Lovelock's theorem revisited

TL;DR

This paper revisits Lovelock's theorem, providing a simplified, self-contained proof of the classification of second-order natural divergence-free symmetric (0,2)-tensors on pseudo-Riemannian manifolds, including the Einstein tensor.

Contribution

It offers a streamlined, formalized proof of Lovelock's theorem, clarifying the structure of natural tensors satisfying specific geometric conditions.

Findings

Explicit classification of second-order natural divergence-free tensors

Simplified proof methodology for Lovelock's theorem

Clarification of the Einstein tensor's role

Abstract

Let (X, g) be an arbitrary pseudo-riemannian manifold. A celebrated result by Lovelock gives an explicit description of all second-order natural (0,2)-tensors on X, that satisfy the conditions of being symmetric and divergence-free. Apart from the dual metric, the Einstein tensor of g is the simplest example. In this paper, we give a short and self-contained proof of this theorem, simplifying the existing one by formalizing the notion of derivative of a natural tensor.