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=??(2q?1)^{?1}, p_t=?p_r, where p_r and p_t are pressures in radial and transverse directions, respectively, and ?>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–Nordstrom one. The global structure of solutions is outlined, giving rise to Carter–Penrose diagram of Reissner–Nordstrom 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=+?, 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.