Computability of the causal boundary by using isocausality

Recently, a new viewpoint on the classical c-boundary in Mathematical Relativity has been developed, the relations of this boundary with the conformal one and other classical boundaries have been analyzed, and its computation in some classes of spacetimes, as the standard stationary ones, has been carried out. In the present paper, we consider the notion of isocausality given by Garc\'ia-Parrado and Senovilla, and introduce a framework to carry out isocausal comparisons with standard stationary spacetimes. As a consequence, the qualitative behavior of the c-boundary (at the three levels: point set, chronology and topology) of a wide class of spacetimes, is obtained.Comment: 44 pages, 5 Figures, latex. Version with minor changes and the inclusion of Figure

On the completeness of impulsive gravitational wave space-times

We consider a class of impulsive gravitational wave space-times, which generalize impulsive pp-waves. They are of the form $M=N\times\mathbb{R}^2_1$, where $(N,h)$ is a Riemannian manifold of arbitrary dimension and $M$ carries the line element $ds^2=dh^2+ 2dudv+f(x)\delta(u)du^2$ with $dh^2$ the line element of $N$ and $\delta$ the Dirac measure. We prove a completeness result for such space-times $M$ with complete Riemannian part $N$.Comment: 13 pages, minor changes suggested by the referee

Isocausal spacetimes may have different causal boundaries

We construct an example which shows that two isocausal spacetimes, in the sense introduced by Garc\'ia-Parrado and Senovilla, may have c-boundaries which are not equal (more precisely, not equivalent, as no bijection between the completions can preserve all the binary relations induced by causality). This example also suggests that isocausality can be useful for the understanding and computation of the c-boundary.Comment: Minor modifications, including the title, which matches now with the published version. 12 pages, 3 figure

The Generalized Jacobi Equation

The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analyzed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed $2^{-1/2}c\approx 0.7 c$ is pointed out. The astrophysical implications of this result for the terminal speed of a relativistic jet is briefly explored.Comment: LaTeX file, 4 PS figures, 28 pages, revised version, accepted for publication in Classical and Quantum Gravit

On the Alexandrov Topology of sub-Lorentzian Manifolds

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which is linked to the property of strong causality of a space-time. We studied three possible ways to define the Alexandrov topology on sub-Lorentzian manifolds, which usually give different topologies, but agree in the Lorentzian case. We investigated their relationships to each other and the manifold's original topology and their link to causality.Comment: 20 page

A Causal Order for Spacetimes with C0C^0 Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves

We recast the tools of ``global causal analysis'' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call $K^+$, and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime \M which is free of causal cycles, one may define a causal curve simply as a compact connected subset of \M which is linearly ordered by $K^+$. Our definitions all make sense for arbitrary $C^0$ metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general $C^0$ metric, the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive energy theorem, and may also prove useful in the study of topology change. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry.Comment: Two small revisions to accomodate errors brought to our attention by R.S. Garcia. No change to chief results. 33 page

Spacetime Splitting, Admissible Coordinates and Causality

To confront relativity theory with observation, it is necessary to split spacetime into its temporal and spatial components. The (1+3) timelike threading approach involves restrictions on the gravitational potentials $(g_{\mu \nu})$, while the (3+1) spacelike slicing approach involves restrictions on $(g^{\mu \nu})$. These latter coordinate conditions protect chronology within any such coordinate patch. While the threading coordinate conditions can be naturally integrated into the structure of Lorentzian geometry and constitute the standard coordinate conditions in general relativity, this circumstance does not extend to the slicing coordinate conditions. We explore the influence of chronology violation on wave motion. In particular, we consider the propagation of radiation parallel to the rotation axis of stationary G\"odel-type universes characterized by parameters $\eta > 0$ and $\lambda > 0$ such that for $\eta 1$) chronology is protected (violated). We show that in the WKB approximation such waves can freely propagate only when chronology is protected.Comment: 25 pages, 3 figures; v2: minor typos corrected, accepted for publication in Phys. Rev.

Hyperfast Interstellar Travel in General Relativity

The problem is discussed of whether a traveller can reach a remote object and return back sooner than a photon would when taken into account that the traveller can partly control the geometry of his world. It is argued that under some reasonable assumptions in globally hyperbolic spacetimes the traveller cannot hasten reaching the destination. Nevertheless, it is perhaps possible for him to make an arbitrarily long round-trip within an arbitrarily short (from the point of view of a terrestrial observer) time.Comment: The final version, close to (but better than) what will be published in Phys. Rev. D. The explanatory part is made more detaile

Morse index and causal continuity. A criterion for topology change in quantum gravity

Studies in 1+1 dimensions suggest that causally discontinuous topology changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n-1 Morse points in topology changing spacetimes built from Morse functions. We establish a weaker form of this conjecture. Namely, if a Morse function f on a compact cobordism has critical points of index 1 or n-1, then all the Morse geometries associated with f are causally discontinuous, while if f has no critical points of index 1 or n-1, then there exist associated Morse geometries which are causally continuous.Comment: Latex, 20 pages, 3 figure