Chaos of charged particles near a renormalized group improved Kerr black hole in an external magnetic field

TL;DR

This work analyzes chaotic dynamics of charged particles around a renormalization group improved (RGI) Kerr black hole in an external magnetic field. By formulating a Hamiltonian with a running Newton constant $G(r)=G_0\left(1-\frac{\xi}{r^2}\right)$ and a Wald-type magnetic potential, the authors demonstrate integrability for neutral motion but nonintegrability for charged motion when the magnetic field is present; they then develop explicit symplectic integrators via a time-transformed three-part Hamiltonian split to enable accurate long-term integration. Chaos indicators (Poincaré sections, Lyapunov exponents, FLIs, and 0-1 tests) reveal a transition from regular to chaotic dynamics as parameters vary, with larger $\xi$ consistently reducing the extent of chaos due to a weakening of the central gravitational attraction. The results show that chaos grows with particle energy $E$ and magnetic field strength $\beta$, but decreases with angular momentum $L$ and black hole spin $a$, aligning with analogous findings in RGI Schwarzschild spacetimes and highlighting how quantum gravity corrections can modulate near-horizon dynamics. Overall, the study provides a computational framework and physical insight into how quantum gravitational corrections influence chaotic motion around magnetized, rotating black holes, offering a route to constrain quantum parameters from dynamical signatures.

Abstract

In a quantum theory of gravity, a renormalization group improved Kerr metric is obtained from the Kerr metric, where the Newton gravitational constant is modified as a function of the radial distance. The motion of neutral test particles in this metric is integrable. However, the dynamics of charged test particles is nonintegrable when an external asymptotically homogeneous magnetic field exists in the vicinity of the black hole. The transition from regular dynamics to chaotic dynamics is numerically traced as one or two dynamical parameters vary. From a statistical point of view, the strength of chaos is typically enhanced as both the particle energy and the magnetic field increase, but it is weakened with increasing the particle angular momentum and the black hole spin. In particular, an increase of the quantum corrected parameter weakens the extent of chaos. This is because the running Newton gravity constant effectively weakens the central gravitational attraction and results in decreasing sensitivity to initial conditions.

Figures (10)

• Figure 1: (a) Hamiltonian errors $\Delta H= \mathcal{H}$ for the two explicit symplectic algorithms acting on Orbit 1 with the initial separation $r=30$. (b) Same as (a) but Orbit 1 is replaced by Orbit 2 with the initial separation $r=80$. Note that the two orbits correspond to the other initial conditions $\theta=\pi/2$ and $p_{r}=0$ and the parameters $E=0.995$, $L=4.6$, $\beta=1\times 10^{-3}$, $a = 0.5$ and $\xi = 0.2$. (c) The approximately equal relation between the proper time $\tau$ and the new time $w$, described by the method S4 solving Orbit 1 or 2.
• Figure 2: (a) Poincaré sections of the two orbits on the plane $\theta=\pi/2$ with $p_{\theta}>0$. (b) The maximal Lyapunov exponents for the two orbits. (c) Fast Lyapunov Indicators (FLIs) for the two orbits. The three methods consistently show regular dynamics of Orbit 1 and chaotic dynamics of Orbit 2.
• Figure 3: (a) The visual of $(p_c, q_c)$ and the correlation coefficient $K_c$ for Orbit 1. (b) Same as (a) but Orbit 1 is replaced by Orbit 2. The regularity of Orbit 1 and the chaoticity of Orbit 2 are shown via the visuals $(p_c, q_c)$ and the correlation coefficients $K_c$.
• Figure 4: (a)-(c): Poincaré sections describing the dynamical behaviors of orbits for three values of the parameter $\xi$. The other parameters are $E=0.995$, $L=4.5$, $\beta=8\times 10^{-4}$ and $a=0.67$ and the tested orbit has its initial separation $r=30$. (d)-(f): The visuals of $(p_c, q_c)$ and the values of $K_{c}$, which respectively correspond to panels (a-c). Chaos occurs for $\xi=0.18$ in panels (a) and (d). Regular dynamics exist for $\xi=5.56$ and $\xi=8.56$ in panels (b), (c), (e) and (f).
• Figure 5: (a) and (b): Same as Fig. 4 but both the maximal Lyapunov exponents $\lambda$ and the FLIs are used. (c): Dependence of the FLI on the parameter $\xi$ for the tested orbit in Fig. 4. For $\xi=0.65$, FLI=5.87 is a threshold between order and chaos.