According to general relativity, in a 3-D universe with time, the gravitational field of all compact objects behaves as if the objects are point masses and the field strength must decline as 1/r2. In a 3-D universe, therefore it is said, it is impossible to support a hyperbolic 1/r gravitational field. But, black holes are different.
Why bother with the whole concept of black holes if they are not different? Collapse of matter into a black hole must not only create a singularity (within the limits imposed by the Heisenberg uncertainty principle) but, the spin rate or orbital frequency of in-falling matter of the black hole must also increase without bound as radius r decreases to values near zero below the event horizon. Attempts to explain away these singularities on the basis of a non-existent quantum gravity scheme are vacuous extrapolations of tentative hypotheses that amount to pure conjecture.
Black hole singularities exist. Einstein through Schwarzchild and others say so. Who claims to be more brilliant than these fellows? I appeal to authority here only because it seems to be the only thing that impresses some. If you want to claim that BH singularities are mere artifacts of an inadequate theory, show me the Math.
Black holes are different. When matter and energy collapse under an infinitely strong gravitational field to a point mass that is as tiny as may be necessary to explain its properties (not necessarily to zero, the true meaning of infinity), the result is a phase change. Spacetime phase changes are S.O.P. in the repertoire of theoretical cosmologists, like Alan Guth. Let us adhere to the hydrodynamics metaphor used by Einstein in his development of GR. Flat spacetime is a massless superfluid. Helium IV is a superfluid but, it is not massless.
To extend the metaphor, it is not hard to imagine that spacetime could undergo a phase change, just as helium IV may. In a black hole this change involves a reduction in dimensionality. This is about the only change available to it. Analysis of the equations of GR shows that gravitational strength, Fg is proportional to 1/r(n-1) where n = number of spatial dimensions. In a 3-D universe, Fg declines as 1/r2. In a 2-D universe, Fg declines as 1/r.
Using basic geometry, GR allows gravitational fields that decline as 1/r(n-1) where n is defined as above. So, if there is to be a hyperbolic SBH gravitational field it must occur in 2-D. The gravitational field strength in its local limit is defined as flux through a surface element of a n-sphere; that surface area behaves as 1/r(n-1). One can find the derivation of this relation in any good geometry textbook.
So, a black hole must use the gravitational energy of in-falling matter to raise its gravitational potential, the gravitational energy level, to the 2-D “state”. We are starting to talk quantum language now.
The shape of this singular BH gravitational field strength diagram, as it is a 2-D entity embedded in a 3-D space, is a nominally flat disk or platter with a potentially infinite radius. Unlike Kerr, I call this topology of the event horizon a “spin disk” because it arises from the infinite rotational and orbital spin rate of matter that has in-fallen toward the singularity. As a spacetime entity, this new phase ignores the event horizon and propagates outward to beyond the edge of the galaxy. It emanates from the central core of the galaxy wherein resides any central supermassive black hole.
Here is one of the non-intuitive consequences of GR. It is known that matter in-falling toward the event horizon must experience time dilation. From our external perspective, we would perceive time for this matter as having slowed and even stopped at the event horizon. Viewed from any point outside the event horizon, time really does stop there. But from its own perspective time does not stop and such matter does indeed drop through the event horizon where it may take part in whatever processes it might (time reversed or not).
There is simultaneously an inverse square gravitational field set up by this time-frozen matter at the event horizon and an inverse gravitational field set up by this same matter that has already in-fallen to the singularity. There is no violation of conservation laws here because no object can feel these separate effects simultaneously. If an object orbits the galactic center in the plane of the galactic disk, it feels the inverse 1/r field. If it orbits on a trajectory not aligned with the galactic plane, if it orbits chaotically, it feels the inverse square 1/r2 field.
This has consequences for the analysis of the orbital motion of close-in Milky Way bulge stars like S2 for the determination of the MW’s supermassive black hole mass according to Kepler’s laws. Kepler is valid for the 2-D case as well as for the 3-D case. But, it has to be adjusted for the 1/r gravitational field as does Newton’s law. No deep relativistic calculations are needed. One can determine what changes must be made in Newton’s law and Kepler’s laws by inspection.
Above, I explained how a 2-D gravitational field can exist in our 3-D universe. It must be associated with a black hole having an infinite spin rate as well as infinite density and infinite gravitational field strength. Within the bounds of Heisenberg uncertainty, these singularities must exist. There is no point in trying to explain them away using some kind of unfalsifiable overly advanced unintelligible gravitational quantum sophistry.
