# Are there issues with Conformal Gravity as a DM solution?

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Gold Member
One approach that has been proposed to replicate the description of dark matter phenomena in galaxies with a gravity modification that is set forth in a simplified manner in the phenomenological toy model theory MOND is to do so through a variation on Einstein's General Relativity known as Conformal Gravity.

My question is whether this is as viable a relativistic generalization of a MOND-like gravity modification as it seems to be, and if it has flaws, what are they.

A recent paper asserts that it does so quite well:

In 2016 McGaugh, Lelli and Schombert established a universal Radial Acceleration Relation for centripetal accelerations in spiral galaxies. Their work showed a strong correlation between observed centripetal accelerations and those predicted by luminous Newtonian matter alone. Through the use of the fitting function that they introduced, mass discrepancies in spiral galaxies can be constrained in a uniform manner that is completely determined by the baryons in the galaxies. Here we present a new empirical plot of the observed centripetal accelerations and the luminous Newtonian expectations, which more than doubles the number of observed data points considered by McGaugh et al. while retaining the Radial Acceleration Relation. If this relation is not to be due to dark matter, it would then have to be due to an alternate gravitational theory that departs from Newtonian gravity in some way. In this paper we show how the candidate alternate conformal gravity theory can provide a natural description of the Radial Acceleration Relation, without any need for dark matter or its free halo parameters. We discuss how the empirical Tully-Fisher relation follows as a consequence of conformal gravity.

James G. O'Brien, et al., "Radial Acceleration and Tully-Fisher Relations in Conformal Gravity" (December 7, 2018) (Submitted to JPCS for the proceedings of the International Association of Relativistic Dynamics 2018 meeting in Merida).

I transcribed and formatted a quotation from that paper setting forth the key equations in a post at my blog (which I am not relying upon as an independent source, just a place where I cut and pasted some the article above) if you'd like to review them, but it took about 40 minutes to do that and I have time to reproduce that tedious formatting in this post (and don't know a way to port the formatting I did from blogger format), I'd welcome the efforts of anyone who did.

An earlier development of the Conformal Gravity theory is found at:

We review some recent developments in the conformal gravity theory that has been advanced as a candidate alternative to standard Einstein gravity. As a quantum theory the conformal theory is both renormalizable and unitary, with unitarity being obtained because the theory is a PT symmetric rather than a Hermitian theory. We show that in the theory there can be no a priori classical curvature, with all curvature having to result from quantization. In the conformal theory gravity requires no independent quantization of its own, with it being quantized solely by virtue of its being coupled to a quantized matter source. Moreover, because it is this very coupling that fixes the strength of the gravitational field commutators, the gravity sector zero-point energy density and pressure fluctuations are then able to identically cancel the zero-point fluctuations associated with the matter sector. In addition, we show that when the conformal symmetry is spontaneously broken, the zero-point structure automatically readjusts so as to identically cancel the cosmological constant term that dynamical mass generation induces. We show that the macroscopic classical theory that results from the quantum conformal theory incorporates global physics effects that provide for a detailed accounting of a comprehensive set of 138 galactic rotation curves with no adjustable parameters other than the galactic mass to light ratios, and with the need for no dark matter whatsoever. With these global effects eliminating the need for dark matter, we see that invoking dark matter in galaxies could potentially be nothing more than an attempt to describe global physics effects in purely local galactic terms. Finally, we review some recent work by 't Hooft in which a connection between conformal gravity and Einstein gravity has been found.

Philip D. Mannheim, "Making the Case for Conformal Gravity" (October 27, 2011) (Presentation at the International Conference on Two Cosmological Models, Universidad Iberoamericana, Mexico City, November 17-19, 2010. Updated final version, contains many new footnotes. To appear in Foundations of Physics).

My question is whether [Conformal Gravity, a.k.a. Weyl Gravity] is as viable a relativistic generalization of a MOND-like gravity modification as it seems to be, and if it has flaws, what are they.

On seeing your post, I decided to start a detailed literature study of this, (which is why I'm rather late in replying).

I'm still in the middle of my study, but I figured it might be useful for subsequent discussion if I post partial summaries as I progress. This is the 1st installment...

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To understand recent (Dec 2018) paper of O'Brien, et al, referenced in the OP, one needs to study the preceding theory and history, which I begin in this installment.

Conformal (Weyl) Gravity ("CG" hereafter) involves 4th-order field equations, rather than the 2nd order field equations of GR. (However, every vacuum solution of GR is also a solution of CG.)

Mannheim & Kazanas [MK-1989] derived a general exact spherical symmetric (vacuum!) solution: $$ds^2 ~=~ -B(r) dt^2 ~+~ B(r)^{-1}\,dr^2 ~+~ r^2 d\Omega ~,~~~~~~(1)$$with$$B(r) ~=~ 1 ~-~ \frac{\beta(2-3\beta\gamma)}{r} ~-~ 3\beta\gamma ~+~ \gamma r ~-~ k r^2 ~,~~~~~(2)$$ and ##\beta, \gamma, k## are integration constants.

