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Kinetic dynamics of neutral spin particles in a spacetime with torsion

Simone Calogero

TL;DR

The paper develops a relativistic kinetic theory for neutral spin particles in a spacetime with torsion (Einstein-Cartan) by introducing a kinetic density $f(x,u,s)$ whose stress-energy tensor $T_{ab}$ and spin current $S_{a b}^c$ are obtained as moments. It derives a Vlasov-type transport equation for $f$ that is compatible with the Einstein-Cartan Bianchi identity and couples to torsion via the spin current through $C_{ab}{}^c=16\pi S_{ab}{}^c$, yielding non-geodesic particle motion in a torsionful background. Because a single spin species does not conserve particle number, the authors formulate a two-species extension with densities $f$ and $\bar f$ and derive a coupled Vlasov system ensuring total particle-number conservation. The framework provides a foundation for modeling spinor dynamics in the early universe (e.g., neutrino ensembles) within EC gravity and sets the stage for further generalizations to charged spin fields.

Abstract

A kinetic model for the dynamics of collisionless spin neutral particles in a spacetime with torsion is proposed. The fundamental matter field is the kinetic density $f(x,u,s)$ of particles with four-velocity $u$ and four-spin $s$. The stress-energy tensor and the spin current of the particles distribution are defined as suitable integral moments of $f$ in the $(u,s)$ variables. By requiring compatibility with the contracted Bianchi identity in Einstein-Cartan theory, we derive a transport equation on the kinetic density $f$ that generalizes the well-known Vlasov equation for spinless particles. The total number of particles in the new model is not conserved. To restore this important property we assume the existence in spacetime of a second species of particles with the same mass and spin magnitude. The Vlasov equation on the kinetic density $\overline{f}$ of the new particles is derived by requiring that the sum of total numbers of particles of the two species should be conserved.

Kinetic dynamics of neutral spin particles in a spacetime with torsion

TL;DR

The paper develops a relativistic kinetic theory for neutral spin particles in a spacetime with torsion (Einstein-Cartan) by introducing a kinetic density whose stress-energy tensor and spin current are obtained as moments. It derives a Vlasov-type transport equation for that is compatible with the Einstein-Cartan Bianchi identity and couples to torsion via the spin current through , yielding non-geodesic particle motion in a torsionful background. Because a single spin species does not conserve particle number, the authors formulate a two-species extension with densities and and derive a coupled Vlasov system ensuring total particle-number conservation. The framework provides a foundation for modeling spinor dynamics in the early universe (e.g., neutrino ensembles) within EC gravity and sets the stage for further generalizations to charged spin fields.

Abstract

A kinetic model for the dynamics of collisionless spin neutral particles in a spacetime with torsion is proposed. The fundamental matter field is the kinetic density of particles with four-velocity and four-spin . The stress-energy tensor and the spin current of the particles distribution are defined as suitable integral moments of in the variables. By requiring compatibility with the contracted Bianchi identity in Einstein-Cartan theory, we derive a transport equation on the kinetic density that generalizes the well-known Vlasov equation for spinless particles. The total number of particles in the new model is not conserved. To restore this important property we assume the existence in spacetime of a second species of particles with the same mass and spin magnitude. The Vlasov equation on the kinetic density of the new particles is derived by requiring that the sum of total numbers of particles of the two species should be conserved.

Paper Structure

This paper contains 13 sections, 2 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Kinetic theory for spinless particles
  3. Introduction to Einstein-Cartan's theory
  4. Manifolds with torsion
  5. Einstein-Cartan equations
  6. General relativistic form of the Einstein-Cartan equations
  7. Kinetic theory of spin particles
  8. Kinetic density of spin particles
  9. Spacetime matter fields
  10. The Vlasov equation for spin particles
  11. The Vlasov system for two species of particles
  12. Conclusions
  13. Appendix

Key Result

Lemma 1

For all $A^\mu,B^\mu$ satisfying firstcond there holds where $s^0=\sigma {\bm \omega}\cdot{\bm u}$, $u^0=\sqrt{1+|{\bm u}|^2}$ and $\slashed{\partial}_{\bm \omega}=(\slashed{\partial}_{\omega^1},\slashed{\partial}_{\omega^2},\slashed{\partial}_{\omega^2})$, $\slashed{\partial}_{\omega^i}=(\delta^j_i-\omega_i\omega^j)\partial_{\omega^j}$, denotes the gra

Theorems & Definitions (10)

  • Remark
  • Remark
  • Remark
  • Remark
  • Lemma 1
  • proof
  • Remark
  • Lemma 2
  • proof
  • Remark