Geometric structures of the classical general relativistic phase space

TL;DR

This paper investigates the geometric properties of the phase space in classical general relativity, focusing on conditions for contact and Jacobi structures arising from Lorentz metrics and connections, including electromagnetic fields.

Contribution

It determines geometric conditions under which the phase space admits contact and Jacobi structures, especially when generated by the metric and additional tensors like electromagnetic fields.

Findings

Identifies conditions for contact structures in relativistic phase space.

Analyzes Jacobi structures related to Lorentz metrics and connections.

Includes special cases with electromagnetic fields.

Abstract

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1--dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialise these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.