Emergence of the Cotton tensor for describing gravity

TL;DR

This paper introduces the Cotton tensor as a novel way to describe gravitational effects beyond general relativity, revealing new solutions and implications for the cosmological constant.

Contribution

It demonstrates that the Cotton tensor can replace the Ricci tensor in describing gravity, leading to new solutions and insights into the structure of gravitational field equations.

Findings

Cotton tensor describes gravity beyond Einstein's theory.

A Schwarzschild-like solution is derived using the Cotton tensor.

Field equations reduce to second-order despite involving third derivatives.

Abstract

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.