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

• DOI: 10.48550/arxiv.2501.16104
• Publication Date: 2025-01-01
• Authors: Gunneberg, Finlay; Gratus, Jonathan; Stanfield, Harvey
• Keywords: High Energy Physics - Theory; High Energy Physics - Theory (hep-th); FOS: Physical sciences; Mathematical Physics (math-ph); Mathematical Physics

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.<br/>28 pages 6 figures<br/>