Conformally Schwarzschild cosmological black holes

We thoroughly investigate conformally Schwarzschild spacetimes in different coordinate systems to seek for physically reasonable models of a cosmological black hole. assume that conformal factor depends only on the time and spacetime is asymptotically flat Friedmann-Lema\^{\i}tre-Robertson-Walker universe filled by perfect fluid obeying linear equation state $p=w\rho$ with $w>-1/3$. In this class spacetimes, McClure-Dyer spacetime, constructed terms isotropic coordinates, Thakurta standard are identical do not describe contrast, Sultana-Dyer Culetu classes Kerr-Schild Painlev\'{e}-Gullstrand respectively, case, corresponding matter field general relativity can be interpreted as combination homogeneous an inhomogeneous null fluid, which valid everywhere unlike Sultana Dyer's interpretation. anisotropic fluid. However, both cases, total energy-momentum tensor violates all energy conditions at finite value radial late times. As consequence, holes $-1/3<w\le 1$ cannot evolution primordial hole after its horizon entry.