A discussion on the symmetry of relativistic Vlasov gas and its accretion in Kerr-Newman black hole

TL;DR

This work develops a kinetic-theory treatment of collisionless Vlasov gas in Kerr-Newman spacetime, showing that spacetime symmetries enforce a distribution function $f(m,E,L_z,L)$ and enabling complete integrability via action-angle variables. Observables such as the particle current and energy-momentum tensor are computed in the Locally Non-Rotating Frame, with horizon regularization and explicit asymptotic expressions for a Jüttner-distributed plasma. Numerical results demonstrate that both mass and energy accretion rates are reduced as rotation $a$ and charge $Q$ grow, while the angular-momentum accretion rate increases with $a$ but decreases with $Q$, leading to a growth toward electrical neutrality and reduced spin. These findings suggest a universal tendency for KN black holes in collisionless plasmas to evolve toward Schwarzschild configurations, with implications for kinetic accretion in extreme spacetimes. All mathematical expressions are presented with explicit dependence on $E$, $L_z$, $L$, and Carter constants, highlighting the role of hidden symmetries in shaping accretion dynamics.

Abstract

We investigate the kinetic properties of collisionless Vlasov gas in Kerr-Newman spacetime, analyzing how spacetime symmetries constrain the distribution functions. The distribution function is shown to depend solely on the constants of motion ($m$, $E$, $L_z$, $L$), reflecting the complete integrability of the system. Within the Locally Non-Rotating Frame, we compute particle number density, energy density, principal pressures, and accretion rates, deriving explicit asymptotic expressions for Jüttner-distributed plasma. Numerical results for the relative (normalized) mass and energy accretion rates reveal an identical parametric dependence: both are suppressed as the black hole's rotation $a$ and charge $Q$ increase. Conversely, the magnitude of the normalized angular momentum accretion rate (which is negative) increases with $a$ but decreases with $Q$. Accretion of weakly charged plasma drives charged black holes toward electrical neutrality while reducing angular momentum, ultimately favoring evolution toward Schwarzschild configurations. These findings provide new insights into kinetic accretion processes in extreme spacetime geometries.

Figures (5)

• Figure 1: Critical parameters for a Kerr-Newman black hole ($M=1, a=0.1, Q=0.1, \kappa=0$): $E_c$ as a function of periastron radius $r$, and $\bar{L}_{c}(r)/5$ as a function of $E_{c}(r)$ on the equatorial plane ($\theta = \pi/2$), with $\sigma = \pi/2$. For visualization purposes, the $E_c$-$r$ relationship is inverted and the $\bar{L}_c$ values are scaled.
• Figure 2: (a) Normalized particle number density $n_{J}/n_{J\infty}$ in the comoving frame and (b) normalized particle number density $n/n_{\infty}$ in the LNRF, both evaluated on the equatorial plane. Dashed lines in (b) display the rescaled quantity $\tilde{n}$ regularized for horizon divergence.
• Figure 3: (a) Energy density $\varepsilon$ and rescaled energy density $\tilde{\varepsilon} \equiv e^{-2\nu} \varepsilon$. (b) Principal pressures $\mathcal{P}_{i}$ and rescaled radial pressure $\tilde{\mathcal{P}}_1 \equiv e^{-2\mu_1} \mathcal{P}_1$, illustrating the anisotropic nature of Vlasov gas.
• Figure 4: Normalized (a) mass and (b) energy accretion rate parameters for characteristic black holes, scaled by their Schwarzschild counterparts.
• Figure 5: Angular momentum accretion rate parameters for characteristic black holes, normalized by the asymptotic particle number density $n_{J\infty}$. The distinct behavior compared to mass/energy rates highlights the different physical mechanisms governing angular momentum transfer.