n   +  1 formalism of f (Lovelock) gravity

Abstract

In this note we perform the n+1 decomposition, or Arnowitt-Deser-Misner (ADM) formulation of f (Lovelock) gravity theory. The hamiltonian form of Lovelock gravity was known since the work of C. Teitelboim and J. Zanelli in 1987, but this result had not yet been extended to f (Lovelock) gravity. Besides, field equations of f (Lovelock) have been recently computed by P. Bueno et al., though without ADM decomposition.We focus on the non-degenerate case, ie. when the Hessian of f is invertible. Using the same Legendre transform as for f (R) theories, we can identify the partial derivatives of f as scalar fields, and consider the theory as a generalised scalar-tensor theory. We then derive the field equations, and project them along a n + 1 decomposition. We obtain an original system of constraint equations for f (Lovelock) gravity, as well as dynamical equations. We give explicit formulas for the f (R, Gauss-Bonnet) case.