Covariant Vlasov Approach

• The covariant Vlasov approach is a collisionless kinetic formulation where the evolution of the one-body distribution is expressed directly in spacetime and momentum, ensuring manifest covariance.
• It encompasses multiple formulations—tensorial with differential forms, Wigner-function based semiclassical transport, and parameter-free bivector kinematics—each offering unique insights into collective modes.
• Applications include modeling dense nuclear matter and neutron stars, influencing studies of dispersion relations, spinodal instabilities, crust–core transitions, and neutrino transport.

The covariant Vlasov approach is a family of collisionless kinetic formulations in which the Vlasov or Liouville dynamics is written in a manifestly covariant form and coupled self-consistently to background or dynamical fields. In the literature summarized here, the approach appears in at least three closely related forms: a tensorial Liouville–Vlasov formulation based on differential forms and Lie derivatives (Drivotin, 2016), a gauge-invariant Wigner-function derivation that yields relativistic semiclassical transport equations in relativistic mean-field models (Avancini et al., 2018), and a parameter-free formulation built from a Vlasov bivector on an 8-dimensional conic subbundle of the tangent bundle (Gunneberg et al., 27 Jan 2025). In nuclear and neutron-star applications, it is used to derive dispersion relations, collective modes, dynamical spinodals, and crust–core transition properties in neutron-proton-electron or neutron-proton-electron-muon matter, including the presence of strong magnetic fields (Rabhi et al., 2024, Rabhi et al., 30 May 2026, Rabhi et al., 23 Sep 2025).

1. Conceptual scope and covariant character

A defining feature of the covariant Vlasov approach is that the one-body distribution is evolved by a transport equation written directly in spacetime and momentum variables, rather than by a non-covariant phase-space prescription. In its relativistic kinetic form, the collisionless equation can be written as

$$p^\mu \partial_\mu^x f_j +\left[g_s M_j^*(x)\partial_x^\mu \phi(x) +g_v p_\nu \Omega^{\mu\nu}(x) +\frac12 g_\rho \tau_j p_\nu B^{\mu\nu}(x) +Q_j p_\nu F^{\mu\nu}(x)\right]\partial_\mu^p f_j =0,$$

for species $j=n,p,e,\mu$, with $\Omega^{\mu\nu}$, $B^{\mu\nu}$, and $F^{\mu\nu}$ the $\omega$, $\rho$, and electromagnetic field tensors, respectively (Rabhi et al., 30 May 2026).

A related semiclassical form, obtained from a gauge-invariant Wigner construction and expansion to $O(\hbar)$, is

$$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$

with $\mathbf v=\mathbf p/E_p$ and $j=n,p,e,\mu$0 (Avancini et al., 2018). In the magnetized relativistic asymmetric nuclear-matter literature, the same laboratory-frame equation is written for the species-dependent effective fields $j=n,p,e,\mu$1 and $j=n,p,e,\mu$2 (Rabhi et al., 23 Sep 2025).

The covariant character is expressed differently across formulations. In the tensorial treatment of Liouville and Vlasov theory, conservation of the phase-density form $j=n,p,e,\mu$3 along the one-particle flow $j=n,p,e,\mu$4 is written as

$j=n,p,e,\mu$5

with $j=n,p,e,\mu$6 the Lie derivative, allowing arbitrary coordinates and curved manifolds (Drivotin, 2016). In the parameter-free theory, covariance is strengthened by avoiding a choice of mass shell or time slicing: one works on an 8-dimensional conic subbundle $j=n,p,e,\mu$7, with a simple, horizontal, integrable bivector $j=n,p,e,\mu$8 and a particle-density 6-form $j=n,p,e,\mu$9 satisfying $\Omega^{\mu\nu}$0, $\Omega^{\mu\nu}$1, and $\Omega^{\mu\nu}$2 (Gunneberg et al., 27 Jan 2025).

A common misconception is that “covariant Vlasov” denotes a single formalism. The cited literature instead shows a broader category: tensorial differential-form formulations (Drivotin, 2016), Wigner-based semiclassical transport in relativistic mean-field theory (Avancini et al., 2018), and parameterization-free bivector kinematics (Gunneberg et al., 27 Jan 2025).

2. Microscopic derivation and phase-space structures

In relativistic mean-field applications, the kinetic equation is derived from Dirac dynamics. For neutron-proton-electron matter, the starting point is a Walecka-type Lagrangian

$\Omega^{\mu\nu}$3

with fermion terms

$\Omega^{\mu\nu}$4

species-dependent covariant derivatives, and effective masses $\Omega^{\mu\nu}$5, $\Omega^{\mu\nu}$6 (Avancini et al., 2018). An analogous RMF Lagrangian for $\Omega^{\mu\nu}$7 matter is written as

$\Omega^{\mu\nu}$8

with $\Omega^{\mu\nu}$9 and charge-neutral, $B^{\mu\nu}$0-equilibrated constraints $B^{\mu\nu}$1, $B^{\mu\nu}$2 (Rabhi et al., 30 May 2026).

