This modified gravity proposal explains galactic rotation curves without dark matter, it's relativistic, and its key parameter beyond general relativity is determined on a very consistent basis from data from nine different galaxies. It bears some general similarities to other modified gravity proposals that do the same thing.
The author's conjecture that the reason we don't have a workable quantum gravity theory is that the standard equations of general relativity that we're trying to quantize aren't quite right also seems plausible.
This candidate isn't as mature as some of the competing modified gravity proposals, so it hasn't be tested against the cosmic microwave background, galaxy formation rates, in non-spiral galaxies, and in galaxy clusters yet. But its a promising proposal that deserves further attention.
A modification of the Einstein-Hilbert Lagrangian by introducing a coupling between the Weyl tensor and the stress-energy tensor was proposed to explain flat galactic rotation curves without the exotic (non-baryonic) dark matter (DM). The proposed coupling constant was previously determined by fitting the rotational velocities of the Milky Way and M31 modeled with constant density, yielding the same coupling constant for both. In this work, we have modified the formalism for a variable density by modeling the galactic systems with realistic, spherically symmetric and radially varying density profiles for the baryonic matter and this analysis is applied to seven edge-on spiral galaxies of the local cluster and the Milky Way.
Asghar Qadir, Ashmal Shahid, Noraiz Tahir, "The Galactic Halo Rotation by Weyl Incorporated Gravity" arXiv:2604.01643 (April 2, 2026) (Arabian Journal of Mathematics (2026)).
The introduction to the paper is also encouraging, although some of the summary of the criticisms of MOND are overstated. The explicit treatment of the effect of the gravitational field, similar to the approach of Deur, is particularly notable. It says:
One of the most striking observations in galactic dynamics is the discrepancy between the predicted and observed rotational velocities of galaxies. According to the standard theories of gravity, the rotational velocity of the galaxies should decrease sharply at large radii where visible matter becomes sparse. However, observations of their rotation curves remain nearly flat out to very large distances [11–13]. Other dynamic considerations had already led Zwicky [14] to propose the existence of DM, but this evidence was much stronger. Rubin’s investigation was extended to galactic clusters [15, 16] providing yet stronger evidence. The observations of the cosmic microwave background (CMB) had already provided minimum and maximum values for baryonic matter in the Universe according to the standard model of particle physics (SMpp). The observations required a value well beyond the limit of the baryonic matter [17]. This has led to various suggestions for exotic (non-baryonic) DM, but there is no direct evidence for any of the proposed candidates. Nevertheless, CMB observations also indicate that ≃ 5% of the Universe should be made up of baryons (the usual protons and neutrons), but observations of the luminous parts of the galaxies show only half of these baryons, this is the “missing baryon problem”. Since the baryons are dark, we call this the baryonic DM. It is proposed that a significant fraction of this baryonic DM is present in the galactic halos [18–23].
For the non-baryonic DM an alternative suggestion was that the standard law of gravity should be modified instead of looking for other forms of matter. The first such suggestion came from Milgrom [24], who proposed the modification of Newton’s law by inserting a Yukawa-like term to damp gravity at large distances, Modified Newtonian Dynamics (MOND). It was not able to explain the dynamics at different scales, especially of single galaxies and clusters, or provide for the formation of structure in the early Universe [25–31]. Most of all, the damping term was totally ad-hoc and was embedded in an obsolete Newtonian framework, which could not be converted to General Relativity (GR) [29]. Apart from galactic dynamics, arguably the most outstanding problem of fundamental physics is the incompatibility of GR and Quantum Theory. In particular, the Renormalization Group Equation of ’t Hooft and Veltman demonstrated that Dirac quantization of GR produced a non-renormalizable theory [32], leading to the well-known “Quantum Gravity (QG)” problem. To avoid these separate problems various ad-hoc modifications of GR involving arbitrarily many new parameters have been proposed [24, 33–36].
