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”
image #100 in this series
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
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?
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
image 98A in this image series
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.