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

TL;DR

This work introduces a parameter-free Vlasov framework built on an 8D conic bundle $U$ by replacing the traditional Vlasov vector field with a Vlasov bivector $\Psi=\mathcal{R}\wedge W$, enabling kinematics without fixing a time-phase domain. A 6-form particle density $\theta$ on $U$ (and its pullbacks to kinematic domains via suitable indicators) satisfies transport equations $d\theta=0$ and $i_W\theta=0$, unifying Vlasov dynamics across metric and premetric theories. The formalism yields a naturally defined current on spacetime, $\mathcal{J}_E=\pi_{E\varsigma}(\theta_E)$, and a stress-energy 3-form $\tau_E{}_\alpha$, clarifying how standard Vlasov quantities depend on the chosen domain while preserving trajectory equivalence through reparameterisations. The approach accommodates lightlike and ultra-relativistic regimes, non-metric connections, and topologies where conventional kinematic domains fail, with explicit links to sprays, semi-sprays, and Finsler geometry. Overall, the Vlasov bivector establishes a robust, parameterisation-invariant foundation for kinetic theory with broad theoretical and potential astrophysical applications, including Einstein–Vlasov systems and beyond.

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.

Figures (6)

• Figure 1: The Vlasov Bivector integrates to form leaves, like the leaves of a book. The density of the leaves also represent the particle distribution.
• Figure 2: Illustration of integral curves, i.e the prolongations, in the mass shell for the case of a Vlasov field built from a force equation with a metric compatible connection (\ref{['Fig_Mass shell slip compat']}) and a non metric compatible connection (\ref{['Fig_Mass shell slip non compat']}). The mass shell is represented by the blue sheet, integral curves $\eta,\;\hat{\eta}$ of the Vlasov fields $W,\hat{W}$ are given by the green lines. The Vlasov fields themselves are depicted by the orange arrows. The green curves show are initially on the unit hyperboloid but do not remain on it for the case where the Vlasov field is built from a non metric compatible connection.
• Figure 3: The 7-dimensional kinematic domains $E_{\textnormal{H}}$ and $E_{t}$ are given by slices of the 8-dimensional conic bundle $U$. The unit hyperboloid $E_{\textnormal{H}}$ is given by the dark blue hyperbola and a lab time bundle $E_{t}$ by the horizontal black line. The vector fields $W_{E_{\textnormal{H}}}$ and $W_{E_{t}}$ (green and red arrows respectively) are tangent to their respective kinematic domains.
• Figure 4: Diagram of the transport equations on a $(2n-1)$--dimensional kinetic domain $E$ using form submanifolds. The details of submanifolds are described in gratus_pictorial_2017. The form manifolds of $\theta_E\in \Gamma\Lambda^{2n-2} E$ (black lines) do not terminate ($d\theta=0$) and are tangent to the vector field $W_E\in\Gamma TE$, represented by blue arrows ($i_{W_E}\theta_E=0$).
• Figure 5: Sketch of an integrable Vlasov bivector $\Psi$. Here, a possible form for $\Psi$ is $\Psi=\mathcal{R}\wedge W$. Notice that since $\Psi$ is integrable, the bivectors 'knit-together' to from the leaves of a foliation. The ambient space is $U\subset \breve{T}M$. The green line is to indicate the absence of 0-vectors in our space. The density of the leaves correspondence to the velocity density of our one particle distributions function with higher density towards the middle and lower density towards the sides. Viewing this diagram as the particle density form $\theta$, then the observation that the leaves are tangent to $\Psi$ and that they have no boundary, is equivalent to the transport equations given in \ref{['PF Transport Equations']}.

Theorems & Definitions (115)

• Definition 1.1: Pullback and Pushforward
• Definition 1.2: Tangential Vector Fields
• Definition 1.3: Scalar Lift
• Definition 1.4: Conic Sub-Bundle
• Definition 1.5: Causal Indicator
• Definition 1.6: Radial Vector Field
• Definition 1.7: Homogeneity
• Definition 2.1: Kinematic Domain
• Definition 2.2: Vlasov field on $E$
• Definition 2.3: Vlasov Field on $U$