Cotton Gravity: Modified Gravitation

• Cotton gravity is a modified gravitational theory that replaces the Einstein tensor with the three-index Cotton tensor, leading to broader definitions of vacuum.
• It reformulates field equations using both a rank-3 Cotton tensor and an equivalent Codazzi tensor approach, organizing solution spaces for black holes, cosmology, and spherical symmetries.
• Observational analyses indicate the theory can fit galaxy rotation curves with a linear term while facing challenges in weak lensing tests, stimulating debates on its predictive power.

Cotton gravity is a modified theory of gravitation proposed by Harada in which the fundamental field equation is written with the three-index Cotton tensor rather than the Einstein tensor. In this framework, every solution of Einstein’s equations, with or without a cosmological constant, also solves the Cotton-gravity equations, but the converse is not generally true; vacuum is characterized by Cotton-flatness rather than Ricci-flatness, and the cosmological constant enters as an integration constant. The theory has therefore been studied both as an extension of general relativity and as a source of additional non-Einstein solution branches in spherical symmetry, cosmology, and black-hole physics (Harada, 2021).

1. Field-equation structure

Harada’s original formulation is built from the rank-3 tensor equation

$$C^{\nu\rho\sigma}=16\pi G\,\nabla_\mu T^{\mu\nu\rho\sigma},$$

where the Cotton tensor is

$$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$

The matter source is not $T_{\mu\nu}$ directly, but a rank-4 tensor built from $T_{\mu\nu}$ whose trace reproduces the ordinary stress tensor. In this formulation the Cotton tensor is divergence-free, traceless, and antisymmetric in its last two indices, while ordinary matter conservation $\nabla_\mu T^{\mu}{}_{\nu}=0$ is built in by contraction of the field equations (Harada, 2021).

A central conceptual shift is that the vacuum equation is

$C^{\nu\rho\sigma}=0,$

not $R_{\mu\nu}=0$. This makes the vacuum sector broader than in general relativity. Harada emphasized that Ricci-flat manifolds, Einstein manifolds $R_{\mu\nu}=\Lambda g_{\mu\nu}$, and conformally flat manifolds all satisfy the vacuum Cotton equation, but that further non-Einstein solutions also exist (Harada, 2021).

The same paper proposed a variational origin based on the Weyl-squared action, but with variation taken with respect to the connection while keeping the metric fixed. In that treatment, although the Cotton tensor contains third derivatives of the metric, the resulting Euler–Lagrange equations are argued to be effectively second order in $g$. This point has remained important because later discussions of consistency and predictivity repeatedly return to whether the theory should be understood primarily as a third-order metric theory or through an equivalent lower-order reformulation (Harada, 2021).

2. Codazzi reformulation and relation to general relativity

A mathematically equivalent formulation rewrites Cotton gravity through a symmetric rank-2 Codazzi tensor. In the form used in the cosmological and geometric literature,

$$\mathscr{C}_{\mu\nu} = R_{\mu\nu} - 8\pi G T_{\mu\nu} - \frac16 (R-16\pi G T)g_{\mu\nu},$$

subject to

$$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$0

This yields the modified Einstein-like equation

$$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$1

In this parametrization, departures from general relativity are encoded in an additional geometrical source, often interpreted as an effective anisotropic fluid (Xia et al., 2024).

This reformulation has two consequences. First, it makes the inclusion of ordinary $$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$2 explicit, avoiding the interpretive awkwardness of a source tensor built from derivatives of matter. Second, it ties Cotton gravity to the geometry of Codazzi tensors, a classical subject in differential geometry. The Codazzi perspective was developed in detail by Mantica and collaborators, who showed that choosing a Codazzi tensor of physically meaningful algebraic type constrains the admissible spacetime geometry, determines the Ricci tensor, and thereby fixes the stress tensor needed for a Cotton-gravity solution (Mantica et al., 2022).

