Dark Matter is an unnecessary ad hoc fix to fill in the blanks in the Friedmann model under the FLRW metric. Galactic supermassive black-holes exist as true physical singularities according to the Kretschmann invariant and Schwartzchild’s analysis of his spacetime metric under GR. Therefore, as point masses, they must possess a hyperbolic (1/kr) gravitational field, NOT a field that falls off as 1/r2. Now, k = constant = 1m, S.I., for dimensional integrity. It is not true that GR cannot tolerate hyperbolic spacetime geometries. “The universe is hyperbolic.” said Albert Einstein in his classic paper of 1915. An hyperbolic field will give constant orbital acceleration to orbiting bodies as far from the center of a black-hole as we might like to measure. This means that bodies near the periphery of a galaxy should seem to move at constant velocity because rotational acceleration does not drop to near zero there as with a 1/r2 inverse square law, it becomes consant. This constant velocity distribution effect has actually been measured and has given rise to the notion of Dark Matter.
Gravitation does not fall nearest to zero between galaxies in a cluster either. So they too can bend light and affect redshifts in ways that mimic Dark Matter. The rotation of galaxies in clusters is also influenced by the black-holes that they contain with their 1/kr gravitational potential profiles. The not quite counterbalanced redshift effects in the Sunyaev-Zeldovich phenomenon are influenced by the hyperbolic galactic and galactic cluster gravitational fields that exist as light falls out of such clusters and super-clusters into a large void and as it climbs out of it again after the universe has expanded by another billion light years or more.
Scientists are mapping, not Dark Matter, but the huge extent of the network of hyperbolic galactic and super-galactic gravitational fields that behave like Dark Matter because of the mathematical properties of the hyperbolic gravitational field are similar to that expected for Dark Matter.
Primordial massive and supermassive black-holes with their 1/kr galactic gravitational fields can also mimic the “halos” of dark Matter that are postulated to have existed just after the big bang and before the emission of the cosmic microwave background. There is nothing that Dark Matter explains that cannot be accounted for just as well or better by the hyperbolic black hole gravitational field.
The hyperbolic 1/kr supermassive black-hole galactic gravitational field explains “the Dark Matter Effect” without Dark Matter and it is more parsimonious and is a falsifiable hypothesis, unlike Dark Matter which is revised every time no Dark Matter is found.
The conditions for validity of Birkhoff’s Theorem are not met for real black-holes. Therefore, Birkhoff’s Theorem does not apply. It sometimes may be used as a first approximation, but it cannot be depended upon as a rigid rule for precise calculations. “The physics near at the extreme curvature of a black-hole singularity is not well defined”. This covers Birkhoff’s too.
It does too matter how the internal mass is distributed if it is contained within a single point. Then, in fact, it is NOT distributed at all! This is the point of Kretschmann’s invariant and Schwartzschild’s GR analysis of the consequences of his metric. Ordinarily, the distribution would not matter. But, a singularity must be different. If this is not explicitly acknowledged in some way, then to say there is a singularity with such intense curvature of spacetime in its vicinity that the laws of physics must begin to break down is a meaningless fatuous gesture to humility. It is false humility if it has no ameliorating effect on professional arrogance. Please, do not just restate Birkhoff.
I contend there is a loophole here or a gross misinterpretation. The consensus interpretation of Birkhoff and of Schwartzschild/Kretschmann cannot both be true at the same time. There must be a measureable consequence of the presence of a singularity that is beyond imaginary untestable gedanken experiments. The test is the hyperbolic gravitational field. It results in a nonzero constant rotational velocity distribution effect in spiral galaxies, ellipticals, globulars and galactic clusters. This is easier to believe than Dark Matter.
The very same phenomena that are used to argue for Dark Matter can be used to argue for the hyperbolic field. So, it is testable. But, how do we choose between them? I think that Occam ’s razor is the principle of choice here. WIMPS and neutralinos and the other oddball particles that have been proposed require ad hoc additions to theories or their complete rewrites. The hyperbolic field is far simpler. All that is needed is acknowledgement that the black hole singularity is unique. No rewrite of GR. No undetectable new heavy particles that get given self-serving, revised, lower detection limits every time they are determined to be really undetectable.
There seems to be a tendency of cosmologists to think inside the box. They never really consider anything that is outside the consensus. So too do journal editors rely on conventional wisdom. They would all have been supremely comfortable with the Pope’s decision to censor Galileo.
“Cosmologists are always wrong, but never in doubt.” Lev Landau
Tags: acceleration, asymptotic value, constant velocity, cosmology, dark matter, galactic cluster, general relativity, hubble expansion, hyperbolic spacetime, inverse square potential, supermassive black holes, testable consequences, universe, velocity distribution