The study of perturbation in magnetized Schwarzschild anti-de Sitter spacetime and dark energy profile

We are going to investigate the geodesic motion of charged particles in the vicinity of Schwarzschild anti-de-Sitter (S-AdS) spacetime with topological defects that admit temporal perturbation. We used the approximate Noether symmetry equation to insert the time conformal factor in the black hole without losing its symmetry structure. This type of insertion is necessary because the black hole radiates its energy and momentum in the form of gravitational waves and Hawking radiation. Along with the temporal perturbation, the S-AdS black hole (BH) is immersed in an external magnetic field. We conduct an in-depth examination of the dynamics of charged particles near a weakly magnetized and time conformal S-AdS BH. Our analysis involves calculating the shift in the position of the innermost circular orbit (ISCO) caused by both temporal perturbation and the presence of a magnetic field. Furthermore, we explore the influence of dark energy (DE) and angular momentum on the stability of these orbits. Additionally, we determine the effective force and escape velocity for a charged particle orbiting around the perturbed magnetized S-AdS BH. The application of time-dependent perturbation theory can extend our understanding to investigate the quasinormal modes (QNMs) of BH mergers.