Universes encircling five-dimensional black holes

We clarify the status of two known solutions to the five-dimensional vacuum Einstein field equations derived by Liu, Mashhoon, and Wesson (LMW) and Fukui, Seahra, and Wesson (FSW), respectively. Both 5-metrics explicitly embed four-dimensional Friedman–Lemaı̂tre–Robertson–Walker cosmologies with a wide range of characteristics. We show that both metrics are also equivalent to five-dimensional topological black hole (TBH) solutions, which is demonstrated by finding explicit coordinate transformations from the TBH to LMW and FSW line elements. We argue that the equivalence is a direct consequence of Birkhoff’s theorem generalized to five dimensions. Finally, for a special choice of parameters we plot constant coordinate surfaces of the LMW patch in a Penrose–Carter diagram. This shows that the LMW coordinates are regular across the black and/or white hole horizons.