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.