Presymplectic Geometry and the Problem of Time. Part 1

TL;DR

This paper develops a presymplectic geometric framework to address the problem of time in General Relativity, reformulating Barbour's timeless theory within phase space for various constrained systems.

Contribution

It introduces a presymplectic phase space reformulation of Barbour's timeless approach, extending it to general constrained reparametrization-invariant theories.

Findings

Reformulation of Barbour's theory using presymplectic geometry.

Application to Jacobi mechanics and relational particle mechanics.

Demonstration of the framework on the free relativistic particle.

Abstract

An effective mathematical framework based on Presymplectic Geometry for dealing with the "phase space picture" of timeless dynamics in General Relativity is presented. In General Relativity, the presence of the scalar Hamiltonian constraint which vanishes leads to the problem of time, which can be solved, up to an extent by adopting a timeless formalism. This has been done by Carlo Rovelli and Julian Barbour et al. In this paper we present a phase space reformulation of Barbour's theory. The Presymplectic dynamics of general totally constrained, reparametrization invariant theories is developed, then applied to Jacobi mechanics, relational particle mechanics and the dynamics of the free relativistic particle.