Symmetries of first-order Lovelock gravity

TL;DR

This paper explores the gauge symmetries of first-order Lovelock gravity, revealing new symmetries and their algebraic structure, and clarifying invariances under various transformations in different dimensions.

Contribution

It identifies new gauge symmetries in Lovelock gravity and analyzes their algebraic structure, extending understanding of its invariance properties.

Findings

Existence of a new gauge symmetry under specific conditions.

Invariance under local translations with non-zero cosmological constant.

Closure of the gauge algebra with structure functions.

Abstract

We apply the converse of Noether's second theorem to the first-order $n$-dimensional Lovelock action, considering the frame rotation group as both $SO\left(1,n-1\right)$ or as $SO(n)$. As a result, we get the well-known invariance under local Lorentz transformations or $SO(n)$ transformations and diffeomorphisms, for odd- and even-dimensional manifolds. We also obtain the so-called `local translations' with nonvanishing constant $\Lambda$ for odd-dimensional manifolds when a certain relation among the coefficients of the various terms of the first-order Lovelock Lagrangian is satisfied. When this relation is fulfilled, we report the existence of a new gauge symmetry emerging from a Noether identity. In this case the fundamental set of gauge symmetries of the Lovelock action is composed by the new symmetry, local translations with $\Lambda \neq 0$ and local Lorentz transformations or $SO(n)$ transformations. The commutator algebra of this set closes with structure functions. We also get the invariance under local translations with $\Lambda=0$ of the highest term of the Lovelock action in odd-dimensional manifolds. Furthermore, we report a new gauge symmetry for the highest term of the first-order Lovelock action for odd-dimensional manifolds. In this last case, the fundamental set of gauge symmetries can be considered as Poincar\'e or Euclidean transformations together with the new symmetry. The commutator algebra of this set also closes with structure functions.