MOND from MacDowell-Mansouri geometrical formulation

[kodama]

TL;DR Summary

. Moreover we make contact with the Modified Newtonian Dynamics (MOND) approach for gravity

One poster is a strong promoter of Deur and Deur theory of how to get MOND out of GR via self-interaction and analogy to QCD.
There is intense skepticism of Deur's approach, which has 0 citations other than the author.

I saw this paper,

High Energy Physics - Theory​

[Submitted on 8 Oct 2022]

Non-relativistic gravity theories in four spacetime dimensions​

Patrick Concha, Evelyn Rodríguez, Gustavo Rubio

In this work we present a non-relativistic gravity theory defined in four spacetime dimensions using the MacDowell-Mansouri geometrical formulation. We obtain a Newtonian gravity action which is constructed from the curvature of a Newton-Hooke version of the so-called Newtonian algebra. We show that the non-relativistic gravity theory presented here contains the Poisson equation in presence of a cosmological constant. Moreover we make contact with the Modified Newtonian Dynamics (MOND) approach for gravity by considering a particular ansatz for a given gauge field. We extend our results to a generalized non-relativistic MacDowell-Mansouri gravity theory by considering a generalized Newton-Hooke algebra.

Comments:	20 pages
Subjects:	High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
Cite as:	arXiv:2210.04101 [hep-th]

on section 3.3 page 8- 9 the authors derive MOND from the cosmological constant and generalized non-relativistic MacDowell-Mansouri gravity theory,

the observed cosmological constant is an input, is this convincing way to derive MOND from MacDowell-Mansouri gravity theory?

for background,

General Relativity and Quantum Cosmology
[Submitted on 30 Nov 2006 (v1), last revised 15 May 2009 (this version, v2)]
MacDowell-Mansouri gravity and Cartan geometry
Derek K. Wise

The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous "model spacetime", including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A "Cartan connection" gives a prescription for parallel transport from one "tangent model spacetime" to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as "rolling" the model spacetime along physical spacetime. I explain Cartan geometry, and "Cartan gauge theory", in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its more recent reformulation in terms of BF theory, in the context of Cartan geometry.

Comments: 34 pages, 5 figures. v2: many clarifications, typos corrected
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Differential Geometry (math.DG)
Cite as: arXiv:gr-qc/0611154

 

[strangerep]

TL;DR Summary: . Moreover we make contact with the Modified Newtonian Dynamics (MOND) approach for gravity

Alas, I'm going to call this out as a fudge.

On the 2nd half of p9, they simply introduce an Ansatz, roughly equivalent to $\Gamma^i_{00} = [\mu(x) + 1]\, \partial^i \varphi(x)$ , with $\varphi(x) = \phi(x) - \frac{9}{2\ell^2} x^k x_k$.

The $\mu(x)$ is intended to be the MOND interpolation function, but they don't derive it -- they assume it.

Sheesh. That's a couple of hours of my life I'll never get back.

 

[kodama]

Alas, I'm going to call this out as a fudge.

On the 2nd half of p9, they simply introduce an Ansatz, roughly equivalent to $\Gamma^i_{00} = [\mu(x) + 1]\, \partial^i \varphi(x)$ , with $\varphi(x) = \phi(x) - \frac{9}{2\ell^2} x^k x_k$.

The $\mu(x)$ is intended to be the MOND interpolation function, but they don't derive it -- they assume it.

Sheesh. That's a couple of hours of my life I'll never get back.

The $\mu(x)$ is intended to be the MOND interpolation function, but they don't derive it -- they assume it.

could you or anyone else derive it

 

[strangerep]

The $\mu(x)$ is intended to be the MOND interpolation function, but they don't derive it -- they assume it.

could you or anyone else derive it

Heh, if I had a solid foundational theory that yielded MOND-like behaviour, I'd be publishing papers about it -- not lazing around twiddling my thumbs here on PF.

