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Register a new userConstruction of initial data sets for Lorentzian manifolds with lightlike parallel spinors
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Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on $M$ with a parallel spinor gives rise to a solution of the constraint equations on $Mtimes (a,b)$ (resp. $Mtimes S^1$).
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Let $(overline M,overline g)$ be a time- and space-oriented Lorentzian spin manifold, and let $M$ be a compact spacelike hypersurface of $overline M$ with induced Riemannian metric $g$ and second fundamental form $K$. If $(overline M,overline g)$ satisfies the dominant energy condition in a strict sense, then the Dirac--Witten operator of $Msubseteq overline M$ is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial on $M$ satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchins $alpha$-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac--Witten operator may be non-invertible, and we will study the kernel of this operator in this case. We will show that the kernel may only be non-trivial if $pi_1(M)$ is virtually solvable of derived length at most $2$. This allows to extend the index theoretical methods to spaces of initial data, satisfying the dominant energy condition in the weak sense. We will show further that a spinor $phi$ is in the kernel of the Dirac--Witten operator on $(M,g,K)$ if and only if $(M,g,K,phi)$ admits an extension to a Lorentzian manifold $(overline N,overline h)$ with parallel spinor $barphi$ such that $M$ is a Cauchy hypersurface of $(overline N,overline h)$, such that $g$ and $K$ are the induced metric and second fundamental form of $M$, respectively, and $phi$ is the restriction of $barphi$ to $M$.
In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.
We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The r^ole of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT$(k)$-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.
On a time-oriented Lorentzian manifold $(M,g)$ with non-empty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets $Ssubset M$ of sources is uniquely determined by measurements of the intersection of future light cones from points in $S$ with a fixed open subset of the boundary of $M$; here, light rays are reflected at $partial M$ according to Snells law. Our proof is constructive, and allows for interior conjugate points as well as multiply reflected and self-intersecting light cones.
The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.
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