Be sure to see the primary article in this series at
http://neocosmology.blogspot.com/2014/03/spinning-supermassive-black-holes-part-1_22.html
Galactic gravitational
fields
Abstract
The set of
gravitational fields postulated herein centers around supermassive black holes
of the kind seen at the centers of spiral galaxies. The concept is not limited
to such particular supermassive black holes because all black holes should
display these 2-D hyperbolic (1/r) and hyperparabolic (1/r3)
gravitational field “surfaces” embedded in 3-D + t spacetime. It is
necessary to postulate a full set of gravitational fields based on unique
properties of black holes. A single type of field does not work in descriptions
of our universe. But allowing more than one type of field solves many vexing
problems such as the anomalous velocity dispersion of stars in galaxies and of
galaxies in clusters and even the Bullet Cluster effect. Using the hyperbolic
(1/r) supermassive black hole galactic gravitational field, one can even derive
the MOND acceleration constant, ao.
Introduction
The small set
of gravitational field types postulated herein centers around supermassive
black holes (SMBH)s of the sort seen at the centers of spiral galaxies. And,
allowing more than one type of field solves many vexing problems such as the
anomalous velocity dispersion (AVD) of stars in galaxies and of galaxies in
clusters[2], the Sunyaev Zel’dovich,
the integrated Sachs Wolfe and the Bullet Cluster effects. Also, the hyperbolic
(1/r) central SMBH gravitational field will help explain the temperature
distributions across galaxies and the pattern of variations in the CMB[3] (because they are consistent with the values that were
previously determined).
Embedded in
3-D plus t spacetime, the galactic 2-D Hyperbolic (1/r) Gravitational Field
(HGF) complex surface-like shape is a hyperboloid of one sheet[4] (at very large r). Significantly more (or less – when gravity
declines as 1/r3 therein) 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. Helping to evolve large
scale structure[5], 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.
In a very rapidly spinning black hole where
mass-energy has collapsed to a virtual singularity[6] beneath the event horizon[7],
only the z dimension has shrunk, according to the Postulate. Because all
mass-energy entering a black hole must initially have 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[8]
and the 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[9].
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”[10].
It forms a flattened disk of nominally infinite
radius. This spacetime matter-gravitational field disk can ignore the event
horizon (EH). While its mass is still constrained by the EH, the field is not
so limited because it is a spacetime entity[11].
The shapes of a whole set of hyperbolic gravitational fields is depicted in
Figure 1 below.
Collapse in the z dimension of the galactic
central SMBH means that mass‑energy‑spacetime (MES) becomes unified in a new
way. It forms, 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[12]
available to it in 2-D. So, this MES 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.
Relativity theory unites matter, energy,
space and time via E = mc2.
This black hole behavior only confirms Einstein, Schwarzchild and Kretschmann.
It indicates that Einstein’s theory may be UV complete and may require no
revision, update or replacement.
Depiction of how a
network might arise
Figure 1 is meant to depict how a network
might arise from the hyperboloids of one sheet that would exist
surrounding supermassive central BHs in galaxies. Actually, in Fig. 1, the gravitational fields
of each SMBH putatively extend to infinity except whenever a hyperboloid
surface of one SMBH meets a surface of another SMBH almost head-on. Then they
form a preferred gravitational link between galaxies because gravity in this
space between central SMBHs declines only as 1/r. Since these Fig. 1
hyperboloids are viewed from a cosmological perspective, they appear quite
pinched at their centers. But, of course, in fact they are. This network will
become exceedingly complex if clusters or superclusters collide.
The part of a hyperboloid that is not
enclosed within the hyperbolic quasi-surface itself, the cages or baskets and
the circum-hole groove are an “antihyperboloid” of spacetime which is
postulated here to gravitationally decline within it as 1/r3,
perhaps as a separate quantum-like “state” of the black hole. It might be
called a “hyperparaboloid”.
Figure
1
The
edges of the hyperboloids are to be extended indefinitely, not truncated like
this. The surfaces are depicted like this for convenience.
The faster decline of G with r in these galactic
SMBH hyperparabolic (1/r3) spacetime regions of 3-D spacetime are
not gravitationally neutral or maybe not even slightly repulsive so that the
network that forms along with the hyperbolic (1/r) contribution has an “open
cell” or sponge-like structure. Gravitational hyperparabolic (1/r3)
spacetime in conjunction with SMBHs is also permitted by general relativity[13],[14],[15].
