Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors

Abstract

We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well-posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups these flow equations have been investigated in the literature before in other contexts. As an application, the results provide a classification of Riemannian manifolds admitting imaginary generalised Killing spinors. We will also give new local normal forms for Lorentzian metrics with parallel null spinor in any dimension.

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Notes

  1. 1.

    This has been shown in [22] for the vacuum Einstein equations \({\overline{\mathrm {Ric}}}= 0\) and remains valid in our setting, as here the \(\mathrm {Z}\)-term in (3.11) enters only algebraically in the \(b_1\)-term.

  2. 2.

    This system is even equivalent to \(P\eta =0\). Indeed, let a triple \((\eta _{A},k_{A},\eta _{A,i})\) solve (4.21)–(4.23). As \(g^{ij}\) is invertible for g sufficiently close to h, (4.22) is the same as \(\partial _t \eta _{A,i}=\partial _{i}k_{A}\), and (4.21) then gives \(\partial _t (\eta _{A,i}-\partial _{i} \eta _{A})=0\). Appropriate choice of initial data ensures \(\eta _{A,i}=\partial _{i}\eta _{A}\) at \(t=0\) and thus equality everywhere. Then (4.23) is nothing but \(P \eta = 0\).

  3. 3.

    We would like to thank Vincente Cortés for alerting us to this example.

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Acknowledgements

We would like to thank Helga Baum and Vicente Cortés for inspiring discussions. TL would like to thank Nick Buchdahl, Mike Eastwood, Jason Lotay and Spiro Karigiannis for helpful discussions in regards to the open problem in Sect. 7.3. AL would like to thank Todd Oliynyk for discussions about symmetric hyperbolic systems during his visit to Monash University. This work was supported by the Australian Research Council via the Grants FT110100429 and DP120104582

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Leistner, T., Lischewski, A. Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors. J Geom Anal 29, 33–82 (2019). https://doi.org/10.1007/s12220-017-9941-x

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Keywords

  • Lorentzian geometry
  • Holonomy groups
  • Parallel null vector field
  • Cauchy problem
  • Killing spinors
  • Parallel spinors
  • Symmetric hyperbolic system

Mathematics Subject Classification

  • Primary 53C50
  • 53C27
  • 53C29
  • Secondary 53C26
  • 53C44
  • 35L02
  • 35L10
  • 83C05