPhysicsAndMath.html

[28]      Gebhardt, K et al., A Relationship between Nuclear Black Hole Mass and Galaxy Velocity Dispersion, The Astrophysical Journal, 539, 2000:  L13-L16

[29]      http://arxiv.org/pdf/0801.3133v2.pdf    Mordehai Milgrom, MOND 

[30]     http://phys.org/news160726282.html     Study plunges standard Theory of Cosmology into Crisis,   May 05, 2009

[31]      ibid

[32]      Ferrarese, F and Merritt, D, A Fundamental Relation between Supermassive Black Holes and Their Host Galaxies, The Astrophysical Journal, 539, 2000  L9-L12

[33]       Rubin, V, et al, ref. (7), ibid. Inverse square gravity (1/r²) cannot explain the AVD or the M-Sigma effect.

[34]      http://arxiv.org/abs/1204.0144   M-sigma relation and anomalous velocity dispersion 

[35]      Lin, C. C.; Shu, F. H. "On the spiral structure of disk galaxies". The Astrophysical Journal 140: 646–655 (August 1964). Bulges with mini spiral structures have a small core having characteristic chaotically orbiting stars

[36]     http://www.optcorp.com/articles/galactic-bulge     Chaotic, random nature of inner bulge star orbits. “They are also in orbits that are essentially random compared to the plane of the galaxy, whence the round shape arises.”

[37]      Standard deviation http://mathworld.wolfram.com/StandardDeviation.html

[38]   The gravitational force near the center is so intense that it’s mathematical behavior must be Minkowski or relativistically asymptotically flat there and at the other extreme of r.

[39]      Johansen, Nils Voje; and Ravndal, Finn - On the discovery of Birkhoff's theorem, version of September 6, 2005.

[40]      http://physics.stackexchange.com/questions/30652/what-is-the-2d-gravity-potential/41179#41179    General relativity allows a gravitational field that declines as 1/r only in 2-D spacetime. Such a field, to be a field at all, must be effectively attractive or the term “field” is meaningless.

[41]     Mordehai Milgrom, MOND - a pedagogical review, arXiv:astro-ph/0112069 v1 4 Dec 2001

[42]     Imamura, Jim (August 10, 2006). "Mass of the Milky Way Galaxy" University of Oregon. AVD Milky Way  v_o = up to around 230 km/s

[43]     Ref. (14)

[44]    Ghez,  A M et al.  "Measuring Distance and Properties of the Milky Way's Central Supermassive Black Hole with Stellar Orbits". Astrophysical Journal 689 (2): 1044–1062, December 2008. arXiv:astro-ph/0808.2870  The M_bh got here uses conventional Kepler’s law which will work just fine but gives possibly incorrect M_bh.

[45]    Ref. (10)

[46]    http://garyakent.wordpress.com/2012/01/11/a-potential-energy-diagram-is-perfectly-possible-for-a-hyperbolic-black-hole-gravitational-field/

[47]    George Jorjadze, Włodzimierz Piechocki, Geometry of 2d spacetime and quantization of particle dynamics,   http://arxiv.org/abs/gr-qc/9811094  for example ……………………………………………………….

[48]     The Meaning of Relativity, A. Einstein, Princeton University Press, 1984, p. 90

[49]     http://www.halexandria.org/dward159.htm    Enhanced Heisenberg uncertainty principle extended to chaotic systems.

[50]    Trimble, V. (1987). "Existence and nature of dark matter in the universe". Annual Review of Astronomy and Astrophysics 25: 425472.

[51]     McMillan, P. J. (July 2011). "Mass models of the Milky Way". Monthly Notices of the Royal Astronomical Society 414 (3): 2446–2457

[52]    Dehnen, Walter; McLaughlin, Dean E.; Sachania, Jalpesh, The velocity dispersion and mass profile of the Milky Way, Monthly Notices of the Royal Astronomical Society, Volume 369, Number 4, 11 2006 , pp. 1688-1692 Oxford University Press

[53]    Marcus Chown,  "Gravity may venture where matter fears to tread", New Scientist vol. 2669, 16 March 2009 http://www.newscientist.com/article/mg20126990.400-gravity-may-      venture-where-matter-fears-to-tread.html 

[54]    Hilgevoord, Jan; Uffink, Jos  The Uncertainty Principle   2006

http://plato.stanford.edu/entries/qt-uncertainty/

[55]     J Bain  Against particle/field duality , 2000    http://faculty.poly.edu/~jbain/papers/lsz.pdf

[56]     Inside a Schwarzchild Black Hole http://jila.colorado.edu/~ajsh/insidebh/schw.html

[57]     http://arxiv.org/pdf/1204.0144v2.pdf     M_BH−σ Relation between SMBHs and the velocity dispersion of globular cluster systems,  Raphael Sadoun, Jacques Colin, 26 September 2012

 

[58]   Black Holes and Galaxies  http://jila.colorado.edu/research/astrophysics/black-holes-galaxies

[59]     Physics for Scientists and Engineers, 5th ed., 2000, Saunders College Publishing div. Harcourt College Publishing, Inc., p. 432

[60]    The Distribution of Matter and Energy in the Universe,   January 4th, 2011    http://euclid.caltech.edu/image/13