Einstein manifolds with optical geometries of Kerr type

TL;DR

This paper classifies Ricci flat Lorentzian manifolds with Kerr-like optical structures, revealing existence only in four dimensions and linking certain classes to Kerr metrics and holomorphic functions.

Contribution

It provides a complete classification of Kerr-type Ricci flat Lorentzian manifolds, showing their existence only in four dimensions and explicitly characterizing their geometric structures.

Findings

No such manifolds exist for dimensions greater than four.

Two large classes of four-dimensional manifolds are identified, including Kerr metrics.

One class corresponds to known Kerr metrics with angular momentum parameters.

Abstract

We classify the Ricci flat Lorentzian $n$-manifolds satisfying three particular conditions, encoding and combining some crucial features of the Kerr metrics and the Robinson-Trautman optical structures. We prove that: (a) If $n>4$, there is no Lorentzian manifold satisfying the considered Kerr type conditions, in unexpected contrast with what occurs for the metrics satisfying (very similar) Taub-NUT type conditions; (b) If $n=4$ there are two large classes of such Kerr type manifolds. Each class consists of manifolds fibering over open Riemann surfaces, equipped with a metric of constant Gaussian curvature $\kappa = 1$ or $\kappa = -1$. The first class includes a three parameter family of metrics admitting real analytic extensions to $(\mathbb R^3 \setminus\{0\}) \times \mathbb R = (S^2 \times \mathbb R_+) \times \mathbb R$ and a large class of other metrics not admitting this kind of extensions. The metrics of this first class admitting such extensions are all isometric to the well known Kerr metrics, with the three parameters corresponding to the three space-like components of the angular momentum of the gravitational field. The second class contains a subclass of metrics defined on $\big(\mathbb D\times \mathbb R_+\big)\times \mathbb R$, where $\mathbb D$ is the Lobachevsky Poincar\'e disc. This subclass is in bijection with the holomorphic functions on $\mathbb D$ satisfying an appropriate open condition. These and other results are consequences of a very simple way to construct totally explicit examples of Ricci flat Lorentzian manifolds.