It helps reinforce the open cell lattice‑like structure of the matrix of galactic
superclusters.
Ordinary inverse square (1/r2)
gravity around a galactic central SMBH, being omnidirectional in nature, cannot
produce this kind of structure. But, the highly directional hyperbolic (1/r)
sheet of a central SMBH can do so.
Figure 2
Large scale structure of the universe
http://wn.com/large-scale_structure_of_the_universe
Objections
to all this
One might object to all this
on the basis of Birkhoff’s[16]
and the “no-hair”[17]
theorems and even Newton’s law of universal gravitation itself[18].
The no-hair theorem is not a
mathematical theorem at all. It is a postulate, a tentative assumption that ideal
black hole
solutions of the Einstein equations of gravitation in general relativity can be completely characterized by
three externally observable classical parameters. These are mass, electric charge,
and angular momentum (spin).
Other information (nicknamed "hair”)
about matter which entered
a black hole "disappears" behind the black hole event horizon. Supposedly,
it is therefore inaccessible to external observers. John Archibald Wheeler enunciated this idea as "black
holes have no hair", the source of the name. But, it is now accepted
that no information is lost in the formation of BHs[19].
The HSMBH gravitational field can exist, hair or no hair.
Newton’s law is sidestepped
by postulating a black hole gravitational field that has different dimensionality
in cosmologically scaled space. On cosmological scales, unusual things should
be possible.
With no strict mathematical
proof of the so called no-hair theorem, mathematicians rightly refer to it as
the no-hair conjecture[20].
Even in the simplest case of BHs with intense gravity alone (that is, zero
electric fields, no spin), this conjecture has been only partially supported by
results of research to date.
Researchers have been working
under the additional hypotheses of simple event horizons and difficult to
justify assumptions concerning details of the mathematical and physical reality of spacetime near black holes.
Birkhoff’s theorem does not apply because of the unrealistically simple
conditions he requires. Kerr’s[21]
and other’s treatments are almost equally simplistic. In other words, there is
no theoretical constraint, including from relativity theory, that would obviate
the central galactic, hyperbolic (1/r), supermassive black hole, spin induced
gravitational field.
Bending of background
illumination by galaxies and galactic central SMBHs
The ultra-high-spin induced hyperbolic (1/r)
central galactic SMBH spacetime parcel is composed of a spacetime continuum
segment or relativistic parcel that can behave as if it had a refractive index >1 relative to ordinary gravity-free 3-D
spacetime. So, no matter what pathway is traversed through the parcel by a ray
of light, the 2-D HSMBH gravitational spacetime packet can bend the trajectory
of electromagnetic radiation, as depicted in Fig. 2.
Figure 3
Derived from Fig.1, this depicts how illumination from behind might interact with the gravitational fields of a cluster of supermassive black holes.Regardless of the pathway through a galaxy cluster, light is bent by the hyperboloids, even when the ray passes perpendicular through the thin part of a disk. The hyperbolic (1/r) gravitational fields act like connected parcels of spacetime having a refractive index > 1. The reverse Compton effect acting on the ray is permanent. The redshift process itself is ongoing.
Mass estimates based on gravitational
microlensing suggest that the hyperbolic (1/r)gravitational masses of SMBHs should
be combined or added to the inverse square (1/r2) galaxy masses
unless these masses are recomputed via the Postulate. Such a recalculation
would be preferred. We think that SMBHs should have masses comparable to
galaxies (on the order of 1011 solar masses). We will attend to this
more fully in a future paper (see ref. 21).
For modeling purposes, each galactic SMBH can
be considered like a polyfunctional monomer from which will be constructed a
crosslinked polymer. For crosslinking, such a monomer must have more than 2
functional groups to which other monomers may attach. The two lobed hyperbolic one‑sheet
is ideal for this. A variety of orientations and attachment points are
permitted for conjoining numerous galactic SMBHs. Viewed as a whole, such a polymer will appear
to have random orientations of its SMBH monomers. Its structure would look like
the cobweb or sponge-like pattern such as that seen in diagrams derived from
the Sloan Digital Sky Survey results in Fig. 2[22],
[23].
