Spinor calculus on 5-dimensional spacetimes

Penrose's spinor calculus of 4-dimensional Lorentzian geometry is extended to the case of 5-dimensional Lorentzian geometry. Such fruitful ideas in Penrose's spinor calculus as the spin covariant derivative, the curvature spinors or the definition of the spin coefficients on a spin frame can be carried over to the spinor calculus in 5-dimensional Lorentzian geometry. The algebraic and differential properties of the curvature spinors are studied in detail and as an application we extend the well-known 4-dimensional Newman-Penrose formalism to a 5-dimensional spacetime.Comment: Convention mismatch and minor typos fixed. To appear in Journal of Mathematical Physic

A Note on Non-compact Cauchy surface

It is shown that if a space-time has non-compact Cauchy surface, then its topological, differentiable, and causal structure are completely determined by a class of compact subsets of its Cauchy surface. Since causal structure determines its topological, differentiable, and conformal structure of space-time, this gives a natural way to encode the corresponding structures into its Cauchy surface

Discrete Group Actions on Spacetimes: Causality Conditions and the Causal Boundary

Suppose a spacetime $M$ is a quotient of a spacetime $V$ by a discrete group of isometries. It is shown how causality conditions in the two spacetimes are related, and how can one learn about the future causal boundary on $M$ by studying structures in $V$. The relations between the two are particularly simple (the boundary of the quotient is the quotient of the boundary) if both $V$ and $M$ have spacelike future boundaries and if it is known that the quotient of the future completion of $V$ is past-distinguishing. (That last assumption is automatic in the case of $M$ being multi-warped.)Comment: 32 page

Causal Relationship: a new tool for the causal characterization of Lorentzian manifolds

We define and study a new kind of relation between two diffeomorphic Lorentzian manifolds called {\em causal relation}, which is any diffeomorphism characterized by mapping every causal vector of the first manifold onto a causal vector of the second. We perform a thorough study of the mathematical properties of causal relations and prove in particular that two given Lorentzian manifolds (say $V$ and $W$) may be causally related only in one direction (say from $V$ to $W$, but not from $W$ to $V$). This leads us to the concept of causally equivalent (or {\em isocausal} in short) Lorentzian manifolds as those mutually causally related. This concept is more general and of a more basic nature than the conformal relationship, because we prove the remarkable result that a conformal relation \f is characterized by the fact of being a causal relation of the {\em particular} kind in which both \f and \f^{-1} are causal relations. For isocausal Lorentzian manifolds there are one-to-one correspondences, which sometimes are non-trivial, between several classes of their respective future (and past) objects. A more important feature of isocausal Lorentzian manifolds is that they satisfy the same causality constraints. This indicates that the causal equivalence provides a possible characterization of the {\it basic causal structure}, in the sense of mutual causal compatibility, for Lorentzian manifolds. Thus, we introduce a partial order for the equivalence classes of isocausal Lorentzian manifolds providing a classification of spacetimes in terms of their causal properties, and a classification of all the causal structures that a given fixed manifold can have. A full abstract inside the paper.Comment: 47 pages, 10 figures. Version to appear in Classical and Quantum Gravit

On the invariant causal characterization of singularities in spherically symmetric spacetimes

The causal character of singularities is often studied in relation to the existence of naked singularities and the subsequent possible violation of the cosmic censorship conjecture. Generally one constructs a model in the framework of General Relativity described in some specific coordinates and finds an ad hoc procedure to analyze the character of the singularity. In this article we show that the causal character of the zero-areal-radius (R=0) singularity in spherically symmetric models is related with some specific invariants. In this way, if some assumptions are satisfied, one can ascertain the causal character of the singularity algorithmically through the computation of these invariants and, therefore, independently of the coordinates used in the model.Comment: A misprint corrected in Theor. 4.1 /Cor. 4.

On the construction of a geometric invariant measuring the deviation from Kerr data

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation ---the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant ---however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte

The causal boundary of wave-type spacetimes

A complete and systematic approach to compute the causal boundary of wave-type spacetimes is carried out. The case of a 1-dimensional boundary is specially analyzed and its critical appearance in pp-wave type spacetimes is emphasized. In particular, the corresponding results obtained in the framework of the AdS/CFT correspondence for holography on the boundary, are reinterpreted and very widely generalized. Technically, a recent new definition of causal boundary is used and stressed. Moreover, a set of mathematical tools is introduced (analytical functional approach, Sturm-Liouville theory, Fermat-type arrival time, Busemann-type functions).Comment: 41 pages, 1 table. Included 4 new figures, and some small modifications. To appear in JHE

Bi-conformal vector fields and their applications

We introduce the concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric \G to be scaled by different conformal factors. In particular, we study their infinitesimal version, called bi-conformal vector fields. We show the differential conditions characterizing them in terms of a "square root" of the metric, or equivalently of two complementary orthogonal projectors. Keeping these fixed, the set of bi-conformal vector fields is a Lie algebra which can be finite or infinite dimensional according to the dimensionality of the projectors. We determine (i) when an infinite-dimensional case is feasible and its properties, and (ii) a normal system for the generators in the finite-dimensional case. Its integrability conditions are also analyzed, which in particular provides the maximum number of linearly independent solutions. We identify the corresponding maximal spaces, and show a necessary geometric condition for a metric tensor to be a double-twisted product. More general ``breakable'' spaces are briefly considered. Many known symmetries are included, such as conformal Killing vectors, Kerr-Schild vector fields, kinematic self-similarity, causal symmetries, and rigid motions.Comment: Replaced version with some changes in the terminology and a new theorem. To appear in Classical and Quantum Gravit

Causal Topology in Future and Past Distinguishing Spacetimes

The causal structure of a strongly causal spacetime is particularly well endowed. Not only does it determine the conformal spacetime geometry when the spacetime dimension n >2, as shown by Malament and Hawking-King-McCarthy (MHKM), but also the manifold dimension. The MHKM result, however, applies more generally to spacetimes satisfying the weaker causality condition of future and past distinguishability (FPD), and it is an important question whether the causal structure of such spacetimes can determine the manifold dimension. In this work we show that the answer to this question is in the affirmative. We investigate the properties of future or past distinguishing spacetimes and show that their causal structures determine the manifold dimension. This gives a non-trivial generalisation of the MHKM theorem and suggests that there is a causal topology for FPD spacetimes which encodes manifold dimension and which is strictly finer than the Alexandrov topology. We show that such a causal topology does exist. We construct it using a convergence criterion based on sequences of "chain-intervals" which are the causal analogs of null geodesic segments. We show that when the region of strong causality violation satisfies a local achronality condition, this topology is equivalent to the manifold topology in an FPD spacetime.Comment: 20 pages, 4 figures. Minor revisions. In particular, (i) terminology in one of the Lemmas corrected, (ii) numbering of Lemmas, Theorems, etc. uniformised. To appear in Classical and Quantum Gravit