First of all, note that a potential energy diagram is perfectly possible for a hyperbolic black-hole gravitational field. The only trouble is with convention. Normally, one takes potential energy U as U = 0 at r = infinity. But, U keeps increasing forever with increasing r in the case of the HBH field. It does not level off to an asymptotic value. So, we would need to adopt a different convention, with U = 0 at r = 1. Then, we would have to remember that all U computed for the HBH case will need to be multiplied by -1 in order to be consistent with conventional usage.
We could represent how the ultra-massive universe excited inflaton HBH gravitational field collapses or transitions to the conventional inverse square field, thereby donating its potential energy to what is increasingly an inverse square gravitational universe, accelerating its expansion in the latter 2/3 of its evolution. We might use weighting factors. We could use a linear weighting factor or maybe an inverse square exponential form or even an hyperbolic expression.
Using x = r in the above diagram, let us try weight for the inverse square derived contribution, S = % and weight for the hyperbolic contribution, H = 1.00 – S, 0.00 < S < 1.00, so that U = S( -1/r +1) + (1.00 – S)ln(r), then total U will transition smoothly from the HBH hyperbolic potential energy to the inverse square potential. Here, 1 is a translation amount to let the inverse square derived curve superpose upon the hyperbolicaly derived. The potential energy lost by the HBH phase of the universe is made up by a gain in kinetic energy of expansion in the inverse square phase.
Remember, it is legitimate to think of Hubble expansion of spacetime carrying the objects embedded within it as a kinematic growth process. One need not always regard it as a “stretching” of spacetime, though for some other purposes, this may help.
Obviously, the resulting composite curve will have a significant positive slope on the right, connoting Dark Energy. But, the curves explicitly describe Dark Matter. So there is a strong link between DM and DE.
There have been misstatements and misinterpretations of Birkhoff’s Theorem. For instance, it has been shown by Kristin Schleich and Donald M. Witt (“A simple proof of Birkhoff’s theorem for cosmological constant”, arXiv: 09084110v2, 27oc09) that Birkhoff does not demand staticity in spherically symmetric solutions to Einstein’s vacuum field equations. Static solutions have heretofore been thought of as being required. There may be other misstatements and misinterpretations that are not yet recognized.
For instance, Birkhoff’s Theorem must actually leave black hole singular gravitational fields as an exception to the commonly quoted rigid rule that only asymptotically flat (commonly assumed meaning: inverse square) gravitational fields are allowed. Otherwise there is no way to measure or unequivocally determine that the center of a black hole is a singularity since electric charge and gravity are the only items the influence of which can escape the interior of a black hole. Then the theories of Schwarzschild and Kretschmann that say such singularities are physically real are largely meaningless as unfalsifiable hypotheses.
There simply must be a measurable consequence of a true singularity at the center of a black hole or else its existence cannot be postulated. That the mathematics seems so very precise is not good enough. There must be a way to experimentally verify or falsify the equations.
If the gravitational singularity at the center of a super-massive galactic black hole results in a hyperbolic gravitational field, there is a way. By measuring the velocity distribution of stars in the surrounding galactic disk, it can be determined if they move with a constant velocity, v = (GM)^½ at large r, as they must if they move in a hyperbolic gravitational field. As a matter of fact, the velocities of stars in spiral galaxies do indeed move with constant velocity at large r. This can be seen as proof of a singularity at the center of a spiral galaxy’s black hole.
We often state that “The laws of physics must break down at the incredibly tight curvature of spacetime near the singularity of a black hole”. What does this mean? One thing it could mean is that Birkhoff’s Theorem breaks down too. The metrics to which Birkhoff applies probably are not strictly valid near the singularities that they themselves predict so, the “asymptotically flat” dictum may not be strictly true either. Otherwise, our cautionary statement is meaningless.
Besides, the intense benefit brought by the postulate of a hyperbolic galactic super-massive black hole gravitational field is too great to be ignored. It explains the anomalous stellar velocity distributions in galaxies, anomalous velocity distributions in galactic clusters, galactic lensing phenomena, temperature distributions within galaxies, Bullet Cluster type apparent offsets in the barycenters of colliding galaxy clusters, etc. It does everything that Dark Matter is supposed to do! So, Dark Matter is an unnecessary complication that violates Occam’s Rule.
Cosmologists will not like this idea. The LCDM model would have to be drastically revised. The consensus would have to change. Since journal editors and referees endorse only papers that conform to the consensus (they would have been comfortable with the Pope’s decision to censor Galileo), no-one will publish a paper that challenges the commonly accepted interpretation of Birkhoff’s Theorem. And, if Birkhoff does break down in the vicinity of a gravitational singularity, how can it ever be proven? One would have to develop a whole new physics of the ultra strong curvature near black hole singularities. Unless it also had consequences outside the black hole, such a theory could not be falsified so, it could not be admitted as a part of science.
Catch 22 says “Anyone who wants to get out of combat duty isn’t really crazy.” Hence, pilots who request a mental fitness evaluation are sane, and therefore must fly in combat. We would be better off flying in combat duty than trying to fly the hyperbolic super-massive galactic gravitational field into some journal’s pages.
The amazing thing is that the HBHG field does have consequences outside the BH and the event horizon.