Parallel Null Vector Research Articles

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

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.

On the Splitting Problem for Lorentzian Manifolds with an $${\mathbb {R}}$$ R -Action with Causal Orbits

We study the interplay between the global causal and geometric structures of a spacetime (M, g) and the features of a given smooth \({\mathbb {R}}\)-action \(\rho \) on M whose orbits are all causal curves, building on classic results about Lie group actions on manifolds described by Palais (Ann Math 73:295–323, 1961). Although the dynamics of such an action can be very hard to describe in general, simple restrictions on the causal structure of (M, g) can simplify this dynamics dramatically. In the first part of this paper, we prove that \(\rho \) is free and proper (so that M splits topologically) provided that (M, g) is strongly causal and \(\rho \) does not have what we call weakly ancestral pairs, a notion which admits a natural interpretation in terms of “cosmic censorship.” Accordingly, such condition holds automatically if (M, g) is globally hyperbolic. We also prove that M splits topologically if (M, g) is strongly causal and \(\rho \) is the flow of a complete conformal Killing null vector field. In the second part, we investigate the class of Brinkmann spacetimes, which can be regarded as null analogues of stationary spacetimes in which \(\rho \) is the flow of a complete parallel null vector field. Inspired by the geometric characterization of stationary spacetimes in terms of standard stationary ones (Javaloyes and Sanchez in Class Quantum Gravity 25:168001, 2008), we obtain an analogous geometric characterization of when a Brinkmann spacetime is isometric to a standard Brinkmann spacetime. This result naturally leads us to discuss a conjectural null analogue for Ricci-flat four-dimensional Brinkmann spacetimes of a celebrated rigidity theorem by Anderson (Ann Henri Poincare 1:977–994, 2000) and highlight its relation with a long-standing conjecture by Ehlers and Kundt (Gravitation: an introduction to current research. Wiley, New York, pp 49–101, 1962).

Symmetries of Lorentzian Three-Manifolds with Recurrent Curvature

Locally homogeneous Lorentzian three-manifolds with recurrect curvature are special examples of Walker manifolds, that is, they admit a parallel null vector field. We obtain a full classification of the symmetries of these spaces, with particular regard to symmetries related to their curvature: Ricci and matter collineations, curvature and Weyl collineations. Several results are given for the broader class of three-dimensional Walker manifolds.

Cauchy problems for Lorentzian manifolds with special holonomy

On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzian manifold with a parallel null vector. Similarly, every parallel null spinor on a Lorentzian manifold induces a generalised imaginary Killing spinor on a space-like hypersurface. Then, using the fact that a parallel spinor field induces a parallel vector field, we can apply the first result to prove: every real analytic Riemannian manifold carrying a real analytic, imaginary generalised Killing spinor can be extended to a Lorentzian manifold with a parallel null spinor. Finally, we give examples of geodesically complete Riemannian manifolds satisfying the constraint conditions.

Parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field

We classify parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field. Some families of semi-parallel surfaces are also described.

LORENTZIAN SYMMETRIC THREE-SPACES AND THE CLASSIFICATION OF THEIR PARALLEL SURFACES

We describe a global model for Lorentzian symmetric three-spaces admitting a parallel null vector field, and classify completely the surfaces with parallel second fundamental form in all Lorentzian symmetric three-spaces. Interesting differences arise with respect to the Riemannian case studied in [2]. Our results complete the classification of parallel surfaces in all three-dimensional Lorentzian homogeneous spaces.

Geometry of all supersymmetric four-dimensional 𝒩 = 1 supergravity backgrounds

We solve the Killing spinor equations of ${\cal N}=1$ supergravity, with four supercharges, coupled to any number of vector and scalar multiplets in all cases. We find that backgrounds with N=1 supersymmetry admit a null, integrable, Killing vector field. There are two classes of N=2 backgrounds. The spacetime in the first class admits a parallel null vector field and so it is a pp-wave. The spacetime of the other class admits three Killing vector fields, and a vector field that commutes with the three Killing directions. These backgrounds are of cohomogeneity one with homogenous sections either $\bR^{2,1}$ or $AdS_3$ and have an interpretation as domain walls. The N=3 backgrounds are locally maximally supersymmetric. There are N=3 backgrounds which arise as discrete identifications of maximally supersymmetric ones. The maximally supersymmetric backgrounds are locally isometric to either $\bR^{3,1}$ or $AdS_4$.

The G2 spinorial geometry of supersymmetric IIB backgrounds

We solve the Killing spinor equations of supersymmetric IIB backgrounds which admit one supersymmetry and the Killing spinor has stability subgroup G2 in Spin(9, 1) × U(1). We find that such backgrounds admit a timelike Killing vector field and the geometric structure of the spacetime reduces from Spin(9, 1) × U(1) to G2. We determine the type of G2 structure that the spacetime admits by computing the covariant derivatives of the spacetime forms associated with the Killing spinor bilinears. We also solve the Killing spinor equations of backgrounds with two supersymmetries and -invariant spinors, and four supersymmetries with - and with G2-invariant spinors. We show that the Killing spinor equations factorize in two sets, one involving the geometry and the 5-form flux, and the other the 3-form flux and the scalars. In the and cases, the spacetime admits a parallel null vector field and so the spacetime metric can be locally described in terms of Penrose coordinates adapted to the associated rotation free, null, geodesic congruence. The transverse space of the congruence is a Spin(7) and a SU(4) holonomy manifold, respectively. In the G2 case, all the fluxes vanish and the spacetime is the product of a three-dimensional Minkowski space with a holonomy G2 manifold.

Three-dimensional Lorentz manifolds admitting a parallel null vector field

Curvature properties of three-dimensional Lorentz manifolds admitting a parallel degenerate line field are examined. A complete characterization of those manifolds being locally symmetric or locally conformally flat is obtained. The results of this study show nice families of examples of such properties within the Lorentzian setting.

Nonuniqueness of the metric in Lorentzian manifolds

This paper is concerned with the correspondence between a Lorentzian metric and its Levi-Civita connection. Although each metric determines a unique compatible symmetric connection, it is possible for more than one metric to engender the same connection. This non-uniqueness is studied for metrics of arbitrary signature and for Lorentzian metrics is shown to arise either from a de Rham-Wu decomposition or a local parallel null vector field. A key ingredient in the analysis is the construct of a submersive connection in which a connection passes to a quotient space. Finally, two examples of metrics are given, the first of which shows that the metric may be non-unique even though a null vector field exists only locally

On wave solutions of gauge-invariant generalization of field theories with asymmetric fundamental tensor in a generalized peres space-time

The gauge invariant generalization of field theories with asymmetric fundamental tensor developed by Buchdahl has been considered and its plane wave-like solutions in the sense of Takeno are investigated in generalized Peres space-time, recently considered by the author. It has been shown that under certain conditions these solutions become identical with those of strong field equations of Einstein in the same space-time. It has been also shown that this space-time satisfying the field equations of Buchdahl admits a parallel null vector field and is gravitationally null which further, transforms to other well known forms of space-time under a new time coordinate Z=z-t.