By the way, one wonders whether the universe
may be considered to be actually composed of a medium like a crosslinked
polymer akin to rubber. A rubber ball analogy may go at least as far as the so
called oscillating universe model[24].
The analogy suggests that a negative cosmological constant would be dominant
when the universe is highly compressed whence gravitation would become
repulsive. Not only might this be the source of the energy for a big bang, such
repulsive gravity may be important on very small scales so that gravitational
attraction does not really increase without bound below the quantum level, that
is, as r à0. And, r cannot à0
anyway, by appeal to the uncertainty principle, which is real and not a
mathematical artifact or semantic device. So, relativity may not suffer from
breakdown at small r.
Still, 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 field is quantum
renormalizable[25],[26]
and applicable to the highest energy regimes (see below) at the most extreme
values of independent and dependent variables. Assuming arbitrarily small distance
between two quantum masses, to say that r à 0 is meaningless under
Uncertainty, it is simply not allowed. The 2-D G* metric below the event horizon
and even far above it is Minkowski asymptotically flat[27]
-- a pure spacetime entity in its own right.
Sunyaev
Zel’dovich (SZ) effect
Photon frequency is plotted
on the vertical (magenta) axis, the ordinate. Distance is plotted on the
horizontal (green) axis, the ordinate. Longer wavelength photons would appear nearer
the bottom, shorter wavelengths would lie nearer the top. The source of photons
would be background stars and/or the CMB.
Figure 4
Sunyaev Zel’dovich effect[28]
This figure is also derived from Fig. 1. Regardless of the pathway through a galaxy cluster, light is bent by the hyperboloids, even when the ray passes through the thin part of a disk. The hyperbolic (1/r) gravitational fields act like connected parcels of spacetime having a refractive index > 1. The reverse Compton effect acting on the ray is permanent. The redshift process itself is ongoing.
Integrated Sachs-Wolfe
(ISW) effect
Figure 5 depicts the integrated Sachs-Wolfe
effect[29],[30].
As above, magenta denotes the photon frequency axis. Green denotes the
cosmological distance axis.
Figure 5
Integrated
Sachs-Wolfe Effect
A ray of photons cascades down the
initial galactic hyperbolic gravitational potential gradient that it
encounters, gaining energy with an increase in frequency, going blue. Then it
propels up the associated opposite galactic potential gradient, losing energy
and going red. But, it does not go as red as it had previously gone
blue. Large galactic voids would behave in the opposite manner giving net redshift
effects[31].
A photon traveling along the yellow pathway
propels down the gravitational field potential gradient, gaining energy – going
toward blue. And, when it propels up the next opposite gradient, it loses
energy, but not as much as it had gained, producing a net blueshift effect. The
expansion of the universe had reduced the height of the gradient in the latter
part of the trip so, there was less of a redshift effect that failed to offset
the initial blueshift.
Experimentally, the problem becomes discriminating
between SZ and ISW effects[32].
Bullet
cluster effect
Also, the Bullet cluster effect may be emulated
by the presence of supermassive black holes (see Fig. 6) having hyperbolic
gravitational fields.
Of course, one must allow for the limitations
of this figure. It is merely diagrammatic. It is not meant to represent a
detailed true-to-life animation. We realize that there will be strong
interactions between the galaxies in the cluster, a few of them even colliding
too, with the expected extreme distortions.
There will also be interactions between the
central supermassive black holes, which become stripped from their respective
galaxies. But, the diagram illustrates how the cluster of SMBHs remains roughly
intact, like buckshot passing through a small soft pillow. This intact cluster
will be invisible to telescopes yet, it will have an unexpectedly strong
gravitational field structure [33].
Such a structure will bend light and appear to us as a disembodied Dark Matter
entity by its distortion of the background stars and/or of the CMB
(microlensing).
Figure 6
In these colliding clusters, by simple physics, the
galaxies are stripped of their central SMBHs (fuzzy grey dots) only because
they are supermassive. The blue and red represent the gravitational influence
proffered by the galaxies under the 2-D hyperbolic (1/r) SMBH gravitational
field postulate (blue) and ordinary galactic inverse square gravity (red). Note
the similarity to Figure 7.
