I very much like the approach of a new paper by Gary Nash that expressly includes a stress-energy contribution from the gravitational field itself and basically solved dark matter and dark energy in the process at the classical level. Time is short, however, so I'll just make this quick and dirty cut and paste from the pre-print index page with just bolding and paragraph breaks in the abstract.

# Modified general relativity

(Submitted on 22 Apr 2019)

A modified Einstein equation of general relativity is obtained by using the principle of least action, a decomposition of symmetric tensors on a time oriented Lorentzian manifold, and a fundamental postulate of general relativity.The decomposition introduces a new symmetric tensorΦαβ which describes the energy-momentum of the gravitational field itself. It completes Einstein's equation and addresses the energy localization problem.

The positive part of, the trace of the new tensor with respect to the metric,Φ describes dark energy.The cosmological constant must vanish and is dynamically replaced byΦ . A cyclic universe which developed after the Big Bang is described. The dark energy density provides a natural explanation of why the vacuum energy density is so small, and why it dominates the present epoch of the universe.

The negative part ofΦ describes the attractive self-gravitating energy of the gravitational field.Φαβ introduces two additional terms into the Newtonian radial force equation: the force due to dark energy and the1r "dark matter" force.When the dark energy force balances the Newtonian force, the flat rotation curves and the baryonic Tully-Fisher relation are obtained.The Newtonian rotation curves for galaxies with no flat orbital curves, and those with rising rotation curves for large radii are described as examples of the flexibility of the orbital rotation curve equation.

## 28 comments:

I do not have the expertise to check the details but this all smells correct, especially when Deur's work is folded in. I have been waiting for someone to do a classical, non quantum, version.

also published in April 2019

"

General Relativity and Quantum Cosmology

A new perspective on galactic dynamics

Matteo Tuveri, Mariano Cadoni

(Submitted on 17 Apr 2019)

We derive the radial acceleration of stars in galaxies by using basic features of thermodynamics, statistical mechanics and general relativity. We assume that the "dark" component of the radial acceleration is originated from the reaction of dark energy to the presence of baryonic matter. It can be also explained as the macroscopic manifestation of a huge number of extremely soft bosonic excitations of the dark energy medium with wavelength larger than the size of the cosmological horizon, in thermal equilibrium with de Sitter spacetime. Our formula agrees with the phenomenological relation proposed by McGaugh et al. which, in turns, fits a large amount of observational data and with the MOND theory. We also show that our formula appears as the weak field limit of Einstein's general relativity sourced by an anisotropic fluid.

Comments: 5 pages, no figures, letter

Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Physics - Theory (hep-th)

Cite as: arXiv:1904.08209 [gr-qc]"

Sounds like that Lee Smolin paper

I suspect MOND is correct and there is no dark matter.

Thanks for the tip.

sure,

I don't see Deur' papers listed in the references arXiv:1904.10803v1

are you saying that Deur's graviton self-interactions is classically equivalent to gravitational energy?

I'm surprised in all these decades there's no paper to produce such a term until arXiv:1904.10803v1

@neo This is correct. In GR as currently applied, there is no express term in the stress-energy tensor or otherwise for gravitational energy. Deur's papers and this one both expressly consider gravitational energy.

An important reason that there's been no such term is that some of the authors of one of the leading GR textbooks have purportedly shown via published proofs that gravitational energy should not be considered expressly (and basically that self-interactions have no effect since they aren't an independent source) and that Einstein's equations implicitly address this even though it isn't expressly considered. This treatment also leads to conclusions that, for example, gravitational energy can't be localized. But, a number of authors, mostly working independently of each other, have argued that this proof is flawed and that, in fact, it is necessary to consider the self-interactions of the gravitational field.

I've always been troubled by this, and the fact that considering self-interactions of the gravitational field expressly seems to solve dark matter and dark energy issues further heightens my skepticism.

