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On Walker and para-Hermite Einstein spaces

Adam Chudecki

TL;DR

This work advances the classification of complex para-Hermite Einstein spaces by focusing on the Walker subclass with a type [D] self-dual Weyl tensor and a degenerate, parallel anti-self-dual Weyl tensor. By embedding these spaces into the hyperheavenly framework, the authors derive explicit metric forms across all admissible Petrov-Penrose types, reducing many cases to Abelian/Liouville-type equations and, in particular, presenting a complete set for the $[\textrm{D}]^{ee} \otimes [\textrm{II}]^{n}$ family. The results include a comprehensive symmetry analysis, gauge freedoms, and the explicit construction of metrics that realize the full range of optical configurations $[++,++], [++,--],$ etc., with several cases admitting 2D or 3D isometry algebras. The findings enrich the landscape of exact non-Lorentzian Einstein solutions, offering pathways to real neutral slices and potential applications in mathematical physics, while also outlining open questions for the broader $[\textrm{D}]^{ee} \otimes [\textrm{deg}]^{e}$ sector.

Abstract

A special class of (complex) para-Hermite Einstein spaces is analyzed. For this class of spaces the self-dual Weyl tensor is type-[D] in the Petrov-Penrose classification. The anti-self-dual Weyl tensor is algebraically degenerate, equivalently, there exists an anti-self-dual congruence of null strings. It is assumed that this congruence is parallely propagated. Thus, the spaces are not only para-Hermite but also Walker. A classification of the spaces according to three criteria is given. Finally, explicit metrics of all admitted Petrov-Penrose types are found.

On Walker and para-Hermite Einstein spaces

TL;DR

This work advances the classification of complex para-Hermite Einstein spaces by focusing on the Walker subclass with a type [D] self-dual Weyl tensor and a degenerate, parallel anti-self-dual Weyl tensor. By embedding these spaces into the hyperheavenly framework, the authors derive explicit metric forms across all admissible Petrov-Penrose types, reducing many cases to Abelian/Liouville-type equations and, in particular, presenting a complete set for the family. The results include a comprehensive symmetry analysis, gauge freedoms, and the explicit construction of metrics that realize the full range of optical configurations etc., with several cases admitting 2D or 3D isometry algebras. The findings enrich the landscape of exact non-Lorentzian Einstein solutions, offering pathways to real neutral slices and potential applications in mathematical physics, while also outlining open questions for the broader sector.

Abstract

A special class of (complex) para-Hermite Einstein spaces is analyzed. For this class of spaces the self-dual Weyl tensor is type-[D] in the Petrov-Penrose classification. The anti-self-dual Weyl tensor is algebraically degenerate, equivalently, there exists an anti-self-dual congruence of null strings. It is assumed that this congruence is parallely propagated. Thus, the spaces are not only para-Hermite but also Walker. A classification of the spaces according to three criteria is given. Finally, explicit metrics of all admitted Petrov-Penrose types are found.

Paper Structure

This paper contains 43 sections, 21 theorems, 165 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Structure of the article
  4. Summary of main results
  5. Remarks about real slices
  6. Preliminaries
  7. Petrov-Penrose classification
  8. Congruences of null strings
  9. Para-Hermite spaces
  10. Congruences of null geodesics
  11. Algebraically degenerate para-Hermite Einstein spaces
  12. Hyperheavenly spaces of the types $[\textrm{D}]^{ee} \otimes [\textrm{any}]$
  13. The metric
  14. Gauge freedom
  15. Symmetries
  16. ...and 28 more sections

Key Result

Theorem 2.1

If $(\mathcal{M}, ds^{2})$ is a complex 4-dimensional Einstein space then the following statements are equivalent

Figures (2)

  • Figure 1: Congruences of null strings and congruences of null geodesics in algebraically degenerate para-Hermite Einstein spaces.
  • Figure 2: Congruences of null strings and congruences of null geodesics in type-[D] para-Hermite Einstein spaces.

Theorems & Definitions (51)

  • Theorem 2.1
  • Corollary 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 41 more