Contact Geometry of Relativistic Particle Motion

TL;DR

This paper develops a contact geometric framework for relativistic particle dynamics, enabling a covariant description of particle decay and massless particles without reparametrization.

Contribution

It introduces a novel geometric approach on an extended phase space that handles dissipative processes and redefines relativistic equations in a fully geometric manner.

Findings

Framework accommodates decaying particles and massless particles.

Reformulates relativistic Hamilton equations without proper time parametrization.

Provides a covariant kinetic theory incorporating particle decay.

Abstract

We introduce a new geometric framework for relativistic particle dynamics based on contact geometry and suitable for treating dissipative processes like particle decay. The dynamics is formulated on a nine--dimensional extended phase space consisting of four position coordinates, four momenta, and an additional variable (functioning as a geometric variant of the particle's proper time). In this setting, the evolution is generated by an evolution contact vector field with a contact Hamiltonian encoding the mass shell. By promoting the proper time to an independent variable, the relativistic Hamilton canonical equations are rewritten in a fully geometric form without having to identify the proper time with a parameter along the worldlines. This makes for instance the evolution of massless particles (photons) well-defined without the need of reparametrization. The framework is then applied to decaying particles. Finally, we formulate a covariant kinetic theory and show how decaying particles can be described geometrically in this framework, changing the entropy.