The Vlasov bivector: a parameter-free approach to Vlasov kinematics

Plasma kinematics is typically performed on either the mass shell or a lab bundle, seven-dimensional phase spaces equipped with a Vlasov vector field. This choice of phase space encodes the parametrisation of the differential equations governing particle dynamics. By replacing the Vlasov vector field and related quantities over a phase space with a bivector on an eight-dimensional conic sub-bundle of the tangent bundle, we construct a parameter-free version of Vlasov theory. The advantages of this formalism include compatibility with light-like and ultra-relativistic particles, non-metric connections, and metric-free or pre-metric theories. Additionally, this formalism applies to theories where no time phase space can exist for topological reasons, e.g. when we wish to consider all geodesics, including space-like geodesics. We also use this formalism to derive a simple formula which enables the transformation of a Vlasov field from one phase space to another in such a way that the trajectories of the particles in the base space are unchanged. Our formalism also has implications for numerical simulations. By extending quantities such as the particle density form onto an eight-dimensional conic sub-bundle, we can extend existing numerical integrators, such as the agile numerical integrator, to relativistic scenarios. The extension of the particle density form also allows for the generalisation of the current 3-form onto the conic bundle. We also discuss the complicated relationship between the stress-energy form and phase space, how our formalism illustrates this, and the corresponding challenges in developing a parameter-free Einstein-Vlasov system. The relationship between our formalism and sprays or semi-sprays is explored, and examples from Finsler geometry are given.