Nuts have no hair

TL;DR

This paper proves that under specific geometric conditions, the only Ricci-flat, asymptotically flat Riemannian metrics with certain symmetries on a 4-manifold are the Kerr solutions, highlighting their uniqueness.

Contribution

It establishes a uniqueness theorem for Riemannian Kerr solutions among Ricci-flat, asymptotically flat metrics with nut-type symmetries on a specific 4-manifold topology.

Findings

Riemannian Kerr solutions are unique under the given conditions.

The metrics considered are characterized by a 1-parameter group of isometries with isolated fixed points.

The results apply to metrics with bounded orbit length at infinity.

Abstract

We show that the Riemannian Kerr solutions are the only Riemannian, Ricci-flat and asymptotically flat ${\rm C}^{2}$-metrics $g_{\mu\nu}$ on a 4-dimensional complete manifold ${\cal M}$ of topology ${\rm R}^{2} \times {\rm S}^{2}$ which have (at least) a 1-parameter group of periodic isometries with only isolated fixed points ("nuts") and with orbits of bounded length at infinity.