 

[kodama]

Heh, if I had a solid foundational theory that yielded MOND-like behaviour, I'd be publishing papers about it -- not lazing around twiddling my thumbs here on PF.

let me be more specific
what would be necessary for MacDowell-Mansouri formulation The ux is intended to be the MOND interpolation function, and they don't derive it

they've stated the the Poisson equation in presence of a cosmological constant, so what more need be done

 

[strangerep]

what would be necessary for MacDowell-Mansouri formulation

Imho, appealing to MacDowell-Mansouri in this context is a red herring. Lots of people are interested in the M-M formulation, so a reasonable paper involving something on that subject is likely to get published.

they've stated the the Poisson equation in presence of a cosmological constant,

So what? One can get that by taking a weak field (Newtonian limit) of the Einstein field equations with CC. (A Poisson-like equation is not MONDian -- do you know why?)

so what more need be done

To have any hope of offering you a better (attempt at an) answer, please summarize your current understanding of the central features of MOND, their motivation, and why it works as puzzlingly well as it does. Just a few paragraphs will do -- so I can see at what level I should pitch my answer.

[My radar senses a new MOND thread approaching...]

 

[haushofer]

I haven't read the paper, but for my PhD-project we used used a similar procedure to derive Newton-Cartan gravity, including a cosmological constant (they also reference to my first paper on this subject). The subtlety here is that in the non-relativistic case, you obtain the Poisson equation. And the cosmological constant in this case acts as a spring constant, which can be put into the Newton potential. As I remember, this effectively gives the same theory as when you apply the gauging procedure to the Newton-Hooke algebra, which of course is the symmetry algebra of "a non-relativistic spacetime with cosmological constant". This differs from the relativistic case, where a redefinition of the metric doesn't magically give you a cosmological constant.

I guess these authors did something similar with MOND.

 

[haushofer]

But I have thought about this issue myself. Somehow, the dynamics of MOND should be reflected by the symmetries of the corresponding spacetime. Gauging these symmetries should lead you to MOND, but I'm not acquinted well enough to see whether leads you to the same trivial redefinition of the Newton potential or not.

So we know that gauging the Bargmann algebra leads you to Newtonian gravity; the gauging of which deformation of the Bargmann algebra gives you MOND? When I find the time I'll check whether the paper answers this question :P

 

[strangerep]

I haven't read the paper, but [...]

I guess these authors did something similar with MOND.

I don't think so. (It's usually advisable to read a paper before guessing about it, no?)

 

[strangerep]

But I have thought about this issue myself. Somehow, the dynamics of MOND should be reflected by the symmetries of the corresponding spacetime. Gauging these symmetries should lead you to MOND, but I'm not acquainted well enough to see whether leads you to the same trivial redefinition of the Newton potential or not.

This is a common misunderstanding of newbies-who-seek-a-MOND-foundation.

Focusing just on getting a $ln(r)$ potential (hence a $1/r$ force) ignores how MOND incorporates the Tully-Fisher relationship ($v^4_{tan} \propto M$) from the start.

Focusing just on transitioning to a scale-invariant theory for low accelerations ignores the fact that there are other ways to fudge a scale-invariant force law (e.g., multiplying by $\Upsilon t$, where $\Upsilon$ is some constant related to MOND's $a_0$). But this doesn't give Tully-Fisher either.

So we know that gauging the Bargmann algebra leads you to Newtonian gravity; the gauging of which deformation of the Bargmann algebra gives you MOND?

I doubt this would somehow turn the usual geodesic equation of motion into a new one involving acceleration squared (or a square-root of the metric) -- which is what's needed to mimic how MOND incorporates. But I'm open to being persuaded differently.

 

[haushofer]

I don't think so. (It's usually advisable to read a paper before guessing about it, no?)

Well, ok, I'll keep my misinformed guesses for myself then and see if I can find the time to read it.

 

[haushofer]

This is a common misunderstanding of newbies-who-seek-a-MOND-foundation.

What would you consider to be a good (online paper) introduction to the field?

 

[mitchell porter]

Focusing just on getting a $ln(r)$ potential (hence a $1/r$ force)

... me, just yesterday ...

ignores how MOND incorporates the Tully-Fisher relationship ($v^4_{tan} \propto M$) from the start.

... oops! Thanks for the warning, I didn't realize you knew MOND this well.

