$d+1$ formalism in Einstein-scalar-Gauss-Bonnet gravity

We present the $d+1$ formulation of Einstein-scalar-Gauss-Bonnet (ESGB) theories in dimension $D=d+1$ and for arbitrary (spacelike or timelike) slicings. We first build an action which generalizes those of Gibbons-Hawking-York and Myers to ESGB theories, showing that they can be described by a Dirichlet variational principle. We then generalize the Arnowitt-Deser-Misner (ADM) Lagrangian and Hamiltonian to ESGB theories, as well as the resulting $d+1$ decomposition of the equations of motion. Unlike general relativity, the canonical momenta of ESGB theories are nonlinear in the extrinsic curvature. This has two main implications: (i) the ADM Hamiltonian is generically multivalued, and the associated Hamiltonian evolution is not predictable; (ii) the "$d+1$" equations of motion are quasilinear, and they may break down in strongly curved, highly dynamical regimes. Our results should be useful to guide future developments of numerical relativity for ESGB gravity in the nonperturbative regime.