Each galaxy retains
its overall structure (except for those that collide), at least initially,
because there are potentially millions of large black holes embedded in them. These
black holes are not stripped because they are smaller than SMBHs and they are gravitationally
trapped in the body of the galaxies like flies on flypaper. Over a long time ever
since their formation, these BHs will have come to have their rotations evolve
to be correlated, indeed, their rotations become parallel due to the extended
frame dragging effect that is active in two dimensions and due to the stronger
(1/r) link between them. These BHs will have correlated their rotations with
the central SMBH and thus to each other. With so many large parallel spinning BHs
embedded within them acting as surrogate central SMBHs, the galaxies retain
their nominally flat 2‑D spiral structures unless they undergo head‑on
collisions or extremely close encounters. Almost immediately, new central SMBHs
will form in each galaxy
The bodies of the galaxies themselves are
mutually slowed down greatly by the fact that they contain so much gas, plasma,
dust and stars that their whole environments, the entire parcels of spacetime
in which they exist, are decelerated. To do this, the masses of the gas and
dust must effectively complement the masses of the luminous portions, mainly
stars, and these masses must then compete with the masses of the compact
central SMBHs. The result is that the dense central SMBHs coast straight through
the clouds of stars and gas constituting each galaxy, thus forming new
independent structures of their own[34].
Note that the diagram in Fig. 6 is greatly
simplified but, it is still suggestive of Figure 6.
Bullet Cluster
Figure 7
NASA, HST image
This diagram suggests that there should be trillions
of naked SMBHs floating about in the universe.
The anomalous velocity dispersion and
galactic temperature gradients,
(sections to be added later)
Milgrom’s ao
derived
According to the Postulate, the anomalous
velocity dispersion of stars in galaxies and galaxies in clusters follows the
relation
(1)
avd vo 4 = MGa0 (Milgrom[35]),
ao is the MOND
acceleration
constant,
M = Mbh
the mass of central galactic SMBH or the.. ………………...center of mass in a cluster.
(2) voavd = vσ = (G*Mbh/r*)½ from
the paper[36].
so
(3)
voavd 4 = (G*Mbh/r*)2 = MbhGao
(4)
ao = (G*Mbh/r*)2/MbhG
= Mbh(G* /G½r*)2
or
(5)
Mbh = ao/(G*/G½r*)2 where
r* is the unit vector of
orbital radius r
According to the Postulate outlined in the next blogs (#1 & #2, below), this does not mean that Mbh is a
constant, but that its value doesn’t matter to ao (being close to an hyperbolic asymptotic value
of nonzero orbital acceleration), which will be virtually a constant regardless of Mbh.
Only the minimum hyperbolic radius of curvature will vary strongly with Mbh.
And, we do have eq. (6), that is,
(6)
ao/(G*/G½r*)2 = K where
K is based on the incipient or minimum
black hole mass or else on the hyperbolic field asymptote
which
should have about the same value for almost all black holes.
G*
is estimated in ref. 22.
Perhaps
this means that one might say that once a central galactic black hole forms,
its additional growing mass does not matter to ao. Or, perhaps this particular initial value of
Mbh is the minimum size for a stable central galactic SMBH that can
produce a hyperbolic acceleration constant, ao. It must also be true that the SMBH hyperbolic
gravitational field has field strength approaching an asymptote (one of the
axes) that is independent of black hole mass. All these factors may be in
effect.
Mbh
in eqs. (1) and (2) are allowed to vary as usual.
So,
Milgrom’s ao is
now derived in terms of the central hyperbolic (1/r) SMBH galactic
gravitational field. One can obtain ao
and K independently by theoretical analysis of eqs. (3), (5) and (6).
Under
the Postulate, for ao
to be a fundamental constant under MOND, Mbh must be constant from
galaxy to galaxy. This is unlikely. The previous analysis recognizes this,
especially when it is realized that Mbh affects only the minimum
radius of curvature of the gravitational hyperbolic field strength and not the
non-zero near asymptotic value of gravitational field strength. These hyperbolic
asymptotes are the axes themselves. This nominally virtually constant asymptotic
hyperbolic value of ao
might be used to estimate the “effective” finite size of the black hole central
core. And it might help define the cosmological uncertainty principle.
The
hyperbolic gravitational field is also consistent with the galactic temperature
gradient as a linear distribution[37].
