Non-existence of Asymptotically Flat Geons in 2+1 Gravity

TL;DR

This paper proves that in 2+1 dimensions, asymptotically flat vacuum or Einstein-Maxwell solutions cannot contain geons, unlike in 3+1 dimensions, but such structures can exist in anti-de Sitter spacetimes behind horizons.

Contribution

It establishes the non-existence of asymptotically flat geons in 2+1 gravity and characterizes conditions for their absence, contrasting with higher-dimensional cases.

Findings

No asymptotically flat vacuum geons in 2+1 dimensions.

Geons can exist in anti-de Sitter spacetimes behind horizons.

Provided conditions for non-existence of flat geons.

Abstract

Geons, small topological structures that exhibit particle properties such as charge and angular momentum without the presence of matter sources, have been extensively discussed in 3+1-dimensional general relativity. Given the recent renewal of interest in 2+1 gravity, it is natural to ask whether or not the notion of geons extends to three dimensions. We prove here that, in contrast to the 3+1-dimensional case, there are no 2+1-dimensional asymptotically flat solutions of the vacuum Einstein or Einstein-Maxwell equations containing geons. In contrast, 2+1-dimensional asymptotically anti-de Sitter spacetimes can indeed contain geons; however, the geons are always hidden behind a single black hole horizon. We also prove sufficient conditions for the non-existence of 2+1-dimensional asymptotically flat geon-containing solutions.