Causal symmetries

Based on the recent work \cite{PII} we put forward a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid which contains as its maximal subgroup the set of conformal transformations. We find the necessary and sufficient conditions for a vector field \xiv to be the infinitesimal generator of a one-parameter submonoid of pure causal symmetries. We speculate about possible applications to gravitation theory by means of some relevant examples.Comment: LaTeX2e file with CQG templates. 8 pages and no figures. Submitted to Classical and Quantum gravit

Trapped surfaces, horizons and exact solutions in higher dimensions

A very simple criterion to ascertain if (D-2)-surfaces are trapped in arbitrary D-dimensional Lorentzian manifolds is given. The result is purely geometric, independent of the particular gravitational theory, of any field equations or of any other conditions. Many physical applications arise, a few shown here: a definition of general horizon, which reduces to the standard one in black holes/rings and other known cases; the classification of solutions with a (D-2)-dimensional abelian group of motions and the invariance of the trapping under simple dimensional reductions of the Kaluza-Klein/string/M-theory type. Finally, a stronger result involving closed trapped surfaces is presented. It provides in particular a simple sufficient condition for their absence.Comment: 7 pages, no figures, final version to appear in Class. Quantum Gra

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

Conserved superenergy currents

We exploit once again the analogy between the energy-momentum tensor and the so-called ``superenergy'' tensors in order to build conserved currents in the presence of Killing vectors. First of all, we derive the divergence-free property of the gravitational superenergy currents under very general circumstances, even if the superenergy tensor is not divergence-free itself. The associated conserved quantities are explicitly computed for the Reissner-Nordstrom and Schwarzschild solutions. The remaining cases, when the above currents are not conserved, lead to the possibility of an interchange of some superenergy quantities between the gravitational and other physical fields in such a manner that the total, mixed, current may be conserved. Actually, this possibility has been recently proved to hold for the Einstein-Klein-Gordon system of field equations. By using an adequate family of known exact solutions, we present explicit and completely non-obvious examples of such mixed conserved currents.Comment: LaTeX, 19 pages; improved version adding new content to the second section and some minor correction