The geometric consequences are strong. A perfect-fluid Codazzi tensor exists if and only if the metric is a generalized Stephani universe; under trace restrictions this reduces to a warped spacetime. Static and spherically symmetric current-flow Codazzi tensors force metrics of Nariai or Bertotti–Robinson type, and Yang Pure spacetimes with Codazzi Ricci tensor provide another natural vacuum sector. In this sense, Codazzi tensors do not merely repackage the equations: they organize the solution space and identify entire classes of admissible geometries (Mantica et al., 2022).

3. Spherical sectors and exact spacetime solutions

The first nontrivial exact vacuum solution exhibited in the theory is the static, spherically symmetric metric

$$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$3

Relative to Schwarzschild–(A)dS, the distinctive new term is the linear contribution $$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$4. Harada interpreted $$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$5 as the mass parameter, $$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$6 as the cosmological constant arising as an integration constant, and $$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$7 as the genuinely new Cotton-gravity parameter (Harada, 2021).

Later work enlarged this spherical sector. The paper “General spherically symmetric solution of Cotton gravity” derived a more general static vacuum family in Schwarzschild gauge with four constants $$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$8, and argued that the pure Schwarzschild sector generically develops a genuine curvature singularity at the photosphere $$C^{\nu\rho\sigma} = \nabla^\rho R^{\nu\sigma} - \nabla^\sigma R^{\nu\rho} - \frac16 \left( g^{\nu\sigma}\nabla^\rho R - g^{\nu\rho}\nabla^\sigma R \right).$$9. The associated claim that this singularity may obstruct the formation of ordinary Schwarzschild-radius stellar black holes was presented as an interpretation rather than as a collapse theorem (Gogberashvili et al., 2023).

Matter-coupled spherical solutions have also been constructed. “Black bounces in Cotton gravity” coupled the theory to nonlinear electrodynamics and a phantom scalar field, using the bounce function

$T_{\mu\nu}$0

This produced Simpson–Visser-like and Bardeen-like geometries whose parameter space includes regular black holes, black bounces with event, Cauchy, and cosmological horizons, and traversable wormholes. In those solutions the Kretschmann scalar remains finite at the bounce $T_{\mu\nu}$1, and the Cotton-gravity deformation enters through the same linear term that characterizes the vacuum solution, now promoted to $T_{\mu\nu}$2 (Junior et al., 2024).

A separate NLED study constructed three families of static, magnetically charged black-hole solutions based on $T_{\mu\nu}$3, $T_{\mu\nu}$4, and $T_{\mu\nu}$5. Their horizon structure ranges from ordinary black holes to multi-horizon configurations and naked singularities, and the paper explicitly confronted the associated shadow sizes with Event Horizon Telescope bounds for Sgr A$T_{\mu\nu}$6 (Junior et al., 24 Mar 2025).

4. Cosmology, FLRW ambiguity, and perturbations

Cosmology has exposed both the flexibility and the ambiguity of Cotton gravity. In the original Cotton-tensor formulation, inserting an FLRW metric and a perfect-fluid stress tensor makes the field equation collapse to the tautology $T_{\mu\nu}$7. This means that the homogeneous background expansion is not fixed by the theory in any unique way at that level, and several subsequent papers therefore treat cosmology through the Codazzi formulation or impose additional assumptions by hand (Xia et al., 2024).

One response was to derive nontrivial FLRW models from the Codazzi picture. In that approach the modified Friedmann equation becomes

$T_{\mu\nu}$8

with an additional geometric function $T_{\mu\nu}$9. When one restricts to models compatible with a well posed initial value formulation, the admissible branches are operationally the same as FLRW models in general relativity. Within that restricted sector, the paper “Cotton Gravity: the cosmological constant as spatial curvature” argued that the $T_{\mu\nu}$0CDM model becomes the unique FLRW dust model with constant negative spatial curvature, thereby reinterpreting the cosmological constant geometrically rather than as dark energy (Sussman et al., 2023).

