Chaotic dynamics of charged particles near weakly magnetized black holes in Einstein-ModMax Theory

Abstract: This paper presents a systematic study of the chaotic dynamics of charged test particles around purely magnetically charged black holes immersed in a uniform external magnetic field within the framework of Einstein-ModMax theory. By constructing an explicit symplectic integrator, we obtain high-precision numerical solutions of the equations of motion. Combining the observational constraints from the Event Horizon Telescope (EHT) shadow images, we further restrict the parameter ranges of the model. We apply Shannon entropy and MIPP (mutual information for particle pairs) as effective indicators to identify the chaotic behavior of charged test particles in the spacetime of this black hole. Numerical results indicate that these indicators can clearly distinguish between regular and chaotic motion of orbits in strong gravitational field systems. Further analysis reveals that, compared to the key conserved quantities that determine the global dynamical behavior of the system -- energy $E$ and angular momentum $L$, the sensitivity of the system parameters $e^{-ν}$ and $Q_{m}$ to transitions in the orbital dynamical states is significantly reduced. This study provides a new perspective for a deeper understanding of the characterization and evolution mechanisms of chaotic dynamics in strong gravitational fields.

• The paper shows that introducing an external magnetic field renders the phase space non-integrable, leading to transitions between regular and chaotic orbits.
• The study implements and compares second, fourth, and PRK6(4) symplectic integrators, revealing PRK6(4) as significantly more accurate for long-term energy conservation.
• Observational EHT shadow constraints tie theoretical parameters to physical reality, enhancing the astrophysical relevance of the chaos diagnostics.

Chaotic Dynamics of Charged Particles Near Weakly Magnetized Black Holes in Einstein-ModMax Theory

Problem Definition and Theoretical Framework

This paper delivers a thorough investigation of the nonlinear dynamics and chaos in the motion of charged particles around black holes with purely magnetic charge, embedded in a uniform external magnetic field, utilizing the Einstein-ModMax framework. The spacetime metric is given as a spherically symmetric solution with a modification parameter $e^{-\nu}$ and magnetic charge $Q_m$, which together provide screening for electromagnetic effects and accommodate non-Maxwellian dynamics. The electromagnetic environment is constructed using a generalized Wald solution, and the dynamics incorporate non-integrability introduced by the external field.

Key conserved quantities—energy $E$, angular momentum $L$, and Carter's constant $C_k$—allow the classical system to be integrable in the absence of magnetization. The introduction of an external magnetic field, however, yields a Hamiltonian system whose phase space is no longer integrable and exhibits rich, complex behavior including transitions between regular and chaotic regimes.

Numerical Integrators and Precision Analysis

To resolve the equations of motion for the non-integrable Hamiltonian system, an explicit symplectic integrator is constructed, decomposing the total Hamiltonian into five analytic sub-Hamiltonians. Both second-order ($S_2$), fourth-order ($S_4$), and optimized PRK ($PRK_{6}4$) explicit symplectic schemes are examined for long-term energy conservation and accuracy. Numerical simulations reveal that PRK$_{6}4$ achieves superior precision, reducing Hamiltonian errors by orders of magnitude compared to lower-order methods, thereby ensuring reliable chaos diagnostics even for extended integrations.

Figure 1: Poincar$\acute{e}$ sections and Hamiltonian errors for multiple symplectic integrators, confirming the accuracy and stability of PRK$Q_m$0 over $Q_m$1 timesteps.

Observational Parameter Constraints

The study integrates astrophysical constraints derived from Event Horizon Telescope (EHT) shadow observations of Sgr A*, which limit the permissible values of $Q_m$2 and $Q_m$3 based on the observed shadow radius, providing direct correspondence between theoretical models and empirical data. Only parameter regions consistent with $Q_m$4 within $Q_m$5 to $Q_m$6 are retained for dynamical analysis, which enhances the physical relevance of the chaos exploration.

