Vector-Length Agnosticism (VLA)

• Vector-Length Agnosticism (VLA) is a metric-free, parameter-independent framework in Vlasov kinetics that avoids vector normalization to include all causal types.
• It employs an 8-dimensional conic subbundle and a global bivector to yield parameter-free dynamics that recover traditional mass-shell and laboratory frame formulations.
• The approach naturally derives physical currents and stress–energy tensors, enabling unified treatments across classical, relativistic, and Finsler kinetic theories.

Vector-length agnosticism (VLA) is a foundational property of contemporary Vlasov kinematic frameworks that eschews any prior choice of velocity (or tangent vector) normalization in the geometric and analytic formulation of collisionless particle kinetics. The VLA paradigm stands in direct contrast to traditional approaches that confine dynamics to a prescribed phase space hypersurface—typically the mass-shell $g(u,u) = -m^2$, enforcing a preferred vector length via metric or energy constraints. The VLA formalism, crystallized in the Vlasov bivector approach, achieves a parameterization- and length-scale–agnostic treatment, permitting simultaneous inclusion of all causal types (timelike, spacelike, and null) within a unified geometric structure (Gunneberg et al., 27 Jan 2025).

1. Fundamental Structures: The 8-Dimensional Conic Subbundle

The VLA approach operates on the 8-dimensional conic subbundle $U \subset TM \setminus \{0\}$ of a spacetime manifold $M$, where $TM$ is the tangent bundle. This subbundle satisfies $\pi(U) = M$ (the projection to $M$), and for any $u \in U$ and scalar $\lambda \neq 0$, $\lambda u \in U$, guaranteeing each fiber is a punctured vector space. The structure is independent of any metric, rendering the approach manifestly metric-free and, critically, vector-length agnostic: no normalization conditions such as $g(u,u)=\text{const.}$ or light-cone impositions ever constrain the kinematical space.

Local adapted coordinates on $U$ are induced from any chart $x^\mu$ on $M$, yielding $(x^\mu, \dot{x}^\mu)$, where $\dot{x}^\mu(u) = u\langle x^\mu\rangle$. The radial (vertical) vector field, $R = \dot{x}^\mu \partial/\partial\dot{x}^\mu$, generates fiberwise homotheties, encoding the scale freedom inherent in the formalism (Gunneberg et al., 27 Jan 2025).

2. The Vlasov Bivector and Parameter-Free Dynamics

Instead of selecting a codimension-1 hypersurface (e.g., mass-shell) as the kinematic domain, VLA deploys a global bivector field $\Psi \in \Gamma \Lambda^2(TU)$: $\Psi = R \wedge W,$ where $W \in \Gamma TU$ is a horizontal, radially-quadratic spray field defined by $[R, W] = W$ and $[\pi_* W] = u$. In local coordinates, $$W = \dot{x}^\mu \partial_{x^\mu} + \phi^\mu(x,\dot{x})\,\partial_{\dot{x}^\mu}$$ with $\phi^\mu$ homogeneous of degree 2 in $\dot{x}$, making $\Psi$ a radially-cubic simple bivector (Gunneberg et al., 27 Jan 2025). The leaves of the foliation tangent to $(R, W)$ correspond to unparameterized solutions of the underlying second-order ODE system (such as geodesics or Lorentz force curves), circumventing any specification of parameterization or vector length.

Three equivalent characterizations of $\Psi$ are provided (Theorem 3.12 in (Gunneberg et al., 27 Jan 2025)):

Condition	Description	Consequence
Radial/Horizontal/Integrable	$R\wedge\Psi=0$, horizontal property, integrable two-plane distribution	Simplicity and foliation by solution surfaces
$\Psi=R\wedge W$	For unique radially quadratic $W$	Bivector encodes all parameterizations
Radially cubic	$\Psi\langle\dot{f},\dot{h}\rangle$ scales as $\lambda^3$ under $\dot{x}\to\lambda\dot{x}$	Homogeneity guarantees scale-invariance

3. Particle Density Form and Parameter-Free Transport

On the $2n$-dimensional space $U$, the particle density is encoded in a $(2n-2)$-form $\theta \in \Gamma \Lambda^{2n-2} U$, obeying the parameter-free transport equations: $$i_R \theta = 0, \qquad i_W \theta = 0, \qquad d\theta = 0.$$ These conditions ensure that the level sets of $\theta$ ("form-submanifolds") are $2$-surfaces tangent to both $R$ and $W$ and possess no boundary in $U$. No parameter or metric constraint is imposed at any stage; hence, all possible kinematic domains are recoverable as slices, but the ambient kinematical structure never commits to a vector length or parametrization (Gunneberg et al., 27 Jan 2025).

