Non-existence of asymptotically flat geons in (2 + 1) gravity

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.