The Codazzi parametrization was also promoted as a unifying FRW framework for modified gravity. In that language the extra geometric sector takes the perfect-fluid form

$T_{\mu\nu}$1

and a single free function $T_{\mu\nu}$2 can reproduce the Friedmann equations of $T_{\mu\nu}$3, modified Gauss–Bonnet $T_{\mu\nu}$4, teleparallel $T_{\mu\nu}$5, Einsteinian cubic gravity, $T_{\mu\nu}$6, Conformal Killing gravity, Mimetic gravity, and effective dark-energy parametrizations such as Chevallier–Polarski–Linder and Jassal–Bagla–Padmanabhan (Mantica et al., 2023).

The most direct observational cosmology to date is the linear scalar perturbation analysis performed on a fixed flat-$T_{\mu\nu}$7CDM background. In Newtonian gauge, the central Cotton-gravity modification is the slip relation

$T_{\mu\nu}$8

which the authors specialized to a constant $T_{\mu\nu}$9. After redefining the perturbed fluid variables, the scalar equations take the same form as in general relativity, so the theory behaves as general relativity plus an additional anisotropic-fluid contribution at linear order. Planck 2018 plus SDSS data constrain

$\nabla_\mu T^{\mu}{}_{\nu}=0$0

at the $\nabla_\mu T^{\mu}{}_{\nu}=0$1 level, and the paper concludes that there is no statistically significant departure from $\nabla_\mu T^{\mu}{}_{\nu}=0$2CDM in this phenomenological realization once neutrino anisotropic stress is modeled properly (Xia et al., 2024).

5. Astrophysical tests and observational status

On galactic scales, Cotton gravity was proposed as a dark-matter substitute because the weak-field spherical solution implies the potential

$\nabla_\mu T^{\mu}{}_{\nu}=0$3

so that

$\nabla_\mu T^{\mu}{}_{\nu}=0$4

Using a sample of 84 rotating galaxies, the paper “Cotton gravity and 84 galaxy rotation curves” solved the effective field equation numerically with observed baryons only and reported that the rotation curves can be explained without dark matter. The price is that $\nabla_\mu T^{\mu}{}_{\nu}=0$5 is not universal but a galaxy-specific integration constant; 76 of the 84 galaxies satisfy

$\nabla_\mu T^{\mu}{}_{\nu}=0$6

and no correlation was found between $\nabla_\mu T^{\mu}{}_{\nu}=0$7 and $\nabla_\mu T^{\mu}{}_{\nu}=0$8, which limits predictive power relative to schemes that postulate a universal acceleration scale (Harada, 2022).

Weak lensing produces a different verdict. Using the spherical metric with the linear term $\nabla_\mu T^{\mu}{}_{\nu}=0$9, the paper “Testing Cotton gravity as dark matter substitute with weak lensing” derived the deflection angle through the Gauss–Bonnet theorem and compared the implied galaxy–galaxy lensing signal with SDSS DR7 data. The result was that Cotton gravity on its own has significant deviation from the measured galaxy–galaxy lensing signals and cannot replace the role of dark matter; when combined with an NFW halo, the Cotton term is tightly constrained and statistically disfavored by the Bayesian Information Criterion (Mo et al., 2024).

Strong-field observations around black holes have so far pushed the spherical Cotton parameter into a regime of near-degeneracy with Schwarzschild. For the vacuum metric $C^{\nu\rho\sigma}=0,$0, finite-distance shadow calculations using the Sgr A$C^{\nu\rho\sigma}=0,$1 shadow size inferred by the Event Horizon Telescope yield the bound

$C^{\nu\rho\sigma}=0,$2

at $C^{\nu\rho\sigma}=0,$3. At that level, the photon sphere, deflection angle, thin-disk images, isoradials, and Lyapunov exponent of nearly bound null geodesics are practically indistinguishable from the Schwarzschild values with present and foreseeable interferometric techniques (Junior et al., 2024).

The present empirical situation may be summarized as follows.

