Quasinormal modes of charged black holes surrounded by fluid in Rastall gravity

We investigate the quasinormal modes of charged black holes surrounded by a dust-like field in Rastall gravity. The background spacetime is described by a static and spherically symmetric charged black hole solution, in which the Rastall parameter modifies the radial dependence of the surrounding-field contribution. We focus on the asymptotically flat branches of the solution and analyze the axial perturbations of the system. Due to the presence of the black hole charge, the electromagnetic and gravitational perturbations are coupled, leading to a two-component Schrödinger-like master equation with an effective potential matrix. The quasinormal frequencies are computed by imposing the ingoing boundary condition at the event horizon and the outgoing boundary condition at spatial infinity, and the resulting eigenvalue problem is solved numerically by a spectral method.

We study in detail how the fundamental quasinormal modes depend on the Rastall parameter, the surrounding-field parameter, and the black hole charge. In the small-λ asymptotically flat branch, the real oscillation frequency and the damping rate exhibit systematic variations as the Rastall parameter is increased. In the large-λ branch, when the sign of the surrounding-field parameter is chosen consistently with the energy condition, the damping rate can display a non-monotonic behavior. We also examine the dependence on the surrounding-field strength and on the charge, the latter controlling both the background geometry and the electromagnetic-gravitational coupling strength. These results show that the quasinormal spectrum encodes the combined effects of Rastall gravity, the surrounding dust-like field, and the charge-induced coupling of perturbations.