Figure 2: EHT shadow radius constraints on $Q_m$7, delineating physically allowed and excluded regions for Einstein-ModMax black holes.

Chaos Diagnostics: Methodological Innovations

Three quantitative chaos indicators are adopted:

• Poincar$Q_m$8 sections: Identify ordered/chaotic orbits via intersection pattern.
• Shannon entropy: Quantifies orbit unpredictability; fluctuates strongly for chaotic states.
• Mutual Information for Particle Pairs (MIPP): Measures statistical correlation between nearby trajectories; values near 1 indicate regularity, values near 0 indicate chaos.

These indicators together delineate transitions between dynamical states more sensitively than classical Lyapunov metrics, with MIPP providing a robust measure for parameter scans due to its computational efficiency and clear discrimination.

Influence of Physical and Geometric Parameters

Systematic scanning in the allowed $Q_m$9 space addresses the impact of $E$0, $E$1, $E$2, and $E$3 on chaos:

• Energy ($E$4): Directly expands chaos-dominated regions with increasing $E$5; regular regions contract, showing high sensitivity.
• Angular momentum ($E$6): More complex; increases in $E$7 may suppress chaos, shifting regular regions and generating layered boundaries.
• $E$8 and $E$9: Affect phase space structure less profoundly; regular zones persist near event horizons. Their modulation is comparatively moderate.

These trends are corroborated by scanning diagrams and multi-dimensional phase space plots.

Figure 3: Chaos indicators for varying $L$0, highlighting the threshold between chaotic and regular regime transitions.

Figure 4: Chaos indicators for $L$1 scans, demonstrating complex and non-monotonic influence on orbital phase transitions.

Figure 5: MIPP parameter space scans with increasing energy $L$2, showing expanding chaotic regions and shrinking regular domains.

Figure 6: MIPP parameter space scans for angular momentum $L$3, with shifting boundaries and complex transitions.

Figure 7: MIPP scans for varying $L$4, displaying persistent regular regions near the event horizon and expansion with increased screening.

Figure 8: MIPP scans for $L$5 variation, again verifying moderate impact relative to conserved dynamical quantities.

Physical Interpretation and Theoretical Implications

Analysis of the Hamiltonian reveals dominant contributions from gravitational and inertial terms associated with $L$6 and $L$7, while $L$8 and $L$9 mainly provide moderate metric corrections. At high $C_k$0, stronger gravitational effects enhance global phase space accessibility, increasing chaos. Conversely, increased $C_k$1 augments centrifugal barriers, suppressing chaos.

From an astrophysical perspective, these findings clarify the hierarchy in sensitivity of orbital chaotic transitions to underlying physical and geometric parameters. The prominence of energy and angular momentum in modifying phase space structures posits practical implications for the dynamics of high-energy plasma and charged particles near magnetized compact objects, potentially impacting observable features such as jet launch mechanisms, accretion disk structure, and shadow morphology.

Future Outlook

The methodology of combining high-order symplectic integration and advanced information-theoretic chaos indicators (Shannon entropy, MIPP) establishes a robust framework for future investigations of test particle dynamics in modified gravitational theories. Extending this approach to rotating black holes, multi-charge configurations, and more general spacetime backgrounds can further elucidate the universality and diversity of chaos phenomena in strong gravity. Astrophysical observations (such as EHT/vLTI mass-to-distance ratios, gravitational wave signals) will continue to constrain theoretical parameter spaces and refine the dynamical models.

Conclusion

This paper presents a comprehensive numerical and theoretical study of chaotic particle dynamics near Einstein-ModMax black holes, integrating observational parameter constraints with advanced chaos diagnostics. The dominant role of energy and angular momentum in shaping orbital phase transitions is quantitatively established, with Shannon entropy and MIPP providing effective characterization of complexity. The approach and results lay the groundwork for deepened understanding of chaos in strong field gravitational environments, facilitating further exploration in the context of both fundamental GR extensions and observational astrophysics.