Domain	Representative result	Source
Rotation curves	84 galaxies fitted with observed baryons and galaxy-specific $C^{\nu\rho\sigma}=0,$4	(Harada, 2022)
Weak lensing	Cotton gravity alone cannot replace dark matter	(Mo et al., 2024)
Linear cosmology	$C^{\nu\rho\sigma}=0,$5 at $C^{\nu\rho\sigma}=0,$6	(Xia et al., 2024)
Sgr A$C^{\nu\rho\sigma}=0,$7 shadow	$C^{\nu\rho\sigma}=0,$8 at $C^{\nu\rho\sigma}=0,$9	(Junior et al., 2024)

A further shadow study of Cotton gravity coupled to nonlinear electrodynamics found that several charged spherical models remain compatible with EHT bounds for Sgr A$R_{\mu\nu}=0$0 within restricted parameter regions, even though their causal structure can be substantially richer than that of Einstein–Maxwell black holes (Junior et al., 24 Mar 2025).

6. Predictivity debates, conserved charges, and later developments

The central theoretical controversy concerns predictivity. In “Cotton gravity is not predictive,” Clément and Nouicer argued that under-determination increases with symmetry: static spherically symmetric solutions depend on an arbitrary function of the radial coordinate, anisotropic cosmologies depend on an arbitrary function of time, and FLRW cosmologies are trivially solved because the Cotton tensor vanishes identically. Their conclusion was that the theory fails to determine geometry uniquely from matter, even after symmetry reduction (Clément et al., 2023).

A response paper rejected that conclusion by insisting that Cotton gravity should be interpreted primarily through the Codazzi formulation rather than by identifying vacuum with $R_{\mu\nu}=0$1. In that reply, the spherical sector was reconsidered under the ansatz $R_{\mu\nu}=0$2, and the Schwarzschild-like vacuum solution

$R_{\mu\nu}=0$3

was presented as the physically relevant vacuum solution, with the broader formal family associated to non-vacuum sources compatible with $R_{\mu\nu}=0$4 (Sussman et al., 2024). The dispute therefore concerns not only solution counting but also which formulation provides the correct notion of vacuum.

A second major criticism comes from conserved charges. “Vanishing of Conserved Charges in Cotton Gravity” derived a superpotential current built from the Cotton tensor and concluded that, if there is no matter at the spatial boundary, all asymptotic Killing charges vanish for all solutions, including black holes. In particular, mass/energy and angular momentum are zero. The paper interpreted this either as evidence that the theory is unphysical, because black holes would carry the charges of the vacuum and could be created at no energy cost, or as a confinement-like scenario in which energy and angular momentum fail to appear as asymptotic charges (Altas et al., 2024).

Despite these debates, the subject has continued to develop in several directions. A thermodynamic formulation in the Codazzi parametrization derived modified horizon entropy laws of the form

$R_{\mu\nu}=0$5

for FRW apparent horizons and static event horizons, with the correction controlled by the temporal or anisotropic Codazzi sector (Ghaffari et al., 12 May 2026). A geometric analysis of static spacetimes introduced the notion of a Cotton-$R_{\mu\nu}=0$6-perfect fluid, established local warped-product structure near regular level sets of the lapse under suitable curvature conditions, and again highlighted Codazzi tensors as the central organizing structure (Colombo et al., 12 May 2026). A separate 2026 black-hole study treated a Schwarzschild AdS metric with a Cotton parameter $R_{\mu\nu}=0$7, found Van der Waals-like criticality for positive $R_{\mu\nu}=0$8, and used the island formula to recover a Page curve with late-time entropy saturating at approximately $R_{\mu\nu}=0$9 (Ladghami et al., 12 Mar 2026).

In a distinct three-dimensional line of research, the Cotton tensor has also appeared as the conformal-curvature object replacing the absent Weyl tensor in topologically massive gravity. There the “Cotton double copy”

$R_{\mu\nu}=\Lambda g_{\mu\nu}$0

has been proved for Type N wave solutions. This construction is conceptually separate from Harada’s four-dimensional Cotton gravity, but it underscores the broader role of Cotton curvature in gravitational theory (González et al., 2022).