Symmetries of first-order Lovelock gravity Articles uri icon

publication date

  • December 2018

start page

  • 235015


  • 23


  • 35

International Standard Serial Number (ISSN)

  • 0264-9381

Electronic International Standard Serial Number (EISSN)

  • 1361-6382


  • 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 PoincarĂ© or Euclidean transformations together with the new symmetry. The commutator algebra of this set also closes with structure functions.


  • lovelock gravity; noether's second theorem; local translations; gauge symmetries