They noted that if ##\gamma=0##, their solution is a Schwarzschild solution in a de Sitter background with scalar curvature ##-12k##. In GR, this would only be obtained with a cosmological constant ("CC"), whereas in CG it emerges without the need for a CC. Hence, the ##\gamma## term measures all departures of CG from Einstein-GR-with-CC. At that time (1989) it was unclear whether ##\gamma## should be associated with the interior structure of a source, or the exterior geometry (including background), or both.

By considering the energy-momentum tensor of a general source of radius ##r_0##, Mannheim & Kazanas [MK-1994] deduced that only ##\beta## and ##\gamma## could be associated with properties of such a local source. The ##-kr^2##, being a trivial vacuum solution is not coupled to the local matter source.

The (serious) proposal of a universal linear (or even quadratic) contribution to the potential is rather confronting. Some years later, Mannheim [M-1997] therefore did some analysis on whether galactic rotation curves are really flat, and compared the (albeit rather limited) data to a CG fitting. His graphs at the end of that paper show an apparently good fit, with (modest) evidence of large-distance rises in the rotation curves for the 11 galaxies considered.

Mannheim & O'Brien [MOB-2012] attempted to fit CG to a much larger set of 111 spiral galactic rotation curves whose data points extend well beyond the optical disk. They re-expressed the ##B(r)## coefficient as $$B(r) ~=~ w - \frac{2\beta}{r} + \gamma r - kr^2 ~,~~~~ \mbox{where}~~ w := \sqrt{1 - 6\beta\gamma}~,~~~~~(3).$$ Further analysis of ##T_{\mu\nu}## led them to conclude that a global quadratic potential is possible due to inhomogeneities in the cosmological background. They also showed that when an RW geometry (normally written in comoving coordinates) is transformed to a static coordinate system, a comoving conformal cosmology looks just like a static metric with universal linear and quadratic terms.
Then they did some complicated stuff to deduce that there are 2 linear potential terms: a local one associated with matter in the galaxy, and a global one associated with the cosmological background. (They denote ##\gamma^*## for the former, and ##\gamma_0## for the latter.) [I need more study of this before commenting further.]

Edit: The "fit" in [MOB-2012] uses 1 free parameter per galaxy -- the ##M/L## mass-to-light ratio. Although these are constrained by data in the inner rotation curve region and are all of similar order to the M/L ratio found in the local solar neighborhood, it still leaves me feeling a bit uneasy. [The Dec 2018 O'Brien et al paper referenced in the OP uses the same per-galaxy-fitted M/L ratios as found in their 2012 paper. ]

[That's all for this installment. More to come later.]

References:
MK-1989:
Mannheim P D and Kazanas D,1989 ApJ 342, p635. PDF here.
MK-1994: P. D. Mannheim and D. Kazanas, Gen. Rel. Gravit. 26, 337 (1994).
M-1997: P. D. Mannheim, 1997, ApJ 479, 659. https://arxiv.org/abs/9605085
MOB-2012: Mannheim P D and O’Brien J G, 2012, Phys. Rev. D 85, 124020. arXiv version here.

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jim mcnamara, ohwilleke, Lord Crc and 2 others
In this (2nd) installment, I'll present my summarized understanding of how Mannheim & O'Brien [MOB-2012] get 2 linear terms in their theoretical velocity##^2## profile in section III of that paper.

I already mentioned that the CG field equations are 4th order. For weak static spherically-symmetric fields, they boil down to a 4th-order Poisson equation: ##\nabla^4 B(r) = 0##, where ##B(r)## was introduced in eq(1) of post #3. In contrast, ordinary GR (in the same situation) reduces to the 2nd-order Newtonian Poisson equation ##\nabla^2 \phi(r) = 0##.

I'll now explain the effect of nontrivial matter sources in both cases.

In the Newtonian case, with a spherically symmetric matter source given by ##g(r)##, we have the field equation $$\nabla^2 \phi_N(r) ~=~ g(r) ~~~~~~~ (4).$$ The general particular solution to eq(4) is $$\phi_N(r) ~=~ -\,\frac{1}{r} \int_0^r\!\!ds\, s^2 g(s) ~-~ \int_r^\infty\!\!ds\, s\, g(s) ~~~~~~(5).$$This implies a force on an orbiting test particle of $$\frac{d\phi_N}{dr} ~=~ \frac{1}{r^2} \int_0^r\!\!ds\, s^2 g(s) ~~~~~~~ (6).$$ Notice how part of the ##r##-derivative of the 1st term in ##\phi_N## cancels with the 2nd, with the result that the force is determined only by matter inside the radius ##r##, and is unaffected by matter further out. [Cf. the Newtonian Shell Theorem.] This implies a radial profile of the tangential velocity of the form ##v^2_{\tan}(r) \propto r^{-1}## .