For those curious about how MOND implies Tully-Fisher:

In the low-acceleration regime, the MONDian acceleration is $\sqrt{(a_0 g_N)}$, the geometric mean of the transition acceleration $a_0$ and the Newtonian acceleration $g_N$, i.e. the geometric mean of a constant and an inverse square, thus the $1/r$ force.

This means (following slide 12 here) that at large radius $r$, the MONDian acceleration $a$ = $v^2/r$ = $(G M a_0)^{1/2}/r$, thus $v^2 \propto M^{1/2}$.

So we know that gauging the Bargmann algebra leads you to Newtonian gravity; the gauging of which deformation of the Bargmann algebra gives you MOND?

It's remarkable that the only gauge-theoretic model of MOND that I can find, is the totally obscure "gauge vector-tensor gravity", which may or may not be related to the bimetric approach to MOND... No idea how well it performs.

Again for those interested, here is "Newtonian Gravity and the Bargmann Algebra".

 

[kodama]

This is a common misunderstanding of newbies-who-seek-a-MOND-foundation.

Focusing just on getting a $ln(r)$ potential (hence a $1/r$ force) ignores how MOND incorporates the Tully-Fisher relationship ($v^4_{tan} \propto M$) from the start.

Focusing just on transitioning to a scale-invariant theory for low accelerations ignores the fact that there are other ways to fudge a scale-invariant force law (e.g., multiplying by $\Upsilon t$, where $\Upsilon$ is some constant related to MOND's $a_0$). But this doesn't give Tully-Fisher either.

does Deur derive the Tully-Fisher relationship ($v^4_{tan} \propto M$) from the start.

 

[strangerep]

does Deur derive the Tully-Fisher relationship ($v^4_{tan} \propto M$) from the start.

Well,... do any of his papers mention "Tully-Fisher" at all? If so, where precisely?

 

[strangerep]

[...] Thanks for the warning, I didn't realize you knew MOND this well.

Oh, I wouldn't say I know it "well". No precious, not well at all.

The associated experimental results about galactic astrophysics are... mountainous. It's astonishing how much detail those guys can extract from mere pin pricks (or less) in the sky.

For those curious about how MOND implies Tully-Fisher:

In the low-acceleration regime, the MONDian acceleration is $\sqrt{(a_0 g_N)}$, the geometric mean of the transition acceleration $a_0$ and the Newtonian acceleration $g_N$, i.e. the geometric mean of a constant and an inverse square, thus the $1/r$ force.

This means (following slide 12 here) that at large radius $r$, the MONDian acceleration $a$ = $v^2/r$ = $(G M a_0)^{1/2}/r$, thus $v^2 \propto M^{1/2}$.

Careful! That way of phrasing it fosters misunderstandings. MOND does not truly "imply" Tully-Fisher, but was constructed to ensure that Tully-Fisher is satisfied. Milgrom wanted both spacetime scaling invariance and Tully-Fisher.

Moreover, MOND does not arise from a length scale, but rather an acceleration scale. The relation $v^2/r ~=~ (G M a_0)^{1/2}/r$ already assumes we're in the low-acceleration regime. Tully-Fisher then relates tangential velocity out at the flat part of the GRC with the total baryonic mass. So it's not really "large $r$" that's relevant. If that were so, the size of galaxies would affect the relationship, but (iiuc) that's not observed. What matters is that we're looking at the flat part of the GRC.

For a more authoritative explanation of this, see Sanders & McGaugh 2002, in particular the bottom paragraphs on p3. In fact, they're important enough that I'll quote them here (with my emboldening):

If one wishes to modify gravity in an ad hoc way to explain flat rotation curves, an obvious first choice would be to propose that gravitational attraction becomes more like $1/r$ beyond some length scale which is comparable to the scale of galaxies. So the modified law of attraction about a point mass $M$ would read $$ F ~=~ \frac{GM}{r^2} \; f (r/r_o ) ~,$$ where $r_o$ is a new constant of length on the order of a few kpc, and f(x) is a function with the asymptotic behavior: f(x) = 1 where x ≪ 1 and f (x) = x where x ≫ 1. Finzi (1963),
Tohline (1983), Sanders (1984) and Kuhn & Kruglyak (1987) have proposed variants of this idea. In Sanders’ (1984) version, the Newtonian potential is modified by including a repulsive Yukawa term $(e^{−r/r_o }/r)$ which can yield a flat rotation velocity over some range in $r$. This idea keeps re-emerging with various modern justifications; e.g., Eckhardt (1993), Hadjimichef & Kokubun (1997), Drummond (2001) and Dvali et al. (2001).