I show that the hyperbolic 1/r inverse gravitational field can exist as a spin disk surrounding any black hole with said disk extending far beyond the event horizon toward infinite radius. This explains MOND and the anomalous velocity dispersion because hyperbolic 1/r gravity means that orbital velocity, v, around a galactic center containing a black hole, v = (GMbh)1/2. That is, v becomes constant dependent only on Mbh and G. This v is not only constant for a given galaxy, it is constant from galaxy to galaxy. This means that GMbh must itself be a constant, implying a new fundamental physical constant. Remember that Noether’s theorem concerning invariants, fields and symmetry is engaged whenever a fundamental constant is invoked.
But, G may not be the same G that applies in 3 dimensions. So, I call it G*. Besides by an extension of GR, one might get G* from the M‑sigma relation as well as by the anomalous velocity dispersion. But, the mass of the central galactic supermassive black holes must first be refigured on the basis of the hyperbolic field if very many of the orbits of the bulge stars that were used to get Mbh were coincident with the galactic plane. If all or most of these orbits were chaotic and not aligned with the galactic plane, the BH mass determinations may be okay.
Perhaps the main expression of the Postulate should be written v = (G*Mbh)1/2 from 2-D Newton’s law. Then G* = v2/Mbh for any galaxy with any size supermassive black hole with its associated vavd = vσ = v. In order for G* to be constant, v2 and Mbh must vary inversely. We see that, from the M-sigma relation, they do.
So, we have a relation between Mbh and v as vavd or vσ that is linear. So, the slope is Mbh/v = a constant . Then Mbh/v2 = a similar constant and v2/Mbh = a constant also. The Postulate says G*Mbh = v2 and so G* = v2/Mbh, as above.
When doing any calculation involving the hyperbolic (1/r) gravitational field, remember that Fg = G*Mm/rr1 where r1 is the unit vector of r. This ensures dimensional integrity.
In the M-sigma relation, the exponent of v or σ has been expressed as being equal to either 4 or 5 with constants adjusted as appropriate to compensate for the difference. In the past, this was done arbitrarily, for convenience. There is now theoretical grounds to choose the exponent of v such that exp(v) = 2 when using the M-sigma relation for bulge stars orbiting in the plane of the galactic disk. When analyzing the AVD one finds that vavd = virtually a constant within and among galaxies. When the scatter in the data is accounted for, the Postulate predicts that this will be found to be untrue and vavd2 = G*Mbh, especially when the effect of embedded massive BHs in the body of each galaxy is made up for.
I think that σ is not distant enough from the singular center for the hyperbolic field to overcome chaotic effects. Perhaps v should be based on 2σ or on the velocities of stars orbiting in the plane of the galactic disk just beyond 98% of the average orbital radii of bulge stars. This suggests the wonderful possibility for much more research.
The Postulate explains both the anomalous velocity dispersion and the M‑sigma effect.
It may not be quite this simple because all spiral galaxies have massive black holes embedded within them, besides their central black hole. Because of the flat hyperbolic (1/r) gravitational field’s longer reach, the gravitational, spacetime spin disks of these BHs would tend to align with that of the central black hole’s so that their hyperbolic fields would combine. So G* got from vavd may not be so pure and pristine.
The meaning of the hyperbolic gravitational field of black holes is that MOND (Modified Newtonian Dynamics, suggested by Mordehai Milgrom) is explained without recourse to Dark Matter or to modifications of Newtonian dynamics. Newton and Kepler must be understood in two dimensions, that is all.
All of the observations that are said to support Dark Matter as being, say, a huge halo of WIMPs engulfing galaxies and galactic clusters also support the hyperbolic (1/r) gravitational field postulate, even the Bullet Cluster effect. Dark Matter 3-D maps obtained by analysis of gravitational lensing also follow logically from the Postulate.
The hyperbolic (1/r) supermassive black hole gravitational field is indeed a postulate. This means that there can be no argument against it. It must be taken at face value and carried to its logical extreme whereupon it will be either reduced to absurdity or else found to be correct.
When extrapolated to the entire universe, the hyperbolic field mimics Dark Energy too. If Alan Guth’s inflaton particle originated in 2-D space and began to roll down its own hyper-gravitational super-potential slope toward a lower energy 3-D state, the higher energy 2-D potential energy would be progressively transformed in a time dependent quantum-like transition to the new 3-D “ground state”. This potential energy would show up as apparently increasing kinematic momentum of all stars and galaxies in the universe. That is, the universe would appear to be expanding at an accelerating rate.
This is an exciting idea because the whole universe is thus to be regarded as a quantum object. It may provide a route to a falsifiable certifiable theory of quantum gravity because 2-D gravity does not lead to a gravitational catastrophe as r tends to zero, due to Heisenberg uncertainty’s prior restraint in this case. And, it is renormalizable, a prerequisite for any quantum theory of gravity. This Postulate may point to a means to prove the existence of the multiverse. If Guth is right, Hugh Everett could be right.