On second-order, divergence-free tensors

TL;DR

This paper characterizes second-order divergence-free tensors on pseudo-Riemannian manifolds, establishing isomorphisms with invariant tensor spaces and explicitly identifying bases in specific cases, refining previous results.

Contribution

It provides a comprehensive description of divergence-free, second-order natural tensors via orthogonal group invariants and explicitly computes bases for these tensor spaces.

Findings

Lovelock tensors form a basis for divergence-free, second-order tensors with two indices

Isomorphisms between tensor spaces and invariant tensor spaces are established

Explicit bases can be computed using orthogonal group invariants

Abstract

This paper deals with the problem of describing the vector spaces of divergence-free, natural tensors on a pseudo-Riemannian manifold that are second-order; i.e., that are defined using only second derivatives of the metric. The main result establishes isomorphisms between these spaces and certain spaces of tensors (at a point) that are invariant under the action of an orthogonal group. This result is valid for tensors with an arbitrary number of indices and symmetries among them and, in certain cases, it allows to explicitly compute basis, using the theory of invariants of the orthogonal group. In the particular case of tensors with two indices, we prove the Lovelock tensors are a basis for the vector space of second-order tensors that are divergence-free, thus refining the original Lovelock's statement.