Qadir and Lee took the view that there must be a sound physical basis for any modification of the highly successful GR, and it must be minimal, i.e., it should remain geometric and involve only one free parameter to explain the discrepancies of galactic dynamics at all scales. Further, it should also provide a base for solving the QG problem. In 2019, they proposed an explicit interaction term between matter and the gravitational field λT.C.T, where λ is a new coupling constant, T is the stress-energy tensor, and C is the Weyl tensor, which represents a pure gravitational field [1]. This idea was inspired by the Feynman vertex representing a similar explicit interaction between the source (an electron) and the electrodynamic field in Quantum Electrodynamics (QED), Aµjµ. In QED the electron is given by a spin-half spinor, which comes twice over in the current jµ and the field, Aµ, which appears singly, while in QG the source would be represented by the rank-two tensor T, which comes twice, and the gravitational field by the rank-four tensor C, which comes once. As QED with this source term is renormalizable, it can be hoped that so would this modified QG. It was called Modified Relativistic Dynamics (MORD).
Previously, MORD was tested by checking whether a single value of the new coupling constant λ could reproduce the rotational velocity at the outer rim of two galaxies, the Milky Way and M31, incorporating only the baryonic DM and not any postulated non-baryonic DM, by assuming a simple-minded model in which both the galaxies were represented as a constant density sphere with a peak density of the baryonic matter from the core to the edge of the galaxy [2, 3]. In that study, a single value of λ was indeed found to fit the rotational velocity values for both galaxies. This approach was inherently limited, i.e., it neglected the radial variation of the galactic halo density, and treated the galaxies as idealized, uniform objects. The aim of this paper is to completely modify the previous formalism of a constant density case to a variable density case, where ρ′= 0, and take the next step forward by generalizing the baryonic DM component to spherically symmetric, radially varying density profiles for the galactic halos of eight spiral galaxies [4–10, 37, 38].
This extension is conceptually significant because it will allow us later to test whether the universality of λ persists under physically motivated halo structures at different radii, rather than only at the rim. By moving from a toy model to a realistic halo description, we not only refine the numerical estimate of λ but also provide a more robust and physically meaningful assessment of MORD across multiple spiral galaxies. We stress that while more realistic baryonic distributions, such as double exponential stellar disks combined with bulge components, are commonly used to model luminous matter, these structures are intrinsically non-spherical, and would require extension of the formalism to two or more variables.
The plan of the paper is as follows: in Section 2, we will briefly explain the Weyl modified Einstein field equations for varying spherically symmetric density profiles and demonstrate how the value of the coupling constant λ is obtained for the Milky Way galactic halo. In Section 3, we will use the analysis for seven other spiral galaxies. Finally, in Section 4, the obtained results will be discussed.
The key formulas are as follows:
The Weyl-modified Einstein-Hilbert Lagrangian is [1]
L =√−g (R − 2Λ − kT + λC(αµβν)T^(αβ)T^(µν)), (1)
where √−g is the determinant of the metric, k = 8πG/c^4 is the coupling constant for matter, where G is the Newton’s gravitational constant, c is the speed of light. This leads to the Weyl incorporated Einstein field equation (WIFE)
R(µν) − 1/2g(µν)R + g(µν)Λ = kT(µν) + λI(µν), (2)
[Ed. For comparison the unmodified Einstein field equation is as follows:
So, the only modification is the addition of the λI(µν) term on the RHS.]
where I(µν) is the interaction term given by
I(µν)=
1/4(−g(αβ)g(ρµ)g(σν)−g(ρσ)g(αµ)g(βν)−g(ασ)g(ρµ)g(βν)−g(ρβ)g(αµ)g(σν))□(T^(αβ)T^(ρσ))
+1/6 (g(αβ)g(ρσ)−g(ρβ)g(ασ))(g(µν)□ −∇(µ)∇(ν)) T(αβ)T(ρσ). (3)
The paper sums up its findings in the conclusion:
The very first modified gravity approach to explain the flat rotational curves of galaxies without invoking DM is MOND [24]. However, MOND’s phenomenological success comes with challenges in covariant formulation and cluster-scale dynamics, motivating alternative modifications rooted in GR. Furthermore, a major challenge to MOND has emerged from the analysis of wide binary stars in Gaia DR3. A comprehensive study by Ref. [59] found that the relative velocities of widely separated binaries (2−30 kAU) are inconsistent with the MOND prediction, which expects a ≈ 20% enhancement over Newtonian gravity due to the external field effect. Their analysis, which rigorously modeled the Galactic external field and population uncertainties, excluded MOND at a statistical significance of 16σ in favour of Newtonian dynamics [59, 60]. These challenges motivate the exploration of alternative single-parameter modified gravity theories rooted in a covariant framework.