4. Recovery of Conventional Kinematic Domains

Traditional, parameter-dependent phase spaces (such as the mass-shell $E_H = \{u \in U \mid g(u,u) = -m^2\}$ or laboratory frame sections) are obtained by selecting a 1-homogeneous "kinematic indicator" $F$ on $U$ with $E = \{F=1\}$ as the desired level set. Associated Liouville fields $W_E$ and induced density forms $\theta_E$ are derived via

$$W_E = (\Sigma_E)_* W, \qquad \theta_E = \Sigma_E^* \theta,$$

where $\Sigma_E$ is the section defined by $F|_E=1$. The induced Vlasov (Liouville) equations on any $E$ are fully contained within $\Psi$ and $\theta$ via the pull-push mechanism. All possible reparameterizations of $W_E$ are generated by the transformation

$$\widetilde{W} = W - \frac{W \langle \widetilde{F}\rangle}{R\langle \widetilde{F} \rangle} R,$$

where $\widetilde{F}$ is the new kinematic indicator [(Gunneberg et al., 27 Jan 2025), Theorem 2.18]. This construction demonstrates that VLA encapsulates all conventional Vlasov-Liouville theories within a single parameter-free geometry.

5. Physical Currents, Stress–Energy, and Conservation Laws

The VLA formalism provides natural, slice-independent definitions for the current and stress–energy tensor:

• The current on $M$ is defined via fiber integration,

$J = \pi_* (\chi \wedge \theta),$

with $dJ = 0$ following directly from the properties of $\theta$. This is independent of the choice of 1-form $\chi$ (with $\int\chi = 1$).

• For any 7D "kinematic domain" $E \subset U$, restricting to $E$ recovers the standard form $J = \pi_{E*}(\theta_E)$ [(Gunneberg et al., 27 Jan 2025), Lemma 4.19].
• For the stress–energy tensor, the 3-form

$$\tau_E(\alpha) = \pi_{E*} (\widehat{\alpha}\, \theta_E),$$

with $\widehat{\alpha}(u) = \alpha(u)$, yields in coordinates the familiar expression: $$T^{\mu\nu}(x) = \int_{E_x} (\dot{x}^\mu \dot{x}^\nu)\, f_E(x, \dot{x})\, \frac{\sqrt{-g}}{\dot{x}_0}\, d^3\dot{x},$$ demonstrating equivalence with standard physical fluxes and ensuring the emergence of the stress–energy tensor from the purely parameter-free VLA data [(Gunneberg et al., 27 Jan 2025), Lemma A.22].

A crucial structural point is that at no point do $\Psi$ or $\theta$ depend on vector-length normalization or parameter-choice; all induced physical objects and conservation laws descend naturally when integrating over suitable slices of the ambient kinematic space.

6. Generalizations, Connections, and Implications

The VLA property enables several theoretical extensions and unifies otherwise distinct frameworks:

• No restriction to mass-shells or light-cones: All causal classes—timelike, spacelike, null—are treated on equal footing, circumventing topological obstructions that arise from attempting to foliate $TM \setminus \{0\}$ by constant-length shells.
• Non-metric connections: The spray $W$ may be constructed from any affine connection, accommodating torsion, non-metricity, and premetric theories without modification of the essential kinematical machinery.
• Ultra-relativistic and limiting regimes: The formalism naturally interpolates between massive and massless (null) dynamics, as in the Burton–Gratus–Tucker ultra-relativistic limit, via continuous deformation of the leaves of $\Psi$.
• Sprays, semi-sprays, and Finsler spacetimes: $W$ satisfies the spray property $[R, W] = W$, and every Liouville field $W_E$ is a semi-spray. Embedding semi-sprays into the unique ambient spray associated with $U$ is equivalent to solving $[R, W] = W$. Finsler structures, via any homogeneous function $F > 0$ of degree $k$ on $U$, allow the observer bundle $\{F=1\}$ to act as a 7D kinematic domain; $\Psi$ then induces the corresponding Finsler Liouville spray. The geometric and analytic dependencies on the metric or Finsler norm are thus restricted solely to the choice of phase-space slicing, never to the construction of the kinematical evolution per se (Gunneberg et al., 27 Jan 2025).

7. Significance and Unification

Vector-length agnosticism provides a unifying geometric mechanism for Vlasov theories on arbitrary manifolds, independently of metric, causal type, or auxiliary parameterizations. All conventional phase-space evolutions—mass shell, laboratory frame, null foliation—descend as level sets $F=\text{const.}$ of a single kinematic structure. Physical fluxes and conservation laws associated with currents and stress–energy tensors emerge from the ambient density and bivector via fiber integration, maintaining technical rigor without recourse to any specific normalization constraint. This approach harmonizes disparate treatments of collisionless kinetics in classical, relativistic, and generalized geometric (Finsler, premetric) contexts using a single, length-scale–independent language (Gunneberg et al., 27 Jan 2025).