$n+1$ formalism of $f($Lovelock$)$ gravity

TL;DR

This paper develops the $n+1$ (ADM) decomposition for $f($Lovelock$)$ gravity, extending Hamiltonian formulations and deriving new constraint and dynamical equations, including explicit formulas for the $f(R, ext{Gauss-Bonnet})$ case.

Contribution

It provides the first ADM formulation of $f($Lovelock$)$ gravity, generalizing previous Hamiltonian results and deriving the associated constraint and evolution equations.

Findings

Derived the $n+1$ decomposition for $f($Lovelock$)$ gravity.

Established the Hamiltonian form and constraint equations.

Provided explicit formulas for the $f(R, ext{Gauss-Bonnet})$ case.

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 be 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 $f(\mathrm{R})$ 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 $f(\mathrm{R},$ Gauss-Bonnet$)$ case.