I analyze this and some of the papers by others working along these lines at https://dispatchesfromturtleisland.blogspot.com/2014/08/more-on-gravitational-field-self.html and https://dispatchesfromturtleisland.blogspot.com/2018/08/can-gr-really-be-reasonably.html

The textbook is quoted at length at https://dispatchesfromturtleisland.blogspot.com/2015/03/prospects-for-general-relativity.html it says:

"To ask for the amount of electromagnetic energy and momen tum in an element of 3-volume make sense. First, there is one and only one forula for this quantity. Second, and more important, this energy-momentum in principle "has weight." It curves space. It serves as a source term on the righthand side of Einstein's field equations. It produces a relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is observable. Not one of these properties does "local gravitational energy-momentum" possess. There is no unique formula for it, but a multitude of quite distinct formulas. The two cited are only two among an infinity. Moreover, "local gravitational energy-momentum" has no weight. It does not curve space. It does not serve as a source term on the righthand side of Einstein's equations. It does not produce any relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is not observable.

Anybody who looks for a magic formula for "local gravitational energy-momentum" is looking forthe right answer to the wrong question. unhappily, enormous time and effort were devoted in the past to trying to "answer this question" before investigators realized the futility of the enterprise. Toward the end, above all mathematical arguments, one came to appreciate the quiet but rock-like strength of Einstein's equivalence principle. One can always find in any given locality a frame of reference in which all local gravitational fields" (all Christoffel symbols . . .) disappear. No [Christoffel symbols] means no "gravitational field" and no local gravitational field means no "local gravitational energy-momentum."

Nobody can deny or wants to deny that gravitational forces make a contribution to the mass-energy of a gravitationally interacting system. The mass-energy of the Earth-moon system is less than the mass-energy that system would have if the two objects were at infinite separation. The mass-energy of a neutron star is less than the mass-energy of the same number of baryons at infinite separation. Surround a region of empty space where there is a concentration of gravitational waves, there is a net attraction, betokening a positive net mass-energy in that region of space (see Chapter 35). At issue is not the existence of gravitational energy, but the localizability of gravitational energy. It is not localizable. The equivalence principle forbids it."

Links to the discussion of how traditional GR does or does not treat gravitational self-interaction is at https://dispatchesfromturtleisland.blogspot.com/2013/12/some-physics-conjectures-related-to.html

interesting,

what I like about this is that in MOND there's no good general relativistic generalization, but here it's GR plus some extra concepts.

Deur's approach presumes gravitons which may or may not exist.

where does conformal gravity fit in all this, which also reproduces RAR and BTFR?

the paper i cited is in the spirit of Verlinde and Smolin, that dark energy interacts with baryonic matter to give rise to MOND like effects like BTFR and RAR at the classical level.

so it seems there are multiple approaches to MOND like physics from multiple angles.

to fully replace dark matter, other tests include galaxy clusters, large scale structure formation, gravitational lensing and CMB

Deser is particularly at fault for suggesting that self-interaction can work in the ways that others have supposed.

I read on researchgate that a German physicist called Stefan Bernhard Rüster could prove that Gary's modified gravity is really wrong. You can find what I read in the RG question " Does a gravitational field have an energy density like an electric field?

The disproof of Gary's modified general relativity is found here https://www.yumpu.com/en/document/view/66160748/disproof-of-gary-nashs-modified-general-relativity-mgr

The disproof of modified general relativity is fundamentally wrong. It assumed in the vacuum that the energy-momentum tensor is orthogonal to the Einstein tensor whereas the tensors are equal and opposite.

The dispoof of Gary's modified general relativity is correct and was checked by experts in both physics and mathematics.Also, Gary's theory is really like an imitation of a theory of gravity proposed by two Chinese scientists about ten years ago.Gary even imitated the format used by those two Chinese scientists in their paper.

