Lovelock Tensor as Generalized Einstein Tensor

TL;DR

This paper demonstrates that the Lovelock tensor generalizes the Einstein tensor through a novel trace operator, unifying various homogeneous terms and extending the concept to scalar Gauss-Bonnet gravity.

Contribution

It introduces a generalized trace operator that reveals the Lovelock tensor as a generalized Einstein tensor, applicable to inhomogeneous Euler-Lagrange expressions.

Findings

Lovelock tensor shares a trace relation similar to Einstein tensor.

The generalized trace operator applies to inhomogeneous tensors.

The technique extends to scalar Gauss-Bonnet gravity.

Abstract

We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts, namely the Ricci tensor and the term proportional to the curvature scalar, with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, with the aid of a generalized trace operator which we define, for any inhomogeneous Euler-Lagrange expression that can be spanned linearly in terms of homogeneous tensors. Then, through an application of this generalized trace operator, we demonstrate that the Lovelock tensor analogizes the mathematical form of the Einstein tensor, hence, it represents a generalized Einstein tensor. Finally, we apply this technique to the scalar Gauss-Bonnet gravity as an another version of string-inspired gravity.