The Canonical Lagrangian Approach To Three-Space General Relativity

TL;DR

This paper analyzes the canonical Lagrangian structure of three-space General Relativity, revealing the dynamics, gauge transformations, and symplectic reduction within the BF'O action framework.

Contribution

It provides a detailed study of the symplectic and presymplectic structures, deriving the canonical vector field and discussing observables in three-space GR.

Findings

Derived the canonical Lagrangian vector field for the BF'O action

Established the symplectic reduction of the constrained phase space

Clarified the gauge transformations and physical evolution in three-space GR

Abstract

We study the action for the three-space formalism of General Relativity, better known as the BF\'O (Barbour--Foster--\'O Murchadha) action, which is a square-root BSW (Baierlein--Sharp--Wheeler) action. In particular, we explore the (pre)symplectic structure by pulling it back via a Legendre map to the tangent bundle of the configuration space of this action. With it we attain the canonical Lagrangian vector field which generates the gauge transformations (3-diffeomorphisms) and the true physical evolution of the system. This vector field encapsulates all the dynamics of the system. We also discuss briefly the observables and perennials for this theory. We then present a symplectic reduction of the constrained phase space.