Variational Principles for Natural Divergence-free Tensors in Metric Field Theories

TL;DR

This paper proves that third-order, divergence-free, tensorial differential equations for the metric are Euler-Lagrange equations derived from some Lagrangian, extending classical results on metric field equations.

Contribution

It establishes that such divergence-free, tensorial metric equations are necessarily variational, generalizing classical theorems to higher-order tensors.

Findings

Third-order divergence-free tensors are Euler-Lagrange expressions.

Extension of classical metric field equation results to higher-order tensors.

Provides a characterization of metric field equations via symmetries and conservation laws.

Abstract

Let $T^{ab}=T^{ba}=0$ be a system of differential equations for the components of a metric tensor on $R^m$. Suppose that $T^{ab}$ transforms tensorially under the action of the diffeomorphism group on metrics and that the covariant divergence of $T^{ab}$ vanishes. We then prove that $T^{ab}$ is the Euler-Lagrange expression some Lagrangian density provided that $T^{ab}$ is of third order. Our result extends the classical works of Cartan, Weyl, Vermeil, Lovelock, and Takens on identifying field equations for the metric tensor with the symmetries and conservation laws of the Einstein equations.