All of these modifications attached to a length scale have one thing in common: equating the centripetal to the gravitational acceleration in the limit $r > r_o$ leads to a mass-asymptotic rotation velocity relation of the form $v^2 = GM/r_o\,$. Milgrom (1983a) realized that this was incompatible with the observed TF law, $L \propto v^4$. Moreover, any modification attached to a length scale would imply that larger galaxies should exhibit a larger discrepancy (Sanders 1986). This is contrary to the observations. There are very small, usually low surface brightness (LSB) galaxies with large discrepancies, and very large high surface brightness (HSB) spiral galaxies with very small discrepancies (McGaugh & de Blok 1998a).

 

[strangerep]

What would you consider to be a good (online paper) introduction to [MOND]?

Alas, I have not yet found a "good" one.

But here's a list from which I've struggled to extract a small amount of understanding...

First some papers by Milgrom:

The MOND Paradigm

The MOND paradigm of modified dynamics

Scale Invariance at low accelerations (aka MOND) and the dynamical anomalies in the Universe

Now some papers by McGaugh and colleagues:

MODIFIED NEWTONIAN DYNAMICS AS AN ALTERNATIVE TO DARK MATTER.
(I'm finding this one most useful at the moment, though it's now a bit old as far as the phenomenology goes.)

Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions. (This one also available on Living Reviews in Relativity.)

More recent, seriously impressive, experimental work:

The Radial Acceleration Relation in Rotationally Supported Galaxies

ONE LAW TO RULE THEM ALL: THE RADIAL ACCELERATION RELATION OF GALAXIES.

and...

Stacy McGaugh's blog, in which he talks (sometimes at great length) about experimental results relevant to MOND, and his personal (tortured) journey from DM to MOND. I suspect future generations (if we survive WW3) will look back on Prof McGaugh as one of the "greats", and indeed Prof Milgrom for his brilliant insight before much of current galactic empirical results were known.

​

 

[haushofer]

Alas, I have not yet found a "good" one.

But here's a list from which I've struggled to extract a small amount of understanding...

First some papers by Milgrom:

The MOND Paradigm

The MOND paradigm of modified dynamics

Scale Invariance at low accelerations (aka MOND) and the dynamical anomalies in the Universe

Now some papers by McGaugh and colleagues:

MODIFIED NEWTONIAN DYNAMICS AS AN ALTERNATIVE TO DARK MATTER.
(I'm finding this one most useful at the moment, though it's now a bit old as far as the phenomenology goes.)

Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions. (This one also available on Living Reviews in Relativity.)

More recent, seriously impressive, experimental work:

The Radial Acceleration Relation in Rotationally Supported Galaxies

ONE LAW TO RULE THEM ALL: THE RADIAL ACCELERATION RELATION OF GALAXIES.

and...

Stacy McGaugh's blog, in which he talks (sometimes at great length) about experimental results relevant to MOND, and his personal (tortured) journey from DM to MOND. I suspect future generations (if we survive WW3) will look back on Prof McGaugh as one of the "greats", and indeed Prof Milgrom for his brilliant insight before much of current galactic empirical results were known.

​

Thanks, much appreciated!

 

[ohwilleke]

does Deur derive the Tully-Fisher relationship ($v^4_{tan} \propto M$) from the start.

Deur derives an extremely close approximation MOND (complete with an interpolation function that is not arbitrary) in the spiral galaxy case from the GR Lagrangian and since MOND implies the Tully-Fisher relationship, Deur does so in the spiral galaxy case.

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

F(G) = G(N)M/r2 + c^2(aπG(N)M)^0.5/(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 a(0) in MOND (that should in principle be possible to derive from Newton's constant) which is equal to 4*10^−44 m^−3s^2. Thus, the self-interaction term that it modifies is proportionate to (GNM)^0.5/r and is initially much smaller that the first order Newtonian gravity term in stronger fields, but it declines more slowly than the Newtonian term with distance until it is predominant.

