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Null fluid gravitational fields on Kerr manifolds and optical lifts of Sasaki structures

Masoud Ganji, Cristina Giannotti, Andrea Spiro

TL;DR

The paper provides a coordinate-free local parametrisation of quasi-Einstein metrics on 4D Kerr manifolds, describing them in terms of eight real base-functions subject to five PDEs, and shows that this framework yields a broad enlarged Kerr family containing Kerr and many Ricci-flat Lorentzian metrics. It demonstrates two key consequences: (i) a simple construction method for null-fluid solutions to the Einstein equations, yielding metrics with null fluid energy momentum that are generically of Petrov type II or D; and (ii) resolution of a local existence conjecture for smooth optical lifts of Sasaki CR structures in the Kerr-type setting, extending prior analytic results to smooth Sasaki CR manifolds. The results deepen the link between Sasaki CR geometry, RT optical lifts, and Lorentzian geometry with null fluids, and provide a rich source of explicit examples for testing theoretical and numerical approaches in general relativity. Overall, the work expands the catalog of exact solutions and clarifies the geometric mechanisms by which Kerr-type optical structures support quasi-Einstein, null-fluid spacetimes.

Abstract

Building on the characterisation in [C. D. Hill, J. Lewandowski and P. Nurowski, Indiana Univ. Math. J. 57 (2008), 3131--3176] of 4-dimensional Lorentzian metrics adapted to an optical structure and satisfying the null fluid Einstein equations, we give an explicit parameterisation of this class under the assumption that the optical structure is of Kerr type. As immediate consequences, we obtain: (1) a new method for constructing solutions to the Einstein equations with a null fluid energy momentum tensor, yielding a large family of explicit metrics that naturally includes the classical Kerr black hole metrics and all Ricci flat examples described in [M. Ganji, C. Giannotti, G. Schmalz and A. Spiro, Ann. Physics 75 (2025), Paper No. 169908, 28]; (2) a solution to a conjecture in Hill, Lewandowski and Nurowski's paper on the local existence of smooth optical lifts in the case of Sasaki CR structures.

Null fluid gravitational fields on Kerr manifolds and optical lifts of Sasaki structures

TL;DR

The paper provides a coordinate-free local parametrisation of quasi-Einstein metrics on 4D Kerr manifolds, describing them in terms of eight real base-functions subject to five PDEs, and shows that this framework yields a broad enlarged Kerr family containing Kerr and many Ricci-flat Lorentzian metrics. It demonstrates two key consequences: (i) a simple construction method for null-fluid solutions to the Einstein equations, yielding metrics with null fluid energy momentum that are generically of Petrov type II or D; and (ii) resolution of a local existence conjecture for smooth optical lifts of Sasaki CR structures in the Kerr-type setting, extending prior analytic results to smooth Sasaki CR manifolds. The results deepen the link between Sasaki CR geometry, RT optical lifts, and Lorentzian geometry with null fluids, and provide a rich source of explicit examples for testing theoretical and numerical approaches in general relativity. Overall, the work expands the catalog of exact solutions and clarifies the geometric mechanisms by which Kerr-type optical structures support quasi-Einstein, null-fluid spacetimes.

Abstract

Building on the characterisation in [C. D. Hill, J. Lewandowski and P. Nurowski, Indiana Univ. Math. J. 57 (2008), 3131--3176] of 4-dimensional Lorentzian metrics adapted to an optical structure and satisfying the null fluid Einstein equations, we give an explicit parameterisation of this class under the assumption that the optical structure is of Kerr type. As immediate consequences, we obtain: (1) a new method for constructing solutions to the Einstein equations with a null fluid energy momentum tensor, yielding a large family of explicit metrics that naturally includes the classical Kerr black hole metrics and all Ricci flat examples described in [M. Ganji, C. Giannotti, G. Schmalz and A. Spiro, Ann. Physics 75 (2025), Paper No. 169908, 28]; (2) a solution to a conjecture in Hill, Lewandowski and Nurowski's paper on the local existence of smooth optical lifts in the case of Sasaki CR structures.

Paper Structure

This paper contains 13 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries (I): Optical geometries and optical lifts
  3. Geodesic shearfree null congruences and optical geometries
  4. Robinson and Trautman optical geometries and CR structures
  5. Optical lifts of Sasaki CR structures
  6. Preliminaries (II): Null fluid gravitational fields on Kerr manifolds
  7. An important class of optical geometries: the Kerr structures
  8. Kähler potentials and complex structures on Kerr manifolds
  9. Null fluid gravitational fields and quasi-Einstein metrics
  10. Quasi-Einstein metrics on $4$-dimensional Kerr manifolds
  11. A new large class of examples of null fluid gravitational fields
  12. Existence of quasi-Einstein lifts for Sasaki CR $3$-manifolds of class $\mathcal{C}^\infty$
  13. The proof that \ref{['shearfreemetric']} is equivalent to \ref{['111']}

Key Result

Proposition 2.2

Given an $A$-bundle $\pi: M \to \mathscr{S}$, with $A = \mathbb{R}$ or $S^1$, there is a natural one-to-one correspondence between the RT geometries $\mathscr{Q} = (\mathscr{W}, [h]_{\pm}, \mathscr{K}, \{g\})$ on $M$ with $\mathscr{K} = T^{\operatorname{vert}}M$ and the sub-conformal structures $(\

Theorems & Definitions (12)

  • Remark 2.1
  • Proposition 2.2: AGSS
  • Theorem 2.3
  • Definition 3.1
  • Definition 3.2
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • Definition 5.1
  • Proposition 5.2
  • ...and 2 more