Diagrammatic derivation of Lovelock densities

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.Lovelock theory is a natural classical extension of General Relativity in higher dimensions [1]. With the advent of a renewed interest on extra-dimensional models of gravity, particularly through String/M-Theory [2] and braneworld models [3,4,5,6,7,8], Lovelock gravity has received a lot of attention. The theory assumes the inclusion of an infinite number of scalar densities in the gravitational Lagrangian, apart from the ordinary Einstein-Hilbert term. Although these do not introduce any new dynamics when restricted in four dimensions, they become important when extra dimensions are introduced [9,10]. The first of these Lovelock densities is the well known Gauss-Bonnet scalar, which also appears in perturbative expansions in String theory, giving further support for the relevance of Lovelock theory [11,12].However, calculations in Lovelock gravity are known to be quite involved. The inclusion of higher scalar densities leads to considerably more complicated equations and these technical challenges render very difficult a systematic treatment of the theory to all orders. The derivation of even the scalar densities themselves quickly becomes very complicated once we start increasing the order [13,14,15]. Higher terms include contractions of products of two or more Riemann tensors and the possible combinations lead to a combinatoric explosion in complexity. As a result, up to now only the lowest order Lovelock correction, i.e. the Gauss-Bonnet term, has been extensively studied [16,17], with recent attempts to also include a third and fourth order density [18,19,20,21].In this letter we discuss an alternative method of calculating the first Lovelock densities, based on the use of permutation diagrams, instead of the more conventional way of using the Kronecker permutation tensor. We demonstrate that this method can lead to dramatic simplification of certain calculations, rendering the derivation of the first few scalar densities almost trivial. The method relies on the symmetries of the Riemann tensor in order to simplify and group together diagrams that lead to the same expressions. Essentially, the calculation of a scalar density is thus reduced to writing down the corresponding diagrams using a number of rules and then determining from them the overall numerical factor and sign of each term by inspecting the form of the appropriate diagram.We illustrate this procedure for t...

2-regular Digraphs of the Lovelock Lagrangian

“…For easier visual recognition, the graphs with more than one component are surrounded by a frame. Bogdanos has pointed at a graphical representation of the terms of Lovelock's Lagrange density [9][10][11]: A graph of a particular term which is a product of n Riemann Tensors is constructed as follows: (i) Start with a bottom row of n nodes plus a top row of another n nodes, where nodes are associated left-to-right with a left-toright reading of the R-factors of the product. (ii) Add an edge from the b-th bottom node to the t-th top node if a covariant lower index of factor number b appears as a contravariant index of factor number t. This assignment works because the two contravariant indices are permutations of the covariant indices.…”