All identically conserved gravitational tensors are metric variations of an action

TL;DR

This paper proves that all symmetric, conserved gravitational tensors originate from variations of an invariant action, confirming a longstanding conjecture in gravitational theory.

Contribution

It provides a proof that all such tensors are metric variations of invariant actions, resolving an old open question in gravity research.

Findings

All identically conserved symmetric tensors are metric variations of invariant actions.

The proof simplifies understanding of possible gravity theories.

Confirms that no alternative gravity theories are needed beyond those derived from actions.

Abstract

I prove an old unsolved conjecture, the hard part of its obvious sufficiency, namely that all identically conserved symmetric two-tensors are necessarily metric variations of invariant actions, thus sparing the world more alternative gravity theories. The proof is reasonably simple, if perhaps a physicist's.