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In summary, Deur's theory is a quantum gravity motivated model that Frame's the effects that explain dark matter and dark energy phenomena very naturally, even though it doesn't resort to any uniquely quantum mechanical effects to get those results.

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Hint: what does the N in MOND stand for?

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Instead, one should try to quantize the relativistic completion of MOND to say anything sensible about gravitons.

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Most of the word in gravity modification is formulated as a modification either of classical Newtonian gravity (e.g. MOND) or as a modification of classical General Relativity (e.g. MOG, TeVeS, f(R), etc.).

Quantizing gravity, in general, is a difficult and somewhat intractable problem. Doing so directly from MOND is basically impossible. It is a toy model theory that uses a very crude and unnatural formula that is not relativistic in nature to get there. Even generalizing it to be relativistic (as TeVeS did) is challenging, although some recent efforts have made progress in generalizing it in a manner that can work for cosmology purposes.

Determining whether space-time modifying quantizations of gravity recover either GR or some modification of GR or Newtonian gravity is highly non-trivial. The last time I looked at the particular question in any depths was two or three years ago and at the time it was the subject of debate whether this had been done. Perhaps progress has been made since then.

Alexandre Deur is done quite a bit of work (much of which is published in peer reviewed journals but not widely cited; the link is to my annotated bibliography of his work, a powerpoint he did, and my own brief background and introduction) developing an approach to getting results in a massless graviton model motivated by the quantum gravity of a spin-2 massless graviton, but approximated numerically with a scalar (i.e. spin-0) massless graviton model that is more tractable mathematically. Nothing about his methods are inherently quantum and can be formulated in classical terms, however.

His results differ from those conventionally assumed to flow from General Relativity in a Newtonian weak field approximation that estimates that non-Newtonian effects are minimal in a non-rigorous manner. But, it isn't really clear if his approach is actually contrary to General Relativity or is just producing results different from General Relativity as conventionally applied, because the conventional approximations of GR in an astronomy and cosmology setting ignore important GR effects. Basically, this amounts to considering terms for gravitational field self-interactions formulated in a manner analogous to those of self-interacting gluons in QCD, rather than in the conventional indirect manner that gravitational field self-interactions are considered when calculated using the Einstein Field Equations.

I mention this approach because it largely reproduces the MOND phenomenology in the domains where MOND works (but performs better in circumstances such as the Bullet Cluster and clusters and makes some additional predictions which are supported by observational evidence) as well as explaining dark energy phenomena.

It does so in a quantum gravity motivated model which frames the effects that explain dark matter and dark energy phenomena very naturally, even though it doesn't resort to any uniquely quantum mechanical effects to get those results. I am not aware of other gravity based explanations of dark matter and dark energy phenomena that have such a transparent quantum gravity motivation (I may have even seen others, but they aren't coming to mind at the moment).

This relationship is recaptured in this theory in disk-like masses.

The gravitational Lagrangian that Deur develops is as follows:

ℒGR=[∂ψ∂ψ]+√G[ψ∂ψ∂ψ]+G[ψ2∂ψ∂ψ]+ΣGn/2[ψn∂ψ∂ψ]+√ψμνTμν

This is derived by expanding the ℒGR in term of tensor gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+G1/2ψμν+...

This is compared to the QCD Lagrangian:

ℒQCD=[∂ψ∂ψ]+√4παs[ψ2∂ψ]+ 4παs[ψ4]

The first terms of each are Newtonian gravity and perturbative QCD respectively (in the static case). The next two terms of the respective Lagrangians are field self-interaction terms.

If the mass is confined to a disk, the self-interactions cause the system to reduce from a three dimensional one to a two dimensional one, causing the force to have a 1/r form that we see in the MONDian regime of spiral galaxies.

In the geometries where Deur's approach approximate's MOND, the following formula approximate's the self-interaction term:

F(G) = G(N)M/r^{2} + (c^{2}(√aπG(N)M)/(2√2)r

where F(G) is the effective gravitational force, G(N) is Newton's constant, c is the speed of light, M is ordinary baryonic mass of the gravitational source, r is the distance between the source mass and the place that the gravitational force is measured, and a is a physical constant that is the counterpart of a0 in MOND (that should in principle be possible to derive from Newton's constant) which is equal to 4*10^{−44} m^{−3}s^{2}.

