Posts Tagged ‘inflation’

Black Hole Singularities and the Possibility of Two Dimensional Gravity

March 9, 2013

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.

Advertisements

The Hyperbolic Black Hole Galactic and Universe Gravitational Field

February 4, 2012

The Hyperbolic Black Hole Galactic and Universe Gravitational Field

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

http://www.fotothing.com/Gak/photo/f7c4dd2a76b88a78fa2f590a8751b883/ 100/92-100

 image #100 in this series

http://www.fotothing.com/photos/f7c/f7c4dd2a76b88a78fa2f590a8751b883_fa2.jpg

image enlargement

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

                Busy, Busy, Busy

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

https://garyakent.wordpress.com

 First:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Second:

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

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

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

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

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

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

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

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

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

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

ln(x) = 0 and

-1/x +1 = 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 98 A  universe accelerating hubble expansion guth inflation

http://www.fotothing.com/Gak/photo/dd3000c76d3f16f595af18ef135cfad2/ 98/92-100

image 98A  in this image series

http://www.fotothing.com/photos/dd3/dd3000c76d3f16f595af18ef135cfad2_fab.jpg

image enlargement

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