We mainly focus on the effects of small changes of parameters on the dynamics of charged particles around Kerr black holes surrounded by an external magnetic field, which can be considered as a tidal environment. The radial motions of charged particles on the equatorial plane are studied via an effective potential. It is found that the particle energies at the local maxima values of the effective potentials increase with an increase in the black hole spin and the particle angular momenta, but decrease with an increase of one of the inductive charge parameter and magnetic field parameter. The radii of stable circular orbits on the equatorial plane also increase, whereas those of the innermost stable circular orbits decrease. On the other hand, the effects of small variations of the parameters on the orbital regular and chaotic dynamics of charged particles on the non-equatorial plane are traced by means of a time-transformed explicit symplectic integrator, Poincaré sections and fast Lyapunov indicators. It is shown that the dynamics sensitivity depends on small variations in the inductive charge parameter, magnetic field parameter, energy, and angular momentum. Chaos occurs easily as each of the inductive charge parameter, magnetic field parameter, and energy increases but is weakened as the angular momentum increases. When the dragging effects of the spacetime increase, the chaotic properties are not always weakened under some circumstances.

Gravitational waves, as potential probes of dark matter halos, have received much attention recently. This paper investigates how the annihilation channels of dark matter particles — namely s-wave, p-wave, and d-wave — influence the spatial distribution characteristics of dark matter halos, thereby leaving observable imprints on the gravitational wave signals from Extreme Mass Ratio Inspiral (EMRI) systems. We construct dark matter halo density distribution models incorporating s-wave, p-wave, and d-wave annihilation channels. We derive the phase corrections to the frequency-domain gravitational waveforms by combining these models with the orbital evolution equations of EMRI systems. Our study reveals that s-wave annihilation leads to the formation of a flattened density region in the dark matter halo, while p-wave and d-wave annihilation primarily modify smaller-scale structures within the halo. These distinct dark matter halo configurations induce significantly different phase shifts in the gravitational waveforms, providing a new observational window to distinguish the scattering properties of dark matter particles. Future space-based gravitational wave detectors, such as LISA, are expected to provide constraints on small-scale dark matter halos and the fundamental properties of dark matter particles by accurately measuring the gravitational wave signal of EMRI systems. These findings not only offer new insights into the gravitational wave signatures of dark matter annihilation, but also have potential implications for indirect dark matter detection via gamma-ray or cosmic-ray observations, as the modified halo profiles may affect the expected annihilation signals in the electromagnetic spectrum.

We investigate the potential of extreme mass ratio inspirals (EMRIs) as probes of strongly interacting dark matter (SIDM) spikes around supermassive black holes. Based on Lagrangian formulations of dark matter self-interactions, we analyze the effects of number changing processes, such as $2\to 0$ annihilation and $3\to 2$ scattering, on the inner density profiles, and propose a generalized definition of the dissolution radius, including its dependence on the central black hole mass. Building on this framework, we construct dark matter spike models suitable for phase shift calculations and study the gravitational waves of EMRIs under the circular orbit adiabatic approximation. Our analysis shows that different interaction mechanisms produce distinct dark matter density distributions, resulting in characteristic phase differences in the gravitational waveforms. The magnitudes of these phase shifts are well above the resolution threshold of space-based detectors. We conclude that future missions, such as LISA, will be able to distinguish different SIDM interaction channels through precise waveform measurements, thereby opening a new avenue for probing the microphysics of dark matter in extreme astrophysical environments.

In the curved space-time, the neutral test particle is not affected by any other force except for the influence of the curved space-time. Similar to the free sub in the flat space, the Lagrangian of the test particle only contains the kinetic energy term—the kinetic energy term of the four-dimensional curved space-time. In the case of small space-time curvature, linear approximation can be made. That is, under the weak field approximation, the Lagrangian quantity degenerates into the Lagrangian quantity in the axisymmetric gravitational field in Newtonian mechanics. In this paper, the curved space-time composed of axisymmetric equidistant black holes is taken as a model. We study the geodesic motion of the test particles around three black holes with equal mass and static axisymmetric distribution, including time-like particles and photons. The three extreme Reissner–Nordstrom black holes are balanced by electrostatic and gravitational forces. We first give the geodesic motion equation of particles in Three black holes space-time, give the relativistic effective potential, discuss the possible motion state of particles, and classify their motion trajectories. Then, the particle motion of the special plane (equatorial plane) is studied. The circular orbits of the two types of particles in the symmetric plane are studied, respectively. The circular orbits outside the symmetric plane are also studied, and their stability is also discussed. We will show the influence of the separation distance of the three black holes on the geodesic motion and explore the change of the relativistic effective potential. Then, the relationship between the inherent quantity and the coordinate quantity in space-time is analyzed. Finally, the chaos of the test particle orbit is explored.

We studied the space-time properties of the triangular symmetric black hole in the case of extreme RN black hole. Because the neutral test particle is only affected by space-time in the curved space-time, we chose the triangular symmetric black hole as the model with which to study the motion of the test particle in this case. The curvature tensor and curvature scalar were calculated by giving the metric and the Christoffel Symbol, and then the kinematics equation of the test particle was obtained and analyzed by using these quantities. Then we analyzed the relationship between the coordinate distance and the inherent distance, the relationship between the coordinate time and the inherent time, the inherent velocity and the coordinate velocity of light, and then verified the correctness of general relativity. Next, the one-dimensional effective potential and two-dimensional effective potential of the system under different separation distances were analyzed. Finally, we analyzed and explored the innermost stable circular orbit, calculated all the Lagrange points under this model, and expounded some applications of circular orbit in astrophysics.

We investigate the dynamical stability of circular orbits around a Kerr black hole embedded in a Dehnen-type dark matter halo. The effective spacetime metric of the combined system is constructed using the Newman–Janis algorithm, and the effective potential for test-particle motion in the equatorial plane is derived. The stability of circular orbits is analyzed through the Hessian matrix of the effective potential, while the stability strength and restoring-force distribution are employed to quantify the orbital response to small perturbations. Our results show that the presence of the dark matter halo significantly alters the spatial structure of stable circular orbits, leading to non-continuous stable regions whose location and extent depend sensitively on the halo’s characteristic density, scale radius, and the black hole spin. The innermost stable circular orbit (ISCO) is shifted relative to the vacuum Kerr case, with its position determined by the combined effects of the spin and halo parameters. Two-dimensional heatmaps, parameter scans, and three-dimensional visualizations systematically illustrate how the black hole spin and dark matter halo properties influence the ISCO and the distribution of stable orbits. Finally, we analyze the influence of the dark matter halo on the structure of the black hole event horizon. These results provide a detailed theoretical investigation of orbital dynamics around rotating black holes in dark-matter-rich environments.