Conformal Gravity As A MOND Alternative

Conformal gravity is a quite subtle modification of General Relativity but is sufficient to reproduce MOND/Radical Acceleration Relation/Tully-Fisher relation in galaxies. Like MOND it has a physical constant corresponding to the MOND a(0) constant. (Incidentally, another paper receives a similar outcome by inserting torsion into the General Relativity.)

Also, unlike MOND, it is more than a toy model theory. It is relativistic in the first place, so it doesn't require a relativistic generalization. The introduction to a new paper on this modified gravity theory explains that:

Despite the enormous successes of the Einstein’s theory of gravity, the latter appears to be about “twenty five percent wrong”. So far the scientists proposed two possible solutions of the problem that are known under the name of “dark matter” or “dark gravity”, and both are extensions of the Einstein’s field equations. The first proposal consists on modifying the right side of the Einstein’s equations, while according to the second proposal it is modified the left hand side. Indeed, in order to take into account of all the observational evidences: galactic rotation curves, structure formation in the universe, CMB spectrum, bullet cluster, and gravitational lensing, it seems needed to somehow modify the Einstein’s field equations. However, in this paper we propose the following different approach, namely: “understand gravity instead of modifying it”. 

In this document we do not pretend to provide a definitive answer to the “mystery of missing mass” or “missing gravity in the universe”, but we only focus on the galactic rotation curves. Nevertheless, we believe our result to be quite astonishing on both the theoretical and observational side. 

The analysis here reported, which follows the previous paper*, is universal and apply to any conformally invariant theory, nonlocal, or local, that has the Schwarzschild-metric as an exact and stable solution. 

* The previous seminal paper did not address the issue of conformally coupled matter that completely changes the geometrical interpretation of our proposal underlining the crucial role of the asymptotic but harmless spacetime singularity. Notice that in the previous paper the massive particles break explicitly the conformal invariance, even if slightly, making the solution no longer exact. Moreover, we will show in this paper that in presence of conformally coupled matter we do not need to resort to the global structure of space-time and to invoke the small inhomogeneities on the cosmological scale or the presence of the cosmological constant, which will turn out to be too small to affect on the rotation curves on a galactic scale: “everything will be limited to the single galaxies”.

However, for the sake of simplicity we will focus on Einstein’s conformal gravity, whose general covariant action functional reads: 

which is defined on a pseudo-Riemannian spacetime Manifold M equipped with a metric tensor field ˆg(µν), a scalar field φ (the dilaton), and it is invariant under the following Weyl conformal transformation: 

where Ω(x) is a general local function. In (1) h is a dimensionless constant that has to be selected extremely small in order to have a cosmological constant compatible with the observed value. However, we here assume h = 0 because the presence of a tiny cosmological constant will not affect our result. For completeness and in order to show the exactness of the solutions that we will expand later on, we here remind the equations of motion for the theory (1) for h = 0, 

The Einstein-Hilbert action for gravity is recovered whether the Weyl conformal invariance is broken spontaneously in exact analogy with the Higgs mechanism in the standard model of particle physics. One possible vacuum of the theory (1) (exact solution of the equations of motion (3)) is φ = const. = 1/k(4) = 1/√16πG, together with the metric satisfying R(µν) ∝ gˆ(µν). Therefore, replacing φ = (1/√16πG)+ϕ in the action (1) and using the conformal invariance to eliminate the gauge dependent Goldstone degree of freedom ϕ, we finally end up with the Einstein-Hilbert action in presence of the cosmological constant, 

where Λ is consistent with the observed value for a proper choice of the dimensionless parameter h in the action (1). Ergo, Einstein’s gravity is simply the theory (1) in the spontaneously broken phase of Weyl conformal invariance. 

Let us now expand about the exact solutions in conformal gravity. Given the conformal invariance (2), any rescaling of the metric ˆg(µν) accompanied by a non trivial profile for the dilaton field φ, is also an exact solution, namely 

solve the EoM obtained varying the action (1) respect to ˆg(µν) and φ. 

So far the rescaling (5) has been used to show how the singularity issue disappearance in conformal gravity. However, and contrary to the previous papers, we here focus on a not-asymptotically flat rescaling of the Schwarzschild metric as a workaround to the non-Newtonian galactic rotation curves. Moreover, the logic in this project is literally opposite to the one implemented in the past works and it is somehow anti-intuitive. In fact, here, instead of removing the spacetime’s singularities, we apparently deliberately introduce an unreachable asymptotic singularity. However, as it will be proved later on, the spacetime stays geodetically complete. Indeed, the proper time to reach the singularity at the edge of the Universe will turn out to be infinite. 

Notice that in order to give a physical meaning to the metric (5), conformal symmetry has to be broken spontaneously to a particular vacuum specified by the function Q(x). The uniqueness of such rescaling will be discussed in section III. In the spontaneously broken phase of conformal symmetry observables are still invariant under diffeomorphisms. 

The paper and its abstract are as follows:

We show that Einstein's conformal gravity is able to explain simply on the geometric ground the galactic rotation curves without need to introduce any modification in both the gravitational as well as in the matter sector of the theory. 

The geometry of each galaxy is described by a metric obtained making a singular rescaling of the Schwarzschild's spacetime. The new exact solution, which is asymptotically Anti-de Sitter, manifests an unattainable singularity at infinity that can not be reached in finite proper time, namely, the spacetime is geodetically complete. It deserves to be notice that we here think different from the usual. Indeed, instead of making the metric singularity-free, we make it apparently but harmlessly even more singular then the Schwarzschild's one. 

Finally, it is crucial to point that the Weyl's conformal symmetry is spontaneously broken to the new singular vacuum rather then the asymptotically flat Schwarzschild's one. The metric, is unique according to: the null energy condition, the zero acceleration for photons in the Newtonian regime, and the homogeneity of the Universe at large scales. 

Once the matter is conformally coupled to gravity, the orbital velocity for a probe star in the galaxy turns out to be asymptotically constant consistently with the observations and the Tully-Fisher relation. Therefore, we compare our model with a sample of 175 galaxies and we show that our velocity profile very well interpolates the galactic rotation-curves for a proper choice of the only free parameter in the metric and the the mass to luminosity ratios, which turn out to be close to 1 consistently with the absence of dark matter.

Leonardo Modesto, Tian Zhou, Qiang Li, "Geometric origin of the galaxies' dark side" arXiv:2112.04116 (December 8, 2021).