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It is shown that the Cotton tensor can describe the effects of gravity beyondgeneral relativity. Any solution of the Einstein equations with or without thecosmological constant satisfies the field equations described by the Cottontensor. It implies that the cosmological constant is an integration constant. Avacuum of a theory is represented by the vanishing of the Cotton tensor, ratherthan the vanishing of the Ricci tensor. An exact Schwarzschild-like solutionfor a static and spherically symmetric source is discovered. Although the fieldequations involve the third order of derivative, it is found that they reduceto the second-order differential equations due to a variational principle.

Emergence of the Cotton tensor for describing gravity