**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/r

^{2}) 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/r

^{2}) 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

**Dark Matter. The implication is that maximally spinning black holes are the ultimate source of DM.***IS*
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/r

^{2}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/r

^{2}) 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

**Dark Matter. The implication is that maximally spinning black holes are the ultimate source of DM.***IS*
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

**Dark Energy too. It is a version of quintessence.***IS*
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}m^{3}kg^{-1}s^{-2}so small (see below), whereas classical G[26]^{,[27]}= 6.67384 x 10^{-11}m^{3}kg^{-1}s^{-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

**in this region and its natural connection to statistical σ, the standard deviation[37].**__v__
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 V^{2}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 (V^{2})^{2}:
1.)

__v___{σ}^{2}= (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___{σ}^{2}= G*M_{bh}/*****__r__**with near hyperbolic asymptotic value proportional, if not equal, to Milgrom’s**__r____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 10^{6}M**[44] , M**_{ʘ }**= 1.9891×10**_{ʘ}^{30}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)

*** = G*M**__F___{bh}m_{star}/***r and**__r____a___{o}=__v___{o}^{2}/*****__r__
from Newton’s law of gravitation adapted to a
2-D parcel of spacetime (the main HSBHG-field Postulate). The unit vector,

*** serves to preserve dimensional integrity. There have been many studies concerning 2-D spacetime**__r__^{[47]}. The idea is not new.
Rearranging eq. (1)

(2) G*
=

__v___{o}^{2}***/M**__r___{bh}
And, for the Milky
Way:(2b)
G* =

(230,000 m/s)

^{2}(1 m) / ((1.25 x 10^{12}M**)(1.9891 x 10**_{ʘ}^{30}kg/M**))**_{ʘ}
=
5.29 x 10

^{10}m^{3}s^{-2}/ 2.486 x 10^{42}kg
=
2.128 x 10

^{-32}m^{3}kg^{-1}s^{-2}
Whereas classical G =
6.67384 x 10

^{-11}m^{3}kg^{-1}s^{-2}as given above.
So, the conversion factor

**= G*/G = 3.1884 x 10**

__κ__^{-22}m

^{-1}and

__κ__^{-1}= G/G* = 3.1364 x 10

^{21}m

may be used to convert G into G* and to
correct M

_{bh}s and M_{galaxy}s 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_{bh}s and many values of galactic__v___{o}s. 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_{bh}s 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):

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__**and**__r__**= G*M**

__F__*_{bh}m

_{star}/

*****

__r__

__r__
All stars in the central bulge may be affected
by

*** 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.**__F__
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}= (**GM**__κ___{bh}/***)**__r__^{½}AVD ,__a___{o}=**GM**__κ___{bh}/***r**__r__*** =**

__F__**GM**

__κ___{bh}m

_{star}/

***r**

__r__
and rearranging (4b), we get

(4c) M

_{bh}=*****__r____v___{o}^{2}/**G =**__κ__*****__r____v___{o}^{2}/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/r

^{2}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

*** = the unit vector of**__r__**, for dimensional integrity.)**__r__
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**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.**__v__
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 }m^{2 }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_{bh}s 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:

since M

_{mw}= Milky Way Mass**= 1.25 ×10**^{12}M**,**_{ʘ}
and M

**= 1.9891×10**_{ʘ}^{30}kg the sun is a fairly small star
G* = 2 x 10

^{‑32 }m^{2 }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___{o}^{2}***/G***__r__*** = 1 m**__r__
=
(2.2 x 10

^{5}m/s)^{2}(1m)/ 2 x 10^{‑32 }m^{3 }kg^{‑1 }s^{-2}= 2.42 x 10^{42 }kg
=
1.247 x 10

^{12}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**, 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.

_{bh}^{mw}
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}= (G^{2}M_{1}/**+**__r____κ__^{2}G^{2}M_{bh}/***)**__r__^{½},

__v___{σ}

^{2}= G

^{2}M

_{1}/

**+**

__r__

__κ__^{2}G

^{2}M

_{bh}/

*** with converted G**

__r__
(5)

__v___{σ}= G^{2}M_{bh}/**+**__r____κ__^{2}G^{2}M_{bh}/*** = (G**__r__^{2}/**+**__r____κ__^{2}G^{2}/***)M**__r___{bh}
(6) M

_{bh}=__v___{σ}/(G^{2}/**+**__r____κ__^{2}G^{2}/***) the mixed result**__r__
Alternately, where M

_{1}≈ 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/r^{2}) 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

**, or semi-major axes, we could determine BH relative momentum, M**__r___{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/r

^{2}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/r

^{2}) 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/r

^{2}) 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/r

^{2}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_{bh}m/r*** where superscripted F* refers to the HSBH G* field.**__r__
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 = mc

^{2}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, M

_{1}= 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 M_{1}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) M

_{1}= naked central supermassive Black Hole M_{bh}via M ‑ sigma with 1/r or 1/r^{2}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}= M_{1}= M_{bh}^{avd}[57]
One can see why it might be necessary to
postulate the 1/r

^{2}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/r

^{2}) 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

_{bh}s 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

_{bh}s 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

Be sure to see the follow up paper

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

_{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

Bibliography

[a] The factor γ
by which lengths contract and time dilates is known as the Lorentz
factor and is given by γ = (1 − v

^{2}/c^{2})^{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 = mc^{2}. If t becomes -t, c^{2}becomes -c^{2}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,

**= x, as approximately continuous variables. In the M ‑ sigma relation, this “σ” or**__v____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

#
[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/r

^{2}) cannot explain the AVD or the M-Sigma effect.
[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)

#
[47] George Jorjadze, Włodzimierz Piechocki, Geometry of 2d spacetime and
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