Reproducing MOND with Conformal Gravity

MOND is a phenomenological toy model that reproduces the dynamical behavior of galaxies entirely from the distribution in baryonic matter in those galaxies, something that suggests that phenomena attributed to dark matter may actually be due to a deviation in reality from the predictions of general relativity as conventionally applied in very weak gravitational fields. But, MOND is not itself a relativistic theory.

One possibility to explain the discrepancy is that MOND is a quantum gravity effect. And, some theories are easier to prove than others.

A new paper illustrates one modified gravity theory that reproduces MOND's phenomenological successes relativistically, called conformal gravity.

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).

The conclusion to this article states:

In McGaugh et al. [8] the RAR in galactic rotation curves was established via a set of 2693 total points. In this work we have shown that conformal gravity can universally fit the g_OBS versus g_NEW data in an even larger 6377 data point sample. Conformal gravity has successfully fitted over 97% of the 6377 data points across 236 galaxies without any filtering of points, fixing of mass to light ratios or modification of input parameters. Further, conformal gravity is shown to satisfy the v⁴∝ M relation consistently found in rotation curve studies, while also providing a derivation and extension of the TF relation. We conclude by noting that there is a great deal of universality in rotation curve data. This universality does not obviously point in favor of dark matter, and is fully accounted for by the alternate conformal gravity theory.

The raw equations used were as follows:

[C]onformal gravity is derived from an action based on the square of the Weyl tensor as 

I_W = −α_g times the indefinite integral of d⁴x(−g)^{1/2*C_λµνκCλµνκ} , (3) 

where 

C_λµνκ = R_λµνκ − 1/2(g_λνR_µκ − g_λκR_µν − g_µνR_λκ + g_µκR_λν) + 1/6(R^α_α)(g_λνg_µκ − g_λκg_µν) (4)

is the conformal Weyl tensor with dimensionless coupling constant α_g. The resulting field equations, 

4α_g (2∇_κ∇_λC^µλνκ − C^µλνκR_λκ) = T^µν

were solved by Mannheim and Kazanas in the region exterior to a static, spherically symmetric source [11], where it was shown that a point stellar mass produces a potential V^∗(r) = −β^∗c²/r + γ^∗c²r/2. To go from the single star to the prediction for rotational velocities of entire spiral galaxies, it was shown [12] that the resulting galactic velocity expectation is given by 

v_CG(R) = (v²_NEW(R) + (M/M⊙)(γ^∗c²R²/2R₀)*I₁(R/2R₀)*K₁(R/2R₀) + γ₀c²R/2 − κc²R²)^1/2 , (5) 

where M is the mass of the galaxy in solar mass units (M⊙), R₀ is the galactic disk scale length, and v_NEW(R) is the standard Freeman formula for a Newtonian disk: 

v_NEW(R) = ((M/M⊙)(β^∗c²R²/2R³₀)(I₀(R/2R₀)*K0(R/2R₀) − I1(R/2R₀)*K₁(R/2R₀))^1/2 . (6) 

Conformal gravity, while quite similar in outcome in most respects to Einstein gravity, is more naturally made a quantum gravity theory. A more complete development of the theory (which is simply compared to data after a thumbnail summary in the new paper) can be 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).

From the body of the 2011 paper:

The non-renormalizable Einstein-Hilbert action is expressly forbidden by the conformal symmetry because Newton’s constant carries an intrinsic dimension. However, as noted above, this does not prevent the theory from possessing the Schwarzschild solution and Newton’s law of gravity. In addition, the same conformal symmetry forbids the presence of any intrinsic cosmological constant term as it carries an intrinsic dimension too; with conformal invariance thus providing a very good starting point for tackling the cosmological constant problem. 

Now we recall that the fermion and gauge boson sector of the standard SU(3)×SU(2)×U(1) model of strong, electromagnetic, and weak interactions is also locally conformal invariant since all the associated coupling constants are dimensionless, and gauge bosons and fermions get masses dynamically via spontaneous symmetry breaking. Other than the Higgs sector (which we shall shortly dispense with), the standard model Lagrangian is devoid of any intrinsic mass or length scales. And with its associated energy-momentum tensor serving as the source of gravity, it is thus quite natural that gravity should be devoid of any intrinsic mass or length scales too. Our use of conformal gravity thus nicely dovetails with the standard SU(3) × SU(2) × U(1) model. To tighten the connection, we note that while the standard SU(3)×SU(2)×U(1) model is based on second-order equations of motion, an electrodynamics Lagrangian of the form F^µ∂_α∂^αF_µν would be just as gauge and Lorentz invariant as the Maxwell action, and there is no immediate reason to leave any such type of term out. Now while an F^µν∂_α∂^αF_µν theory would not be renormalizable, in and of itself renormalizability is not a law of nature (witness Einstein gravity). However, such a theory would not be conformal invariant. Thus if we impose local conformal invariance as a principle, we would then force the fundamental gauge theories to be second order, and thus be renormalizable after all. However, imposing the same symmetry on gravity expressly forces it to be fourth order instead, with gravity then also being renormalizable. As we see, renormalizability is thus a consequence of conformal invariance.

The 2011 approach taken to the Higgs sector, before the Higgs boson was definitively discovered, turns out to be a problem rather than a benefit of the theory, but not necessarily an intractable one.