Gravitational Perturbations of Spherically Symmetric Black Holes Surrounded by Perfect Fluid Matter in Rastall Gravity

Abstract This paper investigates the perturbation problem of a spherically symmetric black hole surrounded by perfect fluid matter in Rastall gravity. Assuming that the black hole is subjected to axial perturbations, we derive the governing equation for perturbation evolution, namely the Regge–Wheeler (RW) equation, and discuss the properties of the perturbation potential V RW ( r ) . From this, a constraint condition on the Rastall gravity parameter k λ is deduced. Furthermore, we solve the quasinormal mode frequencies ( QNMs ) of the equation using the WKB approximation. Subsequently, we distinguish two regions: the region near the perturbation center and the region far from the perturbation center. For these two regions, we separately solve the spatial distribution and time evolution functions of gravitational perturbations satisfying the RW equation under the WKB approximation. Among them, with the help of the analysis of the trajectories of equiphase points, we discuss the problem of the finiteness of the perturbation amplitude. Finally, given the specified perturbation frequencies and the spatiotemporal distribution of perturbations, we solve the variation of the particle’s effective potential curve under perturbations and discuss the influence of gravitational perturbations on the corresponding innermost stable circular orbit.