Thus, the self-interaction term that modifies is proportionate to (√G(N)M)/r. So, it is initially much smaller that the first order Newtonian gravity term, but it declines more slowly than the Newtonian term until it is predominant, because one is a larger term that declines as 1/r^{2} and the other is a smaller term that declines as 1/r. The numerators of both terms are constants with respect to any given disk shaped source mass and the total acceleration due to the gravitational force for a particle in the plane of the galaxy is equal to Newton's constant times M/r^{2} + (14.4√M)/r.

By way of example, for the Milky Way, the self-interaction term is equal to 1% of the Newtonian force at 14.64 parsecs from the center of the galaxy in the galactic plane, 10% of the Newtonian force at 146.4 parces, is equal to the Newtonian force at 1464 parsecs, is 10 tens larger than the Newtonian force at 14.64 kiloparsecs, and is 100 times larger than the Newtonian force at 146.4 kiloparsecs. Earth is about 30 kiloparsecs from the center of the Milky Way galaxy where the self-interaction term for the pull our solar system is receiving towards the center of the Milky Way galaxy is abut 20.5 times larger than the Newtonian term.

But, the self-interaction term would be negative at the same 30 kiloparsecs above (i.e. at a right angle to the plane of the galaxy) making the gravitational pull of the galaxy weaker, in Deur's analysis, rather than being spherically symmetric as it is in MOND.

Deur's approach differs from MOND in the cases of spherically symmetric matter distributions (where the self-interaction effects vanish) and in the case of pairs of point-like masses (where you get the gravitational equivalent of a flux tube in QCD that is stronger than MOND which he used to explain clusters).

Quantizing gravity, in general, is a difficult and somewhat intractable problem. Doing so directly from MOND is basically impossible. It is a toy model theory that uses a very crude and unnatural formula that is not relativistic in nature to get there. Even generalizing it to be relativistic (as TeVeS did) is challenging, although some recent efforts have made progress in generalizing it in a manner that can work for cosmology purposes.

Determining whether space-time modifying quantizations of gravity recover either GR or some modification of GR or Newtonian gravity is highly non-trivial. The last time I looked at the particular question in any depths was two or three years ago and at the time it was the subject of debate whether this had been done. Perhaps progress has been made since then.

Alexandre Deur is done quite a bit of work (much of which is published in peer reviewed journals but not widely cited; the link is to my annotated bibliography of his work, a powerpoint he did, and my own brief background and introduction) developing an approach to getting results in a massless graviton model motivated by the quantum gravity of a spin-2 massless graviton, but approximated numerically with a scalar (i.e. spin-0) massless graviton model that is more tractable mathematically. Nothing about his methods are inherently quantum and can be formulated in classical terms, however.

His results differ from those conventionally assumed to flow from General Relativity in a Newtonian weak field approximation that estimates that non-Newtonian effects are minimal in a non-rigorous manner. But, it isn't really clear if his approach is actually contrary to General Relativity or is just producing results different from General Relativity as conventionally applied, because the conventional approximations of GR in an astronomy and cosmology setting ignore important GR effects. Basically, this amounts to considering terms for gravitational field self-interactions formulated in a manner analogous to those of self-interacting gluons in QCD, rather than in the conventional indirect manner that gravitational field self-interactions are considered when calculated using the Einstein Field Equations.

I mention this approach because it largely reproduces the MOND phenomenology in the domains where MOND works (but performs better in circumstances such as the Bullet Cluster and clusters and makes some additional predictions which are supported by observational evidence) as well as explaining dark energy phenomena.

It does so in a quantum gravity motivated model which frames the effects that explain dark matter and dark energy phenomena very naturally, even though it doesn't resort to any uniquely quantum mechanical effects to get those results. I am not aware of other gravity based explanations of dark matter and dark energy phenomena that have such a transparent quantum gravity motivation (I may have even seen others, but they aren't coming to mind at the moment).

