Hamiltonian Dynamics of Doubly-Foliable Space-Times

Cecília Gergely,László Gergely,Zoltán Keresztes

doi:10.3390/universe4010009

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.

Highlights

In the curved space-time of general relativity, gravitational waves propagate with the speed of light, correcting the Newtonian description of gravity
We generalized the formalism of [19,20] by allowing for nonorthogonal foliations. This led to the reestablishment of the full gauge freedom, allowing a generic discussion of perturbations
In the ADM formalism, the induced metric and extrinsic curvature of the hypersurface play the role of Hamiltonian coordinates and momenta

Summary

Introduction

In the curved space-time of general relativity, gravitational waves propagate with the speed of light (the velocity limit), correcting the Newtonian description of gravity. Much simpler 2 + 1 + 1 decomposition formalism, the decomposition is made along a perpendicular double foliation [19,20] This formalism has been employed in the framework of dark matter and dark energy-motivated scalar-tensor gravitational theories in the discussion of the odd sector perturbations of spherically-symmetric gravity in the effective field theory approach [21]. A modified 2 + 1 + 1 decomposition formalism would be desirable, which keeps the relative simplicity of the formalism of [19,20] (as compared to the formalism exploring optical scalars [16]), but employs 10 metric functions instead of nine, becoming suitable for the discussion of the even sector Such a formalism could be worked out at the price of relaxing the perpendicularity requirement [22]. Boldface lower-case (as i) or uppercase (as A) Latin letters count two-dimensional or four-dimensional basis vectors

The Nonorthogonal Double Foliation

Hamiltonian Dynamics

Conclusions