Axial Perturbations of Charged Kiselev Black Holes in Rastall Gravity: Matter Response and Quasinormal Modes
We investigate axial gravitoelectromagnetic perturbations of charged black holes surrounded by dust-like Kiselev-type anisotropic matter in Rastall gravity. We derive the axial matter compatibility condition and obtain two coupled Schrödinger-type master equations under a closure that neglects perturbations of the covariant anisotropy-direction vector. The equations admit an $r$-independent algebraic decoupling in the Reissner--Nordström and general-relativistic dust-like limits, whereas additional radial structure obstructs such a decoupling in generic charged Rastall backgrounds with nonzero surrounding matter. For the nonextremal backgrounds considered, we establish axial mode stability under this closure: a matrix $S$-deformation excludes exponentially growing coupled modes for $\ell\geq2$, and the physical electromagnetic dipole has a positive effective potential. A Chebyshev pseudospectral calculation, checked against the Reissner--Nordström spectrum and spectral convergence, characterizes the fundamental frequencies. Increasing the Rastall coupling lowers both the oscillation frequency and the damping rate in the small-coupling region; in the large-coupling region, the oscillation frequency rises while the damping rate varies nonmonotonically. A larger charge generally raises the oscillation frequency, whereas stronger surrounding-matter contributions tend to produce longer-lived modes. An auxiliary frozen-source truncation generally violates the matter constraint but yields sub-percent frequency differences for the representative configurations examined, showing that spectral proximity does not establish constraint compatibility.