Deur's first article published in a peer reviewed journal spelling this out has the following abstract and citation:

Our present understanding of the universe requires the existence of dark matter and dark energy. We describe here a natural mechanism that could make exotic dark matter and possibly dark energy unnecessary. Graviton-graviton interactions increase the gravitational binding of matter. This increase, for large massive systems such as galaxies, may be large enough to make exotic dark matter superfluous. Within a weak field approximation we compute the effect on the rotation curves of galaxies and find the correct magnitude and distribution without need for arbitrary parameters or additional exotic particles. The Tully-Fisher relation also emerges naturally from this framework. The computations are further applied to galaxy clusters.

A. Deur, “Implications of Graviton-Graviton Interaction to Dark Matter” (May 6, 2009) (published at 676 Phys. Lett. B 21 (2009)).

This analysis is recapped in the introduction portion of A. Deur, "An explanation for dark matter and dark energy consistent with the standard model of particle physics and General Relativity." 79 Eur. Phys. J. C , 883 (October 29, 2019). https://doi.org/10.1140/epjc/s10052-019-7393-0

Some highlights:

In GR, self-interaction becomes important for 𝐺𝑀/𝐿‾‾‾‾‾‾‾√GM/L large enough (L is the system characteristic scale), typically for 𝐺𝑀/𝐿‾‾‾‾‾‾‾√>𝑟𝑠𝑖𝑚10−3GM/L>rsim10−3 as discussed in Ref. [5] or exemplified by the Hulse-Taylor binary pulsar [8], the first system in which GR was experimentally tested in its strong regime, which has 𝐺𝑀/𝐿‾‾‾‾‾‾‾√=10−3GM/L=10−3. As in the case of QCD, self-interaction increases the binding compared to Newton’s theory. Since the latter is used to treat the internal dynamics of galaxies or galaxy clusters, its neglect of self-interaction may contribute to – or even create – the missing mass problem [4, 5, 9].

In Ref. [4] a non-perturbative numerical calculation based on Eq. (2) is applied in the static limit to spiral galaxies and clusters. A non-perturbative formalism (lattice technique) – rather than a perturbative one such as the post-Newtonian formalism – was chosen because in QCD, confinement is an entirely non-perturbative phenomenon, unexplainable within a perturbative approach.

The results of Refs. [4, 5] indicate that self-interaction increases sufficiently the gravitational binding of large massive systems such that no dark matter nor ad-hoc gravity/dynamical law modifications are needed to account for the galaxy missing mass problem. Self-interaction also explains galaxy cluster dynamics and the Bullet cluster observation [10]. Finally, the Tully–Fisher relation [11, 12], an important observation difficult to explain in the dark matter context, was shown in Ref. [4] to be the GR analog to QCD’s Regge trajectories [13]. Accounting for self-interaction automatically yields flat rotation curves for disk galaxies when those are modeled as homogeneous disks of baryonic matter with exponentially decreasing density profiles, which is a good approximation of the observations. In contrast, dark matter halo profiles must be specifically tuned for each galaxy to make its rotation curve flat.

Besides quark confinement, the other principal feature of QCD is a dearth of strong interaction outside of hadrons (i.e., quark bound states) because color confinement keeps the (colored) gluonic field in the hadron. As shown in numerical lattice calculations [14], the field lines – which for a free-field spread isotropically from the source to infinite distances – are for a self-interacting field rearranged in a finite volume roughly contained between the quarks: the collapsed field lines between two quarks form an approximately one dimensional “flux-tube” in which flux lines do not spread. Their density, i.e., the force acting between quarks, is hence constant with the quark separation r. While this confined field produces a binding energy stronger than in the free-field case, such field concentration inside the hadron causes a field depletion outside. This conforms to energy conservation: compared to the free-field case, the increased binding energy in the hadron is compensated by a near absence of potential energy outside the hadron since the field lines have been pulled-in due to self-interaction. Increases of binding energy have also been calculated, with the same numerical lattice technique, for gravity and massive structures [4, 5]. This increased binding must, by energy conservation, weaken the action of gravity at larger scale. This can then be mistaken for a repulsion, i.e., dark energy. Specifically, the Friedmann equation for an isotropic and homogeneous universe is (for a matter-dominated flat Universe) 𝐻2=8𝜋𝐺𝜌/3H2=8πGρ/3, with H the Hubble parameter and 𝜌ρ the density. As massive structures coalesce, gravity is effectively suppressed at scales larger that these structures. This weakening with time results in a larger than expected value of H at early times, as seen by the observations suggesting the existence of dark energy.

