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On pp-waves with lightlike parallel spinors

Bernd Ammann, Jonathan Glöckle, Klaus Kroencke

Abstract

We parametrize pp-wave spacetimes with compact codimension 2 hypersurfaces. In the vacuum case, we show that these spacetimes are locally in one-to-one correspondence with smooth curves of Riemannian Ricci-flat metrics modulo smooth curves of diffeomorphisms. We also prove that this one-to-one correspondence extends to pp-waves with prescribed null Ricci curvature. Moreover, the pp-wave spacetime carries a lightlike parallel spinor if and only if one (and hence all) of the Ricci-flat metrics carries a parallel spinor.

On pp-waves with lightlike parallel spinors

Abstract

We parametrize pp-wave spacetimes with compact codimension 2 hypersurfaces. In the vacuum case, we show that these spacetimes are locally in one-to-one correspondence with smooth curves of Riemannian Ricci-flat metrics modulo smooth curves of diffeomorphisms. We also prove that this one-to-one correspondence extends to pp-waves with prescribed null Ricci curvature. Moreover, the pp-wave spacetime carries a lightlike parallel spinor if and only if one (and hence all) of the Ricci-flat metrics carries a parallel spinor.
Paper Structure (16 sections, 41 theorems, 195 equations)

This paper contains 16 sections, 41 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Properties of pp-wave spacetimes
  3. Geometric conditions for simple pp-waves
  4. Initial data sets
  5. Geodesic coordinates
  6. Lightlike Ricci curvature
  7. Additional curvature vanishing
  8. Classification of simple pp-waves with null Ricci curvature
  9. The role of the leafwise volume
  10. Proving the classification
  11. Discussion and comparison with the AKM-construction
  12. Constructing parallel spinors on simple pp-waves
  13. Parallel spinors on spacetimes and initial data sets
  14. Analyzing parallelism for spinors on initial data sets
  15. Proving parallelism
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $Q$ be a closed spin manifold. Consider a pp-wave of the form eq:AGK_metrics on $\mathbb{R}\times I\times Q$ such that the metrics $g_s$ are Ricci-flat and satisfy eq:momentum_constraint. If there is an $s_0\in I$ such that the Riemannian metric $g_{s_0}$ carries a parallel unit spinor $\varphi$ under the bundle isomorphism eq:IdentifySpinors.

Theorems & Definitions (104)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 94 more