An analysis of the Sultana-Dyer cosmological black hole solution of the Einstein equations

The Sultana-Dyer solution of general relativity representing a black hole embedded in a special cosmological background is analysed. We find an expanding (weak) spacetime singularity instead of the reported conformal Killing horizon, which is covered by an expanding black hole apparent horizon (internal to a cosmological apparent horizon) for most of the history of the universe. This singularity was naked early on. The global structure of the solution is studied as well.