The Vlasov Bivector: A Parameter-Free Approach to Vlasov Kinematics

TL;DR

This paper introduces a parameter-free, bivector-based formulation of Vlasov kinetics that extends applicability to lightlike, ultra-relativistic, and non-metric scenarios, broadening the theoretical framework of plasma physics.

Contribution

It develops a bivector approach to Vlasov theory that is parameterisation invariant and applicable to a wider class of spacetimes and geodesics, including spacelike geodesics.

Findings

Provides a parameter-free Vlasov formulation using bivectors.

Extends the theory to non-metric and premetric contexts.

Connects to Finsler geometry and semi-sprays.

Abstract

Plasma kinetics, for both flat and curved spacetime, is conventionally performed on the mass shell, a 7--dimensional time-phase space with a Vlasov vector field, also known as the Liouville vector field. The choice of this time-phase space encodes the parameterisation of the underling 2nd order ordinary differential equations. By replacing the Vlasov vector on time-phase space with a bivector on an 8--dimensional sub-bundle of the tangent bundle, we create a parameterisation free version of Vlasov theory. This has a number of advantages, which include working for lightlike and ultra-relativistic particles, non metric connections, and metric-free and premetric theories. It also works for theories where no time-phase space can exist for topological topological reasons. An example of this is when we wish to consider all geodesics, including spacelike geodesics. We extend the particle density function to a 6--form on the subbundle of the tangent space, and define the transport equations, which correspond to the Vlasov equation. We then show how to define the corresponding 3--current on spacetime. We discuss the stress-energy tensor needed for the Einstein-Vlasov system. This theory can be generalised to create parameterisation invariant Vlasov theories for many 2nd order theories, on arbitrary manifolds. The relationship to sprays and semi-sprays is given and examples from Finsler geometry are also given.