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.

Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

“…And thus the well-definedness of the Cauchy problem implies the existence of an associated (n + 1)-dimensional Lorentzian with a parallel spinor. Such a relation between families of metrics with special holonomy and solutions of the constraint equations was already conjectured by Leistner and Lischewski, see [29]. We essentially show that the conditions in [29,Table 1] is satisfied if and only if the divergence condition (31) in our Appendix D is satisfied.…”

“…Lorentzian manifolds with parallel spinors was recently achieved by H. Baum, T. Leistner and A. Lischewski [10,29,30], see also [9] for associated lecture notes. In particular, these authors showed the well-posedness of an associated Cauchy problem which we will now describe in more detail and which will be the main topic of the present article.…”

“…In this article we are concerned with the case, that ϕ is a parallel spinor with a light-like Dirac-current V ϕ . This problem was studied in [10], [30], and [29].…”

Parallel spinor flows on three-dimensional Cauchy hypersurfaces

“…In references [1,18] it was proven that the Cauchy problem posed by a parallel spinor is well-posed, implying that every parallel Cauchy pair admits a Lorentzian development carrying a parallel spinor and hence a parallel spinor flow. However, since a Lorentzian metric admitting a parallel spinor is not necessarily flat, it might not be possible to evolve a constrained Ricci flat parallel Cauchy pair (e, Θ) in such a way that both the Ricci-flatness condition and the existence of a parallel spinor are guaranteed.…”

“…This paper is devoted to the study of the evolution problem posed by a parallel real and irreducible spinor field defined on a globally hyperbolic Lorentzian four-manifold (M, g). This problem is well-posed by the results of Leistner and Lichewski, who proved the statement in arbitrary dimension [19,18]. Existence of a parallel spinor field is obstructed since it implies (M, g) to be a solution of Einstein equations with pure radiation type of energy momentum tensor [20].…”