Gravitational dynamics in s+1+1 dimensions II. Hamiltonian theory

TL;DR

This paper develops a Hamiltonian formalism for brane-world gravity in s+1+1 dimensions, identifying gravitational degrees of freedom and their dynamics both on and outside the brane, including constraints and junction conditions.

Contribution

It introduces a Hamiltonian approach to brane-world gravity with a novel decomposition, explicitly deriving evolution equations and constraints for gravitational fields in this setting.

Findings

Derived canonical evolution equations for gravitational variables.

Identified the continuity of graviton and gravi-scalar across the brane.

Linked the momentum jump of the gravi-vector to energy flow on the brane.

Abstract

We develop a Hamiltonian formalism of brane-world gravity, which singles out two preferred, mutually orthogonal directions. One is a unit twist-free field of spatial vectors with integral lines intersecting perpendicularly the brane. The other is a temporal vector field with respect to which we perform the Arnowitt-Deser-Misner decomposition of the Einstein-Hilbert Lagrangian. The gravitational variables arise from the projections of the spatial metric and their canonically conjugated momenta as tensorial, vectorial and scalar quantities defined on the family of hypersurfaces containing the brane. They represent the gravitons, a gravi-photon and a gravi-scalar, respectively. From the action we derive the canonical evolution equations and the constraints for these gravitational degrees of freedom both on the brane and outside it. By integrating across the brane, the dynamics also generates the tensorial and scalar projection of the Lanczos equation. The vectorial projection of the Lanczos equation arises in a similar way from the diffeomorphism constraint. Both the graviton and the gravi-scalar are continuous across the brane, however the momentum of the gravi-vector has a jump, related to the energy transport (heat flow) on the brane.