Diagrammatic Derivation of Lovelock Densities

TL;DR

This paper introduces a diagrammatic method for deriving Lovelock scalar densities, simplifying calculations by translating tensor contractions into permutation diagrams, and applies it to the first few densities.

Contribution

It presents a novel diagrammatic approach to derive Lovelock densities, reducing algebraic complexity and enabling easier identification of combinatorial factors.

Findings

Successfully derived the first Lovelock densities using the diagrammatic method

Simplified the calculation process compared to traditional algebraic methods

Provided explicit forms for densities of order two and three

Abstract

We discuss a method of calculating the various scalar densities encountered in Lovelock theory which relies on diagrammatic, instead of algebraic manipulations. Taking advantage of the known symmetric and antisymmetric properties of the Riemann tensor which appears in the Lovelock densities, we map every quadratic or higher contraction into a corresponding permutation diagram. The derivation of the explicit form of each density is then reduced to identifying the distinct diagrams, from which we can also read off the overall combinatoric factors. The method is applied to the first Lovelock densities, of order two (Gauss-Bonnet term) and three.