The gauge-invariant Wigner operator used in the magnetized $B^{\mu\nu}$3 derivation is

$B^{\mu\nu}$4

with

$B^{\mu\nu}$5

which leads to a Wigner equation involving $B^{\mu\nu}$6 and $B^{\mu\nu}$7, followed by positive-energy projection, Clifford-algebra decomposition, and integration over $B^{\mu\nu}$8 to obtain the one-body phase-space density $B^{\mu\nu}$9 (Avancini et al., 2018).

The geometric literature adopts a different primitive object. Instead of a scalar distribution function on a 7-dimensional time-phase space, the parameter-free approach introduces a particle-density 6-form $F^{\mu\nu}$0 on $F^{\mu\nu}$1, together with the Vlasov bivector $F^{\mu\nu}$2 satisfying radial simplicity, horizontality, and Frobenius integrability (Gunneberg et al., 27 Jan 2025). The differential-form treatment of Liouville theory similarly replaces “distribution function times phase volume” by a single differential form $F^{\mu\nu}$3 of degree $F^{\mu\nu}$4, with $F^{\mu\nu}$5, thereby accommodating continuous, degenerate, and point-particle ensembles in a common language (Drivotin, 2016).

This suggests that “covariant Vlasov approach” can refer either to a covariant transport equation for a scalar Wigner or distribution function, or to a fully geometric reformulation in terms of forms and bivectors.

3. Relativistic mean-field realization in dense matter

In nuclear-matter applications, the covariant Vlasov approach is embedded in Walecka-type or RMF models with scalar, vector-isoscalar, vector-isovector, and electromagnetic sectors. The mesonic and photon pieces are written as

$F^{\mu\nu}$6

$F^{\mu\nu}$7

$F^{\mu\nu}$8

with the $F^{\mu\nu}$9–$\omega$0 mixing term $\omega$1 (Rabhi et al., 30 May 2026). In the corresponding magnetized $\omega$2 formulation, the same field content appears with scalar and vector self-couplings $\omega$3, and $\omega$4 is stated to soften the high-density symmetry energy (Avancini et al., 2018).

The static background is taken to be uniform and at zero temperature. For longitudinal collective-mode studies without magnetic field, the unperturbed distributions are step-function Fermi seas,

$\omega$5

with constant mean fields $\omega$6, $\omega$7, $\omega$8, and $\omega$9 (Rabhi et al., 30 May 2026). In magnetized matter, one takes a uniform magnetic field $\rho$0 in Landau gauge, with $\rho$1 or equivalently $\rho$2 depending on coordinate convention, while the baryonic mean fields remain purely temporal (Avancini et al., 2018, Rabhi et al., 23 Sep 2025).

The equilibrium distribution reflects the electric charge of the species. Neutrons retain the usual Fermi sphere, whereas protons and electrons undergo Landau-level quantization. In the 2025 magnetized asymmetric nuclear-matter treatment, the proton and electron equilibrium distributions are written as sums over Landau levels with Laguerre polynomials $\rho$3, $\rho$4, and

$\rho$5

with $\rho$6 (Rabhi et al., 23 Sep 2025).

These RMF realizations have been studied with several parameter sets. The 2018 magnetar-stability work used NL3 and FSU, described respectively as a stiff symmetry energy, $\rho$7, and a soft symmetry energy, $\rho$8 (Avancini et al., 2018). The 2024 analysis of cold asymmetric nuclear matter considered eleven RMF models in two families: Set I, with a stiffer EOS at high density, including NL3, NL3 $\rho$9, TM1-2, TM1-2 $O(\hbar)$0, and BigApple; and Set II, with a softer EOS at high density, including FSU, FSU2, FSU2R, FSU2H, TM1, and TM1e (Rabhi et al., 2024). The 2026 $O(\hbar)$1 study used NL3, NL3$O(\hbar)$2, and FSU2H (Rabhi et al., 30 May 2026).

4. Linear response, longitudinal and transverse modes

The standard workflow in the RMF covariant Vlasov literature is linearization around the uniform background. One writes

$O(\hbar)$3

and similarly for the $O(\hbar)$4 and electromagnetic fields, followed by Fourier decomposition

$O(\hbar)$5

(Avancini et al., 2018). For longitudinal modes in uniform $O(\hbar)$6 matter with $O(\hbar)$7, the linearized solution has the form

$O(\hbar)$8

with $O(\hbar)$9 and $$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$0 (Rabhi et al., 30 May 2026).

