It is shown that the Cotton tensor can describe the effects of gravity beyond general relativity. Any solution of the Einstein equations with or without the cosmological constant satisfies the field equations described by the Cotton tensor. It implies that the cosmological constant is an integration constant. A vacuum of a theory is represented by the vanishing of the Cotton tensor, rather than the vanishing of the Ricci tensor. An exact Schwarzschild-like solution for a static and spherically symmetric source is discovered. Although the field equations involve the third order of derivative, it is found that they reduce to the second-order differential equations due to a variational principle.

中文翻译：

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用于描述重力的棉花张量的出现

结果表明，棉花张量可以描述超越广义相对论的引力效应。有或没有宇宙常数的爱因斯坦方程的任何解都满足 Cotton 张量描述的场方程。这意味着宇宙常数是一个积分常数。一个理论的真空表现为 Cotton 张量的消失，而不是 Ricci 张量的消失。发现了静态和球对称源的精确 Schwarzschild 式解。虽然场方程涉及三阶导数，但发现由于变分原理，它们可以简化为二阶微分方程。