On quasinormal modes in 4D black hole solutions in the model with anisotropic fluid

We consider a family of four-dimensional black hole solutions from Dehnen et al. (Grav Cosmol 9:153 arXiv:gr-qc/0211049, 2003) governed by natural number q = 1, 2, 3, . . . , which appear in the model with anisotropic fluid and the equations of state: p${}_r$ = -ρ(2q-1)${}^{-1}$, p${}_t$ = - p${}_r$, where p${}_r$ and p${}_t$ are pressures in radial and transverse directions, respectively, and ρ > 0 is the density. These equations of state obey weak, strong and dominant energy conditions. For q = 1 the metric of the solution coincides with that of the Reissner-Nordström one. The global structure of solutions is outlined, giving rise to Carter-Penrose diagram of Reissner-Nordström or Schwarzschild types for odd q = 2k +1 or even q = 2k, respectively. Certain physical parameters corresponding to BH solutions (gravitational mass, PPN parameters, Hawking temperature and entropy) are calculated. We obtain and analyse the quasinormal modes for a test massless scalar field in the eikonal approximation. For limiting case q = +$\infty$, they coincide with the well-known results for the Schwarzschild solution. We show that the Hod conjecture which connect the Hawking temperature and the damping rate is obeyed for all q ≥ 2 and all (allowed) values of parameters.