Exploring temporal effects on particle dynamics in dark energy-immersed Schwarzschild black hole

The space–time curvature undergoes substantial changes due to phenomena such as gravitational waves as well as Hawking radiation and black hole expansion. This paper will analyze the action of time and dark energy on particles, with neutral and electric charge, around a Schwarzschild black hole. In the context of the Schwarzschild space–time, we define the general time-dependent conformal factor of the form [Formula: see text], with a function of time [Formula: see text], and small perturbation parameter [Formula: see text], which increases during the associated space–time. This modification shifts the energy dynamics of the space–time region. We obtain the Noether symmetry equation of the problem from the Lagrangian of the perturbed space–time and get a system of 19 PDEs. The construction of solutions to PDEs requires us to construct perturbed Schwarzschild black hole space–time admitting both Noether symmetries as well as approximate Noether symmetries and their associated conservation laws. In addition, we also examine the behavior of electrically charged and neutral particles near this newly perturbed Schwarzschild black hole.