Gravitational Solitons and Complete Ricci Flat Riemannian Manifolds of Infinite Topological Type

TL;DR

This paper constructs new higher-dimensional gravitational solitons and Ricci flat manifolds with complex topologies, revealing connections between black hole solutions and Ricci-flat geometries through Wick rotation.

Contribution

It introduces novel space-periodic solutions to Einstein equations, including gravitational solitons, and explicitly computes their topologies, linking them to Ricci-flat manifolds of infinite topological type.

Findings

New space-periodic solutions with Kasner asymptotics

Explicit topological descriptions of solutions as connected sums of spheres

Correspondence between solitons and black hole solutions via Wick rotation

Abstract

We present several new space-periodic solutions of the static vacuum Einstein equations in higher dimensions, both with and without black holes, having Kasner asymptotics. These latter solutions are referred to as gravitational solitons. Further partially compactified solutions are also obtained by taking appropriate quotients, and the topologies are computed explicitly in terms of connected sums of products of spheres. In addition, it is shown that there is a correspondence, via Wick rotation, between the spacelike slices of the solitons and black hole solutions in one dimension less. As a corollary, the solitons give rise to complete Ricci flat Riemannian manifolds of infinite topological type and generic holonomy, in dimensions 4 and higher.