In the CG case, we have a biharmonic field equation: $$\nabla^4 \phi_C(r) ~=~ g(r) ~~~~~~~ (7).$$ The particular solution is now more complicated:$$\phi_C(r) ~=~ -\,\frac{r}{2} \int_0^r\!\!ds\, s^2 g(s) - \frac{1}{6r} \int_0^r\!\!ds\, s^4 g(s) - \frac{1}{2} \int_r^\infty\!\!ds\, s^3\, g(s) - \frac{r^2}{6} \int_r^\infty\!\!ds\, s\, g(s) ~~~~~ (8)$$and the derivative now evaluates to $$\frac{d\phi_C}{dr} ~=~ -\,\frac{1}{2} \int_0^r\!\!ds\, s^2 g(s) ~+~ \frac{1}{6r^2} \int_0^r\!\!ds\, s^4 g(s) ~-~ \frac{r}{3} \int_r^\infty\!\!ds\, s\, g(s) ~~~~~~(9).$$ In this case, the 3rd term implies that there is a contribution to the force coming from matter outside the ##r## radius -- which makes a contribution to ##v^2_{\tan}(r)## of order ##r^2##. Moreover, the 1st term implies a constant force from matter inside the ##r## radius, making a contribution to ##v^2_{\tan}(r)## of order ##r##.

So far, we've only considered the particular solution to ##\nabla^4 \phi_C(r) ~=~ g(r)##. There is also a "complimentary" solution as given in eq(2) of post #3. (Edit: I used scare quotes because it's really for a delta fn source at the origin.)

Putting these together leads to a theoretical parameterization of ##v^2_{\tan}(r)## of the form $$v^2_{\tan}(r) ~\sim~ \frac{N^* \beta^* c^2}{r} ~+~ \frac{N^* \gamma^* c^2 r}{2} ~+~ \frac{ \gamma_0 c^2 r}{2} ~-~ \kappa c^2 r^2 ~~~~~~ (10),$$ where ##\beta^*##, ##\gamma^*##, ##\gamma_0## and ##\kappa## are constants, and ##N^*## is the mass of the galaxy in solar mass units. The "##*##" constants thus express universal aspects of the local structure of any source (with ##N^*## being specific to each source/galaxy), whereas ##\gamma_0## and ##\kappa## are of cosmological (or "rest-of-the-universe") origin. Eq(10) is essentially what they try and fit to galactic rotation curve data.

Hopefully it's now clearer why [MOB-2012] use 2 linear potentials in their theoretical velocity##^2## profile. The one with ##N^*## is generated locally from each galaxy, hence is different for each galaxy because they have different ##N^*## values. The other linear term is of cosmological origin and hence has universal parameters.

That's all for this installment. More still to come.

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ohwilleke
In this (3rd) installment, I continue from eq(10) in the previous installment, i.e.,$$v^2_{\tan}(r) ~\sim~ \frac{N^* \beta^* c^2}{r} ~+~ \frac{N^* \gamma^* c^2 r}{2} ~+~ \frac{ \gamma_0 c^2 r}{2} ~-~ \kappa c^2 r^2 ~~~~~~ (10).$$ The ##\beta^*## constant is the familiar ##M_\odot G/c^2 \approx 1.48\times 10^5## cm. In the 1997 paper [M-1997], using a modest sample of 11 galaxies, Mannheim claims good fits, with the 2 universal parameters found to be $$\gamma^* ~=~ 5.42 \times 10^{-41}~ \mbox{cm}^{-1} ~,~~~~~~ \mbox{and}~~~~ \gamma_0 ~=~ 3.06 \times 10^{-30}~ \mbox{cm}^{-1} ~.~~~~~~~ (11).$$These values are so small that there are no measurable modifications to solar system phenomenology, and one must go to galactic scale before their effects become comparable with the Newtonian contribution.

In the paper [MOB-2012], they include the "##\kappa##" quadratic potential term, and use a larger sample of 111 galaxies. By fitting 21 highly extended galaxies, they were able to detect the effect of the ##\kappa## term and extracted a value of $$\kappa ~=~ 9.54 \times 10^{-54}~ \mbox{cm}^{-2} ~\approx~ (100 \mbox{Mpc})^{-2} ~,~~~~~~ (12).$$ At this point, one should note that the sign of the ##\kappa## term in eq(10) is negative. This means that the quadratic effect eventually dominates (downwardly) against the rising linear contribution:-- since ##v^2_{\tan}(r)## cannot go negative, it means that beyond a distance ##R \sim (N^*\gamma^* + \gamma_0)/2\kappa## (i.e., around 100kpc for a galaxy with ##N^*=\gamma_0/\gamma^* = 5.65 \times 10^{10}##), there can no longer be any bound circular orbits. [Cf. the Milky Way has ##N^* \sim 10^{12}##, with a nominal radius around 50kpc.]

The collection and organization of relevant data sounds quite complicated and laborious, involving photometric luminosities, optical disk length scales, HI gas masses, and various other forms of massaging, which is beyond my pay grade to comment on.

In the next installment I'll move onto the most recent paper mentioned in the OP. I also need to mention at least one (claimed) theoretical refutation of CG, as well as examining how it performs in Gravitational Lensing.

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ohwilleke
Gold Member
$$\kappa ~=~ 9.54 \times 10^{-54}~ \mbox{cm}^{-2} ~\approx~ (100 \mbox{Mpc})^{-2} ~,~~~~~~ (12).$$

What I am missing? The first expression ought to be barely more than the Planck Length squared. Is there a sign error in the exponent?