Recently, as a generalization of general relativity, a gravity theory has

been proposed in which gravitational field equations are described by the

Cotton tensor. That theory allows an additional contribution to the

gravitational potential of a point mass that rises linearly with radius as

$\Phi = -GM/r + \gamma r/2$, where $G$ is the Newton constant. The coefficients

$M$ and $\gamma$ are the constants of integration and should be determined

individually for each physical system. When applied to galaxies, the

coefficient $\gamma$, which has the dimension of acceleration, should be

determined for each galaxy. This is the same as having to determine the mass

$M$ for each galaxy. If $\gamma$ is small enough, the linear potential term is

negligible at short distances, but can become significant at large distances.

In fact, it may contribute to the extragalactic systems. In this paper, we

derive the effective field equation for Cotton gravity applicable to

extragalactic systems. We then use the effective field equation to numerically

compute the gravitational potential of a sample of 84 rotating galaxies. The 84

galaxies span a wide range, from stellar disk-dominated spirals to

gas-dominated dwarf galaxies. We do not assume the radial density profile of

the stellar disk, bulge, or gas; we use only the observed data. We find that

the rotation curves of 84 galaxies can be explained by the observed

distribution of baryons. This is due to the flexibility of Cotton gravity to

allow the integration constant $\gamma$ for each galaxy. In the context of

Cotton gravity, “dark matter” is in some sense automatically included as a

curvature of spacetime. Consequently, even galaxies that have been assumed to

be dominated by dark matter do not need dark matter.