An important point for the present article is that the morphology of the massive structures in which gravity may be trapped determines how effective the trapping is: the less isotropic and homogeneous a system is, the larger the trapping is. For example, this implies a correlation between the missing mass of elliptical galaxies and their ellipticities. The correlation was predicted in [4] and subsequently verified in [9]. The role of the system spacial symmetry is also supported by the relation 𝐽=𝜖𝑀𝛾J=ϵMγ describing both the galactic Tully–Fisher observation and the hadronic Regge trajectories.Footnote3 Here J is the angular momentum, M the system mass, and 𝜖ϵ a constant depending on the type of galaxy or hadron family considered. Inside a less symmetric system, the force is more enhanced and 𝛾γ is larger than that of a more symmetric system, as observed: Regge trajectories apply to hadrons (flux tubes of 1-dimension) and have 𝛾=2γ=2, while for the Tully–Fisher relation which applies to disk galaxies (2-dimensional systems), 𝛾=1.26±0.07γ=1.26±0.07.

The possible effects of the Universe’s inhomogeneity have been discussed in the past to explain cosmological observations without requiring dark energy [15]. In particular, the possible importance of backreactions, of same origin as field self-interaction, has been pointed out [16]. The calculations carried out so far are typically perturbative. Thus they are blind to the non-perturbative phenomena that are critical in the analogous QCD phenomenology. Previous non-perturbative attempts have been inconclusive [17]. The present approach, while remaining within GR’s description of the universe evolution, see Sect. 3, folds the effects of inhomogeneities into a generic function 𝐷𝑀DM that expresses the large distance consequences of the non-perturbative effects, and which functional form is modeled from general considerations, see Sect. 4. That this approach differs from others using backreactions or inhomogeneity is illustrated by the identification of an explicit mechanism (field trapping) that is not perturbative, and by the direct connection between dark energy and dark matter that this work exposes.

In summary, traditional analyses of internal galaxy or cluster dynamics employ Newton’s gravity that neglects the self-interaction terms in Eq. (2), and this may explain the need for dark matter [4, 5]. Traditional analyses of universe evolution do use GR, but under the approximations of isotropy and homogeneity, which suppress the effects of the self-interaction terms [4, 9], and would by definition disregard any local phenomenon that could affect gravity’s field, such as field trapping. The weakening of gravity at large distance due to these terms is thus neglected, which may be why dark energy seems necessary. This was conjectured in Ref. [4] and the present article investigates this possibility.

Deur concludes that MOND effects vanish in truly spherically symmetric systems (which few galaxies are).

He also concludes MOND is an underestimation of gravitational self-interaction effects in extremely massive two body systems like the gravitational pulls between distant individual galaxies in galaxy clusters (where MOND underestimates inferred dark matter phenomena by roughly a factor of two).

But his crude metric for estimating the magnitude of self-interaction effects suggests that these effects should not appear in much less massive wide binary star systems.

Recent research suggests, however, that some of the MOND effect underestimate in galaxy clusters (which is too small by a factor of about two) is due to poor modeling of the galaxy clusters to which MOND is applied leading past estimates of MOND effects in galaxy clusters to be understated. See M. Lopez-Corredoira, et al., "Virial theorem in clusters of galaxies with MOND" arXiv:2210.13961 (October 25, 2022) (accepted for publication in MNRAS).

 

[Davephaelon]

This is a most interesting thread. I don't recall coming across the MacDowell-Mansouri geometric formulation before. I printed the thread out so that I can read it at my leisure while away from a desktop computer and to digest it more thoroughly.