Particle motion and epicyclic frequencies in the einstein special unitary matrices sigma model of black holes

In this study, we test the dynamics of test particles around static and spherically symmetric BHs arising in the Einstein SU ( N ) non-linear sigma model. By showing the influence of general relativity and the metric function topologically in the SU ( N ) group, the spacetime is illustrated as dependent on the internal symmetry parameter α , significantly modifying the BH geometry. In addition, we derive analytical expressions for the energy and angular momentum of circular equatorial orbits and investigate the behavior of the effective potential to determine the orbital stability. The innermost stable circular orbits (ISCOs) are computed and their sensitivity to variations in α and the cosmological constant Λ is analyzed. In addition, we calculate the effective radial force acting on the particles and show how the parameters SU ( N ) influence its attractive or repulsive nature. By perturbing the stable circular orbits, we derive the radial, vertical, and orbital epicyclic frequencies for both local and distant observers. In this case, our results show that the internal gauge structure inherent in the SU ( N ) model has a pronounced effect on the orbital dynamics and oscillatory behavior of test particles, offering potential observational signatures in strong-field astrophysics.