The comment above is erroneous and is likely written by the person or his associate who tried prove the theory wrong. The Chinese scientists mention by Nash, Ma and Wang, only dealt with a Riemannian spacetime, whereas spacetime is clearly Lorentzian and thus the orthogonal decomposition developed by them does not generally apply. In fact, their decomposition involved the covariant derivative of the partial derivative of a scalar whereas York and Deser previously showed that a symmetric tensor on a Riemammian manifold is decomposed as the covariant derivative of a vector field. Only when the vector field is the derivative of a scalar would Ma and Wang's result be equivalent to Deser's and York's. Nash's orthogonal decomposition theorem is constructed from the Lie derivative of a Lorentzian metric and a product of unit line element vectors (which is equivalent to the Lie derivative of a Riemannian metric) along the flow of a nonzero line element vector field. It generalized York and Deser's results. So whoever is trying to tell a bogus story about modified general relativity is a truly uniformed soul. MGR did initially have a non-material error in its orthogonal decomposition theorem which was corrected by Nash as shown in the recently published paper "Modified general relativity and quantum theory inn curved spacetime" in the International Journal of Modern Physics A Vol. 36, No. 29 (2021). This thread is supposed to involve a reliable discussion of physics and mathematics, but it is turning into a medium to spew mistruths rather than a source of correct and informative comments.

I think Gary's theory lacks a lot of physical logic and common sense .He adds a function called Phi that he took from Man and Wang's paper and claimed that it represents the gravitational field, and this Phi vanishes when there is a Killing vector so there is no more gravitational field. This is really physicallying ridiculous.

The misinformed person who is now obviously Stefan Ruster or his associate is so clueless that he does not know the difference between a Riemannian and a Lorentzian spacetime. As stated Ma and Wang tried to describe gravitational energy-momentum in a Riemannian spacetime with a tensor defined by the covariant derivative of the derivative of a scalar. Modified GR is uniquely more general involving the Lie derivative. The comment above is true that Phi_ab vanishes if and only if a Killing vector is involved. This exposes the ignorance of the previous poster because a Killing vector exists iff a symmetry exists which in general is not the case.

I agree that the theory of Gary is an imitation or copy from Ma-Wang theory.I also agree that gravity is not about Killing vectors. Gary's theory is surely wrong and he was not successful in what he proposes. I also read the disproof of Rüster and I think he shows good point of view in his two first equations.I agree with Stefan Bernhard Rüster.

Anonymous is likely Stefan Ruster or alternatively is influencing an associate, and yes, I am Dr. Gary Nash defending against Ruster's perpetual mistruths. He/they seem to have the umlaut on their computer which suggests they are from Germany or an associated country. The comments from Ruster et al. as evidenced above prove they do not understand the difference between Riemannian and Lorentzian spacetimes and many other things. MGR is the natural extension of GR by developing a connection invariant covariant symmetric tensor Phi_ab that is the part of the total energy-momentum tensor that represents the energy-momentum of the gravitational theory in a Lorentzian spacetime. Only a sadly mathematically misinformed person would agree with Ruster's erroneous attempt to prove that Phi_ab is always zero. In fact it vanishes iff X is a Killing vector, which they now sidestep by saying "gravity is not about Killing vectors". Actually, a lot of gravity is about Killing vectors when one wants to expose a particular symmetry such as the maximally symmetric Universe described by the FRLW metric. However, Phi_ab of MGR has nothing to do with symmetries in general.

They now state that "he was not successful in what he proposes" and yet in "Modified general relativity and quantum theory in curved spacetime" published in the journal "International Journal of Modern Physics A" Oct. 2021 the unification of gravity and QT is geometrically shown by the Lie derivative of the metric which is the result of symmetrizing the asymmetric Klein-Gordon spin-1 wave equation in curved spacetime. It was shown that the particle theory of QT and the geometric theory of GR/MGR are not alien to one another, but the geometric approach solves the hierarchy problem and introduces entanglement from the diffeomorphic invariance of the scalar Lagrangians involved.

So go ahead and say anything you want Ruster and friend (or conversely, whatever the case may be) because there is nothing intelligent that either of you (or you) will ever be able to say about MGR, a theory you or both of you do not and probably never will understand. Ruster has already admitted he doesn't understand differential geometry; in his words, it is too dry to learn. However, that just means it is too hard for him and his "buddy" (if he exists) to learn. You should know better than to spew mistruths as there are legal implications of doing so that I would not hesitate to pursue.

I am the one who wrote the comment just before the last one. In case you are Gary Nash who wrote the last comment, it is surely wrong to say that when there is a gravitational field,there is no Killing vector, and when there is a Killing vector,there is no gravitational field.

Why are you so upset and nervous?!

