A Monte Carlo approach to stationary kinetic disks in the Kerr spacetime

We extend a recently proposed Monte Carlo scheme for computing stationary solutions of the general-relativistic Vlasov equation to the Kerr spacetime. As an example, we focus on razor-thin configurations of a gas confined to the equatorial plane and extending to spatial infinity. We consider monoenergetic models as well as solutions corresponding to planar Maxwell-Jüttner distributions at infinity. In both cases, the components of the particle current surface density are recovered within the proposed Monte Carlo framework. Some aspects of razor-thin kinetic disk models, including an analysis of the bulk angular momentum and angular velocity, are briefly covered.

💡 Research Summary

This paper extends a recently introduced Monte‑Carlo scheme for constructing stationary solutions of the general‑relativistic Vlasov (collisionless Boltzmann) equation to the rotating Kerr spacetime. The authors focus on razor‑thin, collisionless gas disks confined to the equatorial plane and extending to spatial infinity. Two families of distribution functions are considered: (i) mono‑energetic models, where every particle shares the same dimensionless energy ε>1, and (ii) planar Maxwell‑Jüttner distributions prescribed at infinity, characterized by a temperature parameter Θ and a chemical potential μ.

The methodology consists of three independent steps. First, the geodesic equations governing individual particle trajectories are solved analytically using the Mino time parametrisation. In the equatorial plane the radial motion reduces to a fourth‑order polynomial R̃(ξ) in the dimensionless radius ξ=r/M. By expressing the solution ξ(s) in terms of the Weierstrass ℘‑function, the authors obtain exact, horizon‑regular expressions for both the radial coordinate and the azimuthal angle φ(s). This formulation naturally distinguishes absorbed trajectories (no turning point outside the horizon) from scattered ones (at least one turning point).

Second, a sampling procedure is devised to generate a representative ensemble of geodesics consistent with the chosen distribution function. For the mono‑energetic case the sampling reduces to drawing the conserved angular momentum λ from the appropriate range; for the Maxwell‑Jüttner case the joint probability density in (E,λ) is proportional to exp