Hamiltonian dynamics of doubly-foliable space-times

TL;DR

This paper develops a Hamiltonian framework for analyzing gravitational perturbations in space-times with a distinguished spatial direction, using a 2+1+1 decomposition that clarifies the dynamics and constraints in such geometries.

Contribution

It introduces a Hamiltonian formulation based on a 2+1+1 foliation, extending previous work to include gauge freedom for even sector perturbations in spherically symmetric space-times.

Findings

Identifies canonical variables and constraints in the 2+1+1 decomposition.

Derives Hamiltonian dynamics using Poisson brackets.

Clarifies gauge freedom in even sector perturbations.

Abstract

The 2+1+1 decomposition of space-time is useful in monitoring the temporal evolution of gravitational perturbations/waves in space-times with a spatial direction singled-out by symmetries. Such an approach based on a perpendicular double foliation has been employed in the framework of dark matter and dark energy motivated scalar-tensor gravitational theories for the discussion of the odd sector perturbations of spherically symmetric gravity. For the even sector however the perpendicularity has to be suppressed in order to allow for suitable gauge freedom, recovering the 10th metric variable. The 2+1+1 decomposition of the Einstein-Hilbert action leads to the identification of the canonical pairs, the Hamiltonian and momentum constraints. Hamiltonian dynamics is then derived via Poisson brackets.