This relationship is recaptured in this theory in disk-like masses.

The gravitational Lagrangian that Deur develops is as follows:

ℒGR=[∂ψ∂ψ]+√G[ψ∂ψ∂ψ]+G[ψ2∂ψ∂ψ]+ΣGn/2[ψn∂ψ∂ψ]+√ψμνTμν

This is derived by expanding the ℒGR in term of tensor gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+G1/2ψμν+...

This is compared to the QCD Lagrangian:

ℒQCD=[∂ψ∂ψ]+√4παs[ψ2∂ψ]+ 4παs[ψ4]

The first terms of each are Newtonian gravity and perturbative QCD respectively (in the static case). The next two terms of the respective Lagrangians are field self-interaction terms.

If the mass is confined to a disk, the self-interactions cause the system to reduce from a three dimensional one to a two dimensional one, causing the force to have a 1/r form that we see in the MONDian regime of spiral galaxies.

In the geometries where Deur's approach approximate's MOND, the following formula approximate's the self-interaction term:

F(G) = G(N)M/r

where F(G) is the effective gravitational force, G(N) is Newton's constant, c is the speed of light, M is ordinary baryonic mass of the gravitational source, r is the distance between the source mass and the place that the gravitational force is measured, and a is a physical constant that is the counterpart of a0 in MOND (that should in principle be possible to derive from Newton's constant) which is equal to 4*10

Thus, the self-interaction term that modifies is proportionate to (√G(N)M)/r. So, it is initially much smaller that the first order Newtonian gravity term, but it declines more slowly than the Newtonian term until it is predominant, because one is a larger term that declines as 1/r

By way of example, for the Milky Way, the self-interaction term is equal to 1% of the Newtonian force at 14.64 parsecs from the center of the galaxy in the galactic plane, 10% of the Newtonian force at 146.4 parces, is equal to the Newtonian force at 1464 parsecs, is 10 tens larger than the Newtonian force at 14.64 kiloparsecs, and is 100 times larger than the Newtonian force at 146.4 kiloparsecs. Earth is about 30 kiloparsecs from the center of the Milky Way galaxy where the self-interaction term for the pull our solar system is receiving towards the center of the Milky Way galaxy is abut 20.5 times larger than the Newtonian term.

But, the self-interaction term would be negative at the same 30 kiloparsecs above (i.e. at a right angle to the plane of the galaxy) making the gravitational pull of the galaxy weaker, in Deur's analysis, rather than being spherically symmetric as it is in MOND.

Deur's approach differs from MOND in the cases of spherically symmetric matter distributions (where the self-interaction effects vanish) and in the case of pairs of point-like masses (where you get the gravitational equivalent of a flux tube in QCD that is stronger than MOND which he used to explain clusters).

Last edited:

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andresB said:

The properties of a massless spin-2 particle that couples proportionately to the mass-energy of a particle (i.e. a "Standard Graviton") is fully constrained and has only one solution, but this solution is difficult to realize in practice in a useable way, to grossly oversimplify, because the math necessary to do it is very hard.

In a classical approximation, the properties of a Standard Graviton are conventionally believed to be equivalent to classical General Relativity, and there is no doubt that this is, at a minimum, very nearly true.

But there could be (and indeed should be) subtle differences between the full quantum gravity behavior of a Standard Graviton and classical General Relativity (e.g. Hawking radiation from black holes).

I am also not entirely convinced that the classical fields that a Standard Graviton approximates the precise conventional Einstein equations of General Relativity, as opposed to some very subtle variant of it, such a conformal gravity or some other technical variant of General Relativity that is well known in General Relativity scholarship.

MOND could arise from classical GR effects that aren't properly estimated in conventional modeling of lambdaCDM with conventional GR, from some subtle flaw in the equations of classical GR in weak gravitational fields, form quantum-specific effects that classical GR doesn't capture, from additional fields with their own carrier bosons (e.g. a set of massless or very nearly massless gravitons of spin-0 and spin-1 as well as spin-2), because the real graviton isn't the Standard Graviton, or because a graviton carrier boson approach isn't actually the proper way to quantize gravity.

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