Hamiltonian formulation of scalar field collapse in Einstein–Gauss–Bonnet gravity

Abstract

We compute the Hamiltonian for spherically symmetric scalar field collapse in Einstein–Gauss–Bonnet gravity in D dimensions using slicings that are regular across future horizons. We first reduce the action to two dimensions using spherical symmetry. We then show that choosing the spatial coordinate to be a function of the areal radius leads to a relatively simple Hamiltonian constraint whose gravitational part is the gradient of the generalized mass function. Next we complete the gauge fixing such that the metric is the Einstein–Gauss–Bonnet generalization of non-static Painlevé–Gullstrand coordinates. Finally, we derive the resultant-reduced equations of motion for the scalar field. These equations are suitable for use in numerical simulations of spherically symmetric scalar field collapse in Gauss–Bonnet gravity and can readily be generalized to other matter fields minimally coupled to gravity.