More about this in a future paper because it is a key indicator.
For the reader,
a problem:
The near asymptotic value of gravitational
field strength occurs at around r = 10, say, in arbitrary units, where the
gravitational strength hyperbola almost flattens out and its average slope
becomes very small. If, at this value of r, the (1/r) hyperbolic gravitational
field strength is nearly = 0 (zero is an asymptote) then the HSMBH gravitational
field can be said to virtually end, at r >> any galaxy radius. But then,
what is the value of this maximum effective r? How big is the DM field,
effectively? Use the definition of a right hyperbola as the full hyperbolic
field, field strength, FG* = 1/r, and scale it in natural units with
the radius of a galaxy = 1 unit. Forget that G*Mbhm and r cannot
simultaneously be =1. This is just an exercise.
Dark
Matter exists
Dark Matter exists. But, its nature is
profoundly different from that which is presumed by the consensus view. Black
holes have been said to be crucial in the formation of the universe. They are
not only crucial to its formation, but also to its detailed appearance.
To be continued
See the related blogs # 1 and #2
http://garyakent.blogspot.com/2014/03/spinning-supermassive-black-holes-and.html
Footnotes and Bibliography,
[1]
Please excuse the author's unfamiliarity with cosmologists' traditional jargon. Mr. Kentgen obtained a Ph.D. equivalent M.S. degree from the
Illinois Institute of Technology in 1985. His nearly complete yet unfinished (due to a funding
issue) doctoral dissertation on inorganic physical biochemistry 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 largely from
galactic and stellar supernovae as do heaviest elements, the evolution of the Postulate
presented here seems natural.
[2] 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
[4]
Hyperboloid of one sheet http://www.math.umn.edu/~rogness/quadrics/hyper1.shtml
[6]
Singularities http://plato.stanford.edu/entries/spacetime-singularities/
[7]
event horizon http://archive.ncsa.illinois.edu/Cyberia/NumRel/BlackHoleAnat.html
[8] R.
A. Serway, R. J. Beichner, Physics for Scientists and Engineers, 5th
ed., Saunders College Publishing, 2000, www.harcourtcollege.com, Kepler’s 3rd law, p432
[9]
Such in-falling material must orbit in
concert. http://jila.colorado.edu/~ajsh/insidebh/schw.html
[10] Brian
Greene, The Fabric of the Cosmos, Alfred A. Knopf, 2004, page 254
[11]
The gravitational field is not so limited
because it is a spacetime entity.
[14] Ibid Gravitational Hyperparabolic (1/r3)
spacetime in conjunction with SMBHs is also permitted by general relativity in
a spatially n-dimensional parcel of the universe where spatial dimension n = 4
and gravity declines as 1/r(n-1) as implied by GR .
[16]
Birkhoff’s theorem http://www.einstein-online.info/dictionary/birkhoffs-theorem
[17]
No hair theorem http://www.ibtimes.com/black-holes-have-hair-say-scientists-wait-what-explaining-hairy-new-developments-1412948
[18] even Newton’s law of universal gravitation
itself http://www.scribd.com/doc/188147252/Gravitational-Forces-Within-Homogenous-Spheres
[21] Kerr
metric http://www.astro.ku.dk/~milvang/RelViz/000_node12.html
[22] 3-D structure
of the universe……………………………………………………………
[23] Sponge,
lattice, open cell or cobweb structure of the universe,………………………
[25] 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
[27]
Minkowski asymptotically flat http://relativity.livingreviews.org/Articles/lrr-2000-4/node6.html
[28] Sunyaev Zel’dovich effect, Inverse Compton effect from hot
intragalactic plasma http://ned.ipac.caltech.edu/level5/Sept05/Carlstrom/Carlstrom2.html
[31]
George Ellis has proposed that the whole
visible universe may be adjacent to a huge void of comparable size, not within
our light horizon. If so, it would affect all cosmological redshift
measurements including those from which accelerating expansion was deduced.
[33] Gary Kentgen, Spinning Supermassive Black
Holes, Hadronic Journal (in press), Actual black hole masses may be equivalent
to that of whole galaxies.
[35]
Mordehai Milgrom arXiv:0801.3133v2 [astro-ph] 3 Mar
2008
