Evaluation of circular geodesics and thermodynamics of a Bumblebee gravity-like black hole

In this study, we explore the thermodynamic and geodesic stability of Reissner-Nordstr & ouml;m (RN) black holes within the framework of Bumblebee gravity (BG), a Lorentz-violating extension of general relativity (GR). By examining the black hole's mass distribution and horizon thermodynamics, we establish that although charged black holes in BG exhibit local stability, they remain globally unstable. This conclusion is substantiated by the positive-definite Gibbs free energy and the negative-definite Hessian matrix, which collectively indicate a fundamental instability in the system's global equilibrium. We utilized the effective potential to derive the Lyapunov exponents for both time-like and null geodesics, assessing their stability. We found that the circular orbits beyond the innermost stable circular orbits in the Schwarzschild regime are stable, which contrasts with the RN black hole solution in GR. Furthermore, our findings demonstrate that a strong Bumblebee field induces unstable orbits. We establish that extreme RN black holes in Einstein gravity do not possess quasi- normal frequencies and instead exhibit stable, constant perturbations. In the non-extremal case, perturbations settle into a steady state, maintaining a constant amplitude. This behavior parallels observations made in the context of the Bumblebee effect.