Apparent horizons of the Thakurta spacetime and the description of
  cosmological black holes
  We discuss the validity of the Thakurta metric to describe cosmological black
holes by analysing the nature of its horizon. By adopting the preferred
foliation of the Thakurta spacetime associated with the Kodama time, we
demonstrate that the Thakurta horizon is indeed a future outer trapping
horizon. Therefore, the respective observers see it as a cosmological black
hole, contrary to some claims in the literature.
