How to bypass Birkhoff through extra dimensions (a simple framework for investigating the gravitational collapse in vacuum)

TL;DR

This paper introduces a new symmetry reduction of vacuum Einstein equations in higher odd dimensions, enabling the study of gravitational collapse in vacuum beyond spherical symmetry, which was previously limited by Birkhoff's theorem.

Contribution

It presents a novel cohomogeneity-two symmetry reduction that bypasses Birkhoff's theorem, allowing for time-dependent vacuum solutions in higher dimensions.

Findings

New reduction evades Birkhoff's theorem in vacuum

Allows study of gravitational collapse beyond spherical symmetry

Provides a 1+1 dimensional framework for vacuum collapse

Abstract

It is fair to say that our current mathematical understanding of the dynamics of gravitational collapse to a black hole is limited to the spherically symmetric situation and, in fact, even in this case much remains to be learned. The reason is that Einstein's equations become tractable only if they are reduced to a 1+1 dimensional system of partial differential equations. Due to this technical obstacle, very little is known about the collapse of pure gravitational waves because by Birkhoff's theorem there is no spherical collapse in vacuum. In this essay we describe a new cohomogeneity-two symmetry reduction of the vacuum Einstein equations in five and higher odd dimensions which evades Birkhoff's theorem and admits time dependent asymptotically flat solutions. We argue that this model provides an attractive 1+1 dimensional geometric setting for investigating the dynamics of gravitational collapse in vacuum.