5-Dimensional Space-Periodic Solutions of the Static Vacuum Einstein Equations

TL;DR

This paper proves the existence of 5-dimensional static vacuum solutions with complex horizon topologies, including black hole configurations and solitons, expanding the understanding of higher-dimensional Einstein equations.

Contribution

It confirms the conjecture of Myers by constructing diverse 5D static vacuum solutions with black holes and solitons, and explores their geometric and topological properties.

Findings

Existence of 5D solutions balancing infinite black holes with Kasner asymptotics

Construction of space-periodic vacuum spacetimes without black holes

New examples of manifolds with large or infinite Betti numbers

Abstract

An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring $S^1\times S^2$ and sphere $S^3$ cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.