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

TL;DR

This work extends a Monte Carlo framework for stationary general-relativistic Vlasov solutions to Kerr spacetime, focusing on razor-thin, equatorial plane disks extending to infinity. By sampling geodesic parameters and averaging over hypersurfaces adapted to spacetime symmetries, the authors reconstruct the particle current surface density for monoenergetic and Maxwell-Jüttner distributions and analyze the resulting angular momentum and angular velocity of the flow. The method is validated against analytic results for J_mu and demonstrates robust agreement, showing its potential for more complex setups including electromagnetic fields and less symmetric flows. The approach provides a practical tool for modeling collisionless kinetic disks around rotating black holes with applications to dark matter phenomenology and beyond.

Abstract

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.

Figures (8)

• Figure 1: The angular velocity $\Omega$ and the azimuthal angular momentum $\ell_z$ for monoenergetic models. Subfigure \ref{['omega12']} illustrates the angular velocity $\Omega$, while Subfig. \ref{['angularenergy123']} shows the difference of $\Omega_{\mathrm{ZAMO}}-\Omega$. Subfig. \ref{['angularmomentum12']} shows the azimuthal angular momentum for three distinct values of the spin parameter $\alpha$. In both cases, the energy is fixed at $\varepsilon_0 =$1.3. In Subfigure \ref{['angularenergy12']} we plot the azimuthal angular momentum for three different values of the energy, with the black hole spin parameter set to $\alpha = 0.8$.
• Figure 2: Time components of particle current surface density $J_{t}$ for the monoenergetic planar model with $\alpha = 0.8$, $\varepsilon_{0} = 1.3$, and $\xi_{\mathrm{out}} = 20$. Left and right panels show the results for $\epsilon_\sigma = +1$ and $\epsilon_\sigma = -1$, respectively. Exact solutions \ref{['jtabsmono']}--\ref{['jtscatmono']} are plotted with solid and dashed lines. Dots (blue and green) represent Monte Carlo estimators \ref{['jtmontecarlomonoabs']}--\ref{['jtmontecarlomonoscat']}. A total of $10^{6}$ trajectories is simulated: in Subfig. \ref{['jtplus']}, $N_{\mathrm{abs}}$ = 166847, $N_{\mathrm{scat}}$ = 833153; in Subfig. \ref{['jtminus']}, $N_{\mathrm{abs}}$ = 424080, $N_{\mathrm{scat}}$ = 575920.
• Figure 3: Angular components of particle current surface density $J_{\varphi}$ for the monoenergetic planar model with $\alpha = 0.8$, $\varepsilon_{0} = 1.3$, and $\xi_{\mathrm{out}} = 20$. Left and right panels show the results for $\epsilon_\sigma = +1$ and $\epsilon_\sigma = -1$, respectively. Exact solutions \ref{['jphiabsmono']}--\ref{['jphiscatmono']} are plotted with solid and dashed lines. Dots (blue and green) represent Monte Carlo estimators \ref{['jphimontecarlomonoabs']}--\ref{['jphimontecarlomonoscat']}. A total of $10^{6}$ trajectories is simulated: in Subfig. \ref{['jphiplus']}, $N_{\text{abs}}$= 166847, $N_{\mathrm{scat}}$ = 833153; in Subfig. \ref{['jphiminus']}, $N_{\mathrm{abs}}$ = 424080, $N_{\mathrm{scat}}$ = 575920.
• Figure 4: Total components (the sums of contributions with $\epsilon_\sigma = \pm1$) of the particle current surface density $J_t$ (left) and $J_\varphi$ (right). Both panels correspond to the planar monoenergetic model with $\alpha = 0.8$, $\varepsilon_{0} = 1.3$, and $\xi_{\mathrm{out}} = 20$. Vertical lines mark locations of circular photon orbits $\xi_{\mathrm{ph}}$ for $\epsilon_\sigma = \pm 1$.
• Figure 5: Time components of the particle current surface density $J_{t}$ for the Maxwell-Jüttner planar model with $\varepsilon_{\text{cutoff}}$ = 10, $\xi_{\mathrm{out}}$ = 20, $\beta=1/10$, and $\alpha$ = 0.8. Left and right panels show the results for $\epsilon_\sigma = +1$ and $\epsilon_\sigma = -1$, respectively. Exact solutions \ref{['jtabsjuttner']}--\ref{['jtscatjuttner']} are plotted with solid and dashed lines. Dots (blue and green) represent Monte Carlo estimators \ref{['jtmontecarlomonoabsju']}--\ref{['jtmontecarlomonoscatju']}. In Subfigure \ref{['jtjuttnerplus']}, $N_{\mathrm{abs}}$ = 32298, $N_{\mathrm{scat}}$ = 272564; in Subfig. \ref{['jtjuttnerminus']}, $N_{\mathrm{abs}}$ = 106663, $N_{\mathrm{scat}}$ = 203118.