In the 2024 cold asymmetric nuclear-matter analysis, eliminating the meson and photon perturbations yields a $$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$1 algebraic system for $$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$2,

$$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$3

where

$$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$4

and the dispersion relation is the vanishing of the determinant (Rabhi et al., 2024). In the $$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$5 extension, one obtains a $$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$6 system $$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$7 for $$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$8, and the determinant factorizes as

$$\partial_t f_\pm + \mathbf v\!\cdot\!\nabla_x f_\pm +\left(\mathbf{\mathcal E}\pm \mathbf v\times \mathbf{\mathcal B}\right)\!\cdot\!\nabla_p f_\pm \mp \frac{M^*}{E_p}\nabla_x M^*\!\cdot\!\nabla_p f_\pm=0,$$9

with the leptonic factors identified as pure electron and muon plasmons and the nuclear sub-determinant as the isoscalar/isovector zero-sound sector (Rabhi et al., 30 May 2026).

In the magnetized $\mathbf v=\mathbf p/E_p$0 formulation of 2018, the Fourier-transformed linearized Vlasov equation contains an azimuthal derivative term,

$\mathbf v=\mathbf p/E_p$1

and is solved in cylindrical coordinates $\mathbf v=\mathbf p/E_p$2 by the Oberman–Ron transform

$\mathbf v=\mathbf p/E_p$3

leading to a closed-form expression for $\mathbf v=\mathbf p/E_p$4 (Avancini et al., 2018). Longitudinal modes correspond to $\mathbf v=\mathbf p/E_p$5, whereas transverse modes have $\mathbf v=\mathbf p/E_p$6 and retain the full Bessel-function structure in the generalized Lindhard integrals (Avancini et al., 2018). The 2025 magnetized asymmetric nuclear-matter work also uses the Oberman–Ron expansion; for $\mathbf v=\mathbf p/E_p$7, only $\mathbf v=\mathbf p/E_p$8 contributes because $\mathbf v=\mathbf p/E_p$9 (Rabhi et al., 23 Sep 2025).

The propagator structure is central to the mode classification. In the magnetized collective-mode study, the isoscalar channel is carried by the $j=n,p,e,\mu$00 and $j=n,p,e,\mu$01 propagators,

$j=n,p,e,\mu$02

the isovector channel by

$j=n,p,e,\mu$03

and the Coulomb coupling by

$j=n,p,e,\mu$04

which couples protons to electrons (Rabhi et al., 23 Sep 2025).

5. Instabilities, collective branches, and magnetic-field effects

Within this framework, instability analysis is performed through the dispersion matrix. In the 2018 magnetized $j=n,p,e,\mu$05 study, the dynamical spinodal is defined by $j=n,p,e,\mu$06 solutions of the determinant condition at fixed $j=n,p,e,\mu$07; one solves the determinant at $j=n,p,e,\mu$08 and plots the boundary between stable and unstable regions in the $j=n,p,e,\mu$09–$j=n,p,e,\mu$10 plane. Equivalently, one may look for purely imaginary $j=n,p,e,\mu$11, where positive $j=n,p,e,\mu$12 are growth rates; analytically, the onset of instability occurs when the smallest eigenvalue of the $j=n,p,e,\mu$13 stiffness matrix first changes sign (Avancini et al., 2018).

At low density, the unmagnetized covariant Vlasov spectrum contains a spinodal unstable isoscalar mode. In the 2024 study of asymmetric cold nuclear matter, a low-density spinodal unstable isoscalar mode appears alone below $j=n,p,e,\mu$14 in all models and defines the dynamical spinodal (Rabhi et al., 2024). Above that regime, isovector-like modes exist for $j=n,p,e,\mu$15, while high-density isoscalar-like modes occur only in models with a stiff symmetric EOS, such as NL3, NL3 $j=n,p,e,\mu$16, BigApple, and TM1-2 (Rabhi et al., 2024).

The presence of leptons modifies these conclusions. In the 2026 $j=n,p,e,\mu$17 analysis, the spectrum around $j=n,p,e,\mu$18 splits into two leptonic plasmons, electron and muon, and a proton-dominated zero-sound mode, with the proton mode in pure $j=n,p,e,\mu$19Coulomb becoming a plasmon at low $j=n,p,e,\mu$20; in $j=n,p,e,\mu$21 it couples chiefly to the electron plasmon, and in $j=n,p,e,\mu$22 an extra muon plasmon appears (Rabhi et al., 30 May 2026). Lepton screening modifies the effective proton-proton interaction $j=n,p,e,\mu$23 by reducing the isovector mode energy and can even suppress propagation if $j=n,p,e,\mu$24 (Rabhi et al., 30 May 2026).

