Universes encircling 5-dimensional black holes

TL;DR

This paper demonstrates that two known 5-dimensional vacuum solutions, which embed 4D cosmologies, are equivalent to 5D topological black holes, highlighting their horizon-regularity and the role of a generalized Birkhoff's theorem.

Contribution

It establishes the equivalence between two cosmological solutions and 5D black holes, and shows their horizon regularity using explicit coordinate transformations.

Findings

Both metrics are equivalent to 5D topological black holes.

LMW coordinates are regular across horizons.

The equivalence stems from a generalized Birkhoff's theorem in 5D.

Abstract

We clarify the status of two known solutions to the 5-dimensional vacuum Einstein field equations derived by Liu, Mashhoon & Wesson (LMW) and Fukui, Seahra & Wesson (FSW), respectively. Both 5-metrics explicitly embed 4-dimensional Friedman-Lemaitre-Robertson-Walker cosmologies with a wide range of characteristics. We show that both metrics are also equivalent to 5-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 5 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.