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 $\mathrm{\Phi }=-GM/r+\gamma r/2$, 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.

中文翻译：

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棉花重力和84个星系自转曲线

最近，作为广义相对论的推广，提出了一种引力理论，其中引力场方程由科顿张量描述。该理论允许对随半径线性上升的点质量的引力势做出额外贡献$\mathrm{\Phi }=-G米/r+\gamma r/2$， 在哪里$G$是牛顿常数。系数$米$和$\gamma$是积分常数，应为每个物理系统单独确定。当应用于星系时，系数$\gamma$，它具有加速度的维度，应该为每个星系确定。这与必须确定质量相同$米$对于每个星系。如果$\gamma$足够小，线性势项在短距离内可以忽略不计，但在远距离时会变得很重要。事实上，它可能有助于河外系统。在本文中，我们推导了适用于河外系统的棉花重力有效场方程。然后，我们使用有效场方程数值计算 84 个旋转星系样本的引力势。这 84 个星系的范围很广，从以恒星盘为主的螺旋星系到以气体为主的矮星系。我们不假设星盘、核球或气体的径向密度分布；我们只使用观察到的数据。我们发现84个星系的自转曲线可以用观测到的重子分布来解释。这是由于棉花重力的灵活性允许积分常数$\gamma$对于每个星系。在科顿引力的背景下，“暗物质”在某种意义上被自动包含为时空曲率。因此，即使是被认为由暗物质主导的星系也不需要暗物质。