Strong magnetic fields introduce qualitatively new effects because proton and electron transverse motion is quantized. In the 2018 magnetized neutron-star matter study, for longitudinal modes with $j=n,p,e,\mu$25, the spinodal region fragments into “bands” due to Landau quantization, and the instability region can extend to significantly higher densities than the $j=n,p,e,\mu$26 case; the larger the slope $j=n,p,e,\mu$27, the stronger the $j=n,p,e,\mu$28-induced shift (Avancini et al., 2018). The most unstable mode at each density, defined by the largest $j=n,p,e,\mu$29, has half-wavelength $j=n,p,e,\mu$30, interpreted as a typical cluster size of order $j=n,p,e,\mu$31–$j=n,p,e,\mu$32 depending on $j=n,p,e,\mu$33 and $j=n,p,e,\mu$34 (Avancini et al., 2018).

The 2025 magnetized asymmetric nuclear-matter work resolves the mode content in greater detail. A strong magnetic field gives rise to low-lying isovector modes that propagate in nuclear matter and are not present in non-magnetized matter. Landau quantization modifies proton-like collective modes, leading to new branches associated with distinct Landau levels, and these branches can propagate even at high densities and exhibit isoscalar or isovector character (Rabhi et al., 23 Sep 2025). By contrast, neutron-like modes are essentially not affected by the presence of a strong magnetic field, and neutron-like isoscalar modes at high density are stated to be insensitive to the presence of a magnetic field (Rabhi et al., 23 Sep 2025).

An important directional effect also appears. In the 2018 study, transverse modes are strongly suppressed for $j=n,p,e,\mu$35: the transverse spinodal shrinks below the $j=n,p,e,\mu$36 case. For $j=n,p,e,\mu$37, the transverse spinodal nearly coincides with the $j=n,p,e,\mu$38 result, so perturbations perpendicular to $j=n,p,e,\mu$39 are almost unchanged by more moderate fields (Avancini et al., 2018).

6. Crust–core transition, currents, and broader generalizations

The covariant Vlasov approach has direct neutron-star applications. Once the dynamical spinodal is known, one overlays the $j=n,p,e,\mu$40-equilibrium equation of state curve $j=n,p,e,\mu$41, satisfying $j=n,p,e,\mu$42, in the $j=n,p,e,\mu$43–$j=n,p,e,\mu$44 plane; its intersection with the spinodal gives the density $j=n,p,e,\mu$45 at which homogeneous $j=n,p,e,\mu$46 matter becomes unstable and marks the inner-crust boundary (Avancini et al., 2018). The 2018 analysis therefore links dynamical instabilities, growth rates, and cluster scales to the location and thickness of the crust–core transition region in magnetars (Avancini et al., 2018). The 2024 study similarly states that the low-density unstable isoscalar mode determines the crust–core transition in neutron stars and impacts crustal composition and mechanical properties (Rabhi et al., 2024).

At higher density, collective modes are connected in the cited literature to transport and emission. The 2026 $j=n,p,e,\mu$47 work states that neutrino opacities and cooling of neutron stars are sensitive to these collective modes and to the high-density behavior of the symmetry energy (Rabhi et al., 30 May 2026). The 2024 work states that neutrino transport and opacity in supernovae and proto-neutron stars depend sensitively on the existence and strength of collective modes, and that high-density isoscalar zero sound can influence bulk viscosities, thermal conductivity, and cooling times in the inner core of cold neutron stars, but only if the EOS is sufficiently stiff (Rabhi et al., 2024). The 2025 magnetized study adds that, in magnetar interiors with $j=n,p,e,\mu$48, the multitude of low-lying proton collective branches could modify transport coefficients, neutrino emissivities via plasmon decay, and the damping of stellar oscillations (Rabhi et al., 23 Sep 2025).

Outside RMF neutron-star matter, the covariant Vlasov approach has been generalized in explicitly geometric directions. In the parameter-free bivector formalism, the conserved spacetime 3-current is defined by

$j=n,p,e,\mu$49

is independent of the choice of support 1-form $j=n,p,e,\mu$50, and satisfies $j=n,p,e,\mu$51 (Gunneberg et al., 27 Jan 2025). On a chosen kinematic domain $j=n,p,e,\mu$52, the corresponding stress-energy tensor is recovered from

$j=n,p,e,\mu$53

which is identified as the standard Einstein–Vlasov stress-energy tensor (Gunneberg et al., 27 Jan 2025).

The earlier tensorial differential-form formulation states the same general ambition in another language: it is valid in both non-relativistic and relativistic cases, allows arbitrary coordinate systems, is convenient for degenerate distributions such as the Kapchinsky–Vladimirsky distribution, and can be applied to mass distributions in curved spacetime (Drivotin, 2016). A plausible implication is that the term “covariant Vlasov approach” now denotes not only a neutron-star linear-response technique, but also a broader geometric program for collisionless kinetic theory across flat spacetime, curved spacetime, and non-standard kinematic domains.