Qadir and Lee’s MORD [1] proposed a modification to the standard Lagrangian by incorporating an interaction coupling constant λ with a term involving the Weyl tensor and the stress-energy tensor, expressed as λC(µνρπ)T^(µν)T^(ρπ) whose purpose was to see whether a single unique value of λ can account for the rotational velocity curves of galaxies by replacing the exotic DM by what we now feel should be called Weyl Incorporated Gravity (WIG), solving the outstanding DM problem with a single new parameter. This was tested using a simplistic constant density model which did have just the one value of the coupling for all galaxies considered [2, 3].
In the present work, we have modeled the galactic halos of eight spiral galaxies, modifying the previous framework for a variable density case to estimate the value of the coupling constant λ. For this purpose we adopt three widely used density profiles, the Navarrow-Frenk-White (NFW), Moore, and Burkert models, normally used for all DM in the halos, but here used only for the baryonic DM to test the robustness of the proposal by verifying that the choice of model makes no difference to the results [40-50].
We find that a tiny range, λ = (6.9546 ± 0.00012) × 10^−18 km^2s^4kg^2, consistently reproduces the observed halo rotational velocities at r = 100 kpc.
For the Milky Way, the fitted rotational velocities span v(rot) ≃ 153–159 km s^−1, compared to the observed value 150 ± 10 km s^−1, with an enclosed halo mass M(h)(≤ 100 kpc) ≃ 1.0 × 10^12 M⊙.
For M31, we obtain v(rot) ≃ 230–232 km s^−1 versus the observed 225 ± 10 km s^−1, corresponding to a halo mass M(h) ≃ 1.4 × 10^12 M⊙.
In the case of M33, the modeled velocities v(rot) ≃ 121–122 km s^−1 agree with the observed 120 ± 5 km s^−1, yielding M(h) ≃ 3.2 × 10^11 M⊙.
For M81 and M82, the fitted velocities lie in the ranges 256-257 km s^−1 and 251–253 km s^−1, respectively, consistent with the observed values 250 ± 15km s^−1 and 250 ± 18 km s^−1, with inferred halo masses M(h) ≃ 1.3 × 10^12 M⊙ and 1.0 × 10^11 M⊙.
Similarly, for NGC 5128, NGC 4594, and M90, the modeled rotational velocities at 100 kpc fall within the observed ranges reported in the literature, with corresponding halo masses M(h) ≃ 4.4 × 10^12 M⊙, 6.3 × 10^13 M⊙, and 3.5 × 10^13 M⊙, respectively. It is clearly seen that these mass and velocity estimates are consistent with the observed values (see Refs. [40, 44], and Table 1).
Before closing the paper, note that we have used spherical symmetry to explain the dynamics of the galactic halo to estimate the value of λ. However, to make a more realistic model of the galaxy, we should take into account the vertical component of the velocity, which is missing in the present geometry. The hope is that one can fit the complete rotational velocity curve and get a more robust model. Indeed, the chosen metric would change, and we may need to consider the Kerr geometry, or a slow rotation approximation of it [61], to incorporate the vertical component, as it accounts for the angular momentum effects [62, 63]. This will be addressed separately later.
As previously discussed, the problem of DM and QG may share a common origin [1]. Addressing observational issues related to DM could provide insights into resolving fundamental difficulties in QG. Instead of assuming an indirect interaction, we have introduced a direct nonlinear coupling between matter and gravity, analogous to the interaction between electromagnetic sources and the electromagnetic field. The modified Lagrangian, given in eq. (1), represents the minimal extension of the Einstein-Hilbert Lagrangian and is proposed as a potential solution to both problems with a single additional parameter. Naturally, the feasibility of this approach must first be tested against the DM problem before seeing if our WIG fits on the messier head of QG.
As a further conjecture of my own, their coupling constant could be parsed out suggestively into X^2/G^2 where G is Newton's constant and X is a coupling constant with units of 1/km^2 (the same units as the cosmological constant), and a numerical value on the order of 10^-20, which is roughly the square root of the cosmological constant's value.
