On the validity of the 5-dimensional Birkhoff theorem: The tale of an exceptional case

TL;DR

This paper examines the limits of the 5-dimensional Birkhoff theorem by analyzing the Gergely-Maartens space-time, showing it relates to black hole horizon metrics despite violating the theorem in its standard form.

Contribution

It demonstrates that the 5d Birkhoff theorem holds in a weaker sense, linking the GM space-time to degenerated horizon metrics of certain black-hole solutions.

Findings

The GM space-time is not a standard class member of the 5d Birkhoff theorem.

The GM space-time relates to the horizon metric of specific black-hole solutions.

The result parallels the relation between extremal Reissner-Nordstrom and Bertotti-Robinson space-times.

Abstract

The 5-dimensional (5d) Birkhoff theorem gives the class of 5d vacuum space-times containing spatial hypersurfaces with cosmological symmetries. This theorem is violated by the 5d vacuum Gergely-Maartens (GM) space-time, which is not a representant of the above class, but contains the static Einstein brane as embedded hypersurface. We prove that the 5d Birkhoff theorem is still satisfied in a weaker sense: the GM space-time is related to the degenerated horizon metric of certain black-hole space-times of the allowed class. This result resembles the connection between the Bertotti-Robinson space-time and the horizon region of the extremal Reissner-Nordstrom space-time in general relativity.