Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Einstein manifolds with optical geometries of Kerr type

Masoud Ganji, Cristina Giannotti, Gerd Schmalz, Andrea Spiro

TL;DR

The paper delivers a decisive no-go result for Einstein metrics on Kerr-type optical geometries in dimensions $n>4$, and a complete 4D classification of Kerr-type backgrounds and their Ricci-flat Kerr families. It builds Kerr-type Lorentzian manifolds from a quantisable Kähler base and uses a canonical datum to parametrize compatible metrics, revealing that the 4D backgrounds are flat and arise from κ=±1 base geometries with a global Kähler potential. The authors construct explicit Ricci-flat Kerr families via a Kerr-Schild deformation constrained by a linear elliptic PDE in two variables, yielding real-analytic families parameterised by holomorphic data and a single parameter $m$. The results connect classical Kerr metrics to a broad, explicit class of Ricci-flat Lorentzian manifolds and highlight a delicate interplay between twisting optical structures, Kähler geometry, and Einstein equations, with potential applications to cosmological and non-stationary gravitational models.

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 $κ= 1$ or $κ= -1$. 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 Poincaré 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.

Einstein manifolds with optical geometries of Kerr type

TL;DR

The paper delivers a decisive no-go result for Einstein metrics on Kerr-type optical geometries in dimensions , and a complete 4D classification of Kerr-type backgrounds and their Ricci-flat Kerr families. It builds Kerr-type Lorentzian manifolds from a quantisable Kähler base and uses a canonical datum to parametrize compatible metrics, revealing that the 4D backgrounds are flat and arise from κ=±1 base geometries with a global Kähler potential. The authors construct explicit Ricci-flat Kerr families via a Kerr-Schild deformation constrained by a linear elliptic PDE in two variables, yielding real-analytic families parameterised by holomorphic data and a single parameter . The results connect classical Kerr metrics to a broad, explicit class of Ricci-flat Lorentzian manifolds and highlight a delicate interplay between twisting optical structures, Kähler geometry, and Einstein equations, with potential applications to cosmological and non-stationary gravitational models.

Abstract

We classify the Ricci flat Lorentzian -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 , 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 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 or . The first class includes a three parameter family of metrics admitting real analytic extensions to 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 , where is the Lobachevsky Poincaré disc. This subclass is in bijection with the holomorphic functions on 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.
Paper Structure (30 sections, 11 theorems, 127 equations)

This paper contains 30 sections, 11 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Kerr families of gravitational fields on Kerr type backgrounds
  3. Optical geometries
  4. Kerr manifolds
  5. Parameterisations of compatible metrics and adapted frame fields
  6. Kerr families of gravitational fields on Kerr type backgrounds
  7. A no-go theorem for the Einstein metrics that are compatible with optical geometries of Kerr type
  8. A dimensional constraint for Kerr manifolds with Kerr type backgrounds
  9. Proof of the dimensional constraint
  10. Notational issues
  11. The Christoffel symbols of the metric \ref{['back*']}
  12. The condition ${\operatorname{Ric}}({\operatorname{p}}_o, {\operatorname{p}}_o) = 0$ as an equation on $\sigma$
  13. The conditions ${\operatorname{Ric}}(\widehat{E}_i, {\operatorname{p}}_o) = 0$ as equations on $\varphi$, $B$ and $\sigma$
  14. The proof that ${\operatorname{Ric}}({\operatorname{p}}_o, {\operatorname{p}}_o) = 0$ and ${\operatorname{Ric}}({\operatorname{p}}_o, \widehat{E}_i) = 0$ are not compatible if $n > 4$
  15. Classification of Kerr $4$-manifolds admitting Kerr type backgrounds
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $M = \mathscr{S} \times I$, $I \subset \mathbb{R}$, be a Kerr manifold of dimension $n = 2k + 2\geq 4$. We denote by $\pi^\mathscr{S}: M = \mathscr{S} \times I \to \mathscr{S}$ the standard projection onto the total space of the $\mathbb{R}$-bundle $\pi: \mathscr{S} = N \times \mathbb{R} \to N

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 10 more