Whoever you are, why are you afraid to expose yourself? You obviously do not understand what you write. You just said "it is surely wrong to say that when there is a gravitational field, there is no Killing vector, and when there is a Killing vector, there is no gravitational field."

If you understood that a connection independent symmetric tensor maintains its value when the connection coefficients vanish, then you would not blurt out the first part of your statement. Phi_ab is constructed from the Lie derivative of both the metric and a product of unit line element vectors, along the line element vector. The Lie derivative has the same value when expressed in covariant or partial derivatives. Second, you say a gravitational field does not exist when a Killing vector exists despite me pointing out to you the exact opposite; a maximally symmetric Universe in 4-D has 10 Killing vectors. You have things backwards which doesn't surprise me in the least. I will not respond to you any further and I am certainly not nervous, but you should be!

Whether you are Gary Nash or not, you are the one and not me who claims that when there is a gravitational field,there is no Killing vector, and when there is a Killing vector,there is no gravitational field.In all this dispatch, this comment is my second so please do not misjudge people. It is clear that what you propose is wrong, and please do not reply to my comment.

What is clear is that you contradict the facts and yourself. Period

What is clear is that you contradict yourself and the proof is that you previously wrote that you would not respond to me but you did. It seems that you are lost in physics and in your actions.

The disproof of Stefan Rüster is well written and clear. What is proposed by Gary in his paper that when there is a Killing vector the gravitatiobal field vanishes is balderdash and illogical and not physical.

I see that the ignorance of Ruster or his associate still festers on this site. Again, we see that they/he maintain/s I said "when there is a Killing vector the gravitational field vanishes", which is not what is stated in the paper "Modified general relativity and quantum theory in curved spacetime" with a proof: "Phi_ab vanishes if and only if X^b is a Killing vector". The gravitational field is described by the connection coefficients, which do not vanish if a Killing vector is involved. The gravitational energy-momentum tensor does vanish if the Lie derivative is along a Killing vector. Then, there is an isometry in the spacetime and the metric does not change along the flow of that vector; neither do the unit line element vectors. The introduction of that Killing vector describes an uninteresting Lorentzian spacetime which cannot achieve a state of maximal symmetry such as that of in the FLRW metric. Obviously, you do not understand the difference between a connection and a gravitational energy-momentum tensor, which must be connection independent to be in agreement with the equivalence principle.

You, whoever you are but if you are not Ruster, how about his buddy or penname "Adam Dutch", sound like a struggling undergraduate who has a very poor understanding of the mathematics required to understand curved spacetime. When you want to criticize another's work, you should at least state who you are so that other's can assess your credibility for themselves.

I do not agree with Gary Nash because a gravitational field can exist when there is a Killing vector even in vacuum. The paper written by Gary looks like an imitation of Ma-Wang publication from which Gary copied many things from their theory.

Unfortunately, the previous poster, who is very likely the same "Ruster" or Adam Dutch or whatever he calls himself, again exposes his ignorance. First, as carefully explained above, it is true that "a gravitational field can exist when there is a Killing vector even in vacuum" which is precisely what the FRLW metric describes with it maximum 10 Killing vectors as noted above. Second, and for the last time, Ma and Wang used a deficient decomposition of orthogonal symmetric tensors in a Riemannian spacetime. As stated, Deser and York properly did the decomposition decades before Ma and Wang tried to do it; they should have been able to at lest duplicate those results but instead chose to pursue Sobelev spaces of tensor fields, which led to their reliance on a scalar rather than a vector field. I used the powerful Berger-Ebin theorem of Riemannian geometry and adapted that to a Lorentzian spacetime. For you to claim that is an imitation of Ma and Wang's work is a completely erroneous and ignorant comment. The only similarity in the papers was that the papers led to different explanations of dark matter and dark energy. Theirs did not invoke the line element field in any way whatsoever, and neither did the work of Deser and York. Obviously you won't expose yourself to avoid making a fool of yourself.

I agree with the disproof of Stefan Bernhard Rüster;it shows that a gravitational field exist even with Killing vectors in vacuum.The Phi and other concepts in Gary theory are taken from Ma-Wang theory.

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