Conserved Matter Superenergy Currents for Hypersurface Orthogonal Killing Vectors

We show that for hypersurface orthogonal Killing vectors, the corresponding Chevreton superenergy currents will be conserved and proportional to the Killing vectors. This holds for four-dimensional Einstein-Maxwell spacetimes with an electromagnetic field that is sourcefree and inherits the symmetry of the spacetime. A similar result also holds for the trace of the Chevreton tensor. The corresponding Bel currents have previously been proven to be conserved and our result can be seen as giving further support to the concept of conserved mixed superenergy currents. The analogous case for a scalar field has also previously been proven to give conserved currents and we show, for completeness, that these currents also are proportional to the Killing vectors.Comment: 13 page

Axial symmetry and conformal Killing vectors

Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others mush vanish or else the symmetry is larger than that originally considered. The results are completely general and do not depend on Einstein's equations or any particular matter content.Comment: 15 pages, Latex, no figure

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

Symmetry-preserving matchings

In the literature, the matchings between spacetimes have been most of the times implicitly assumed to preserve some of the symmetries of the problem involved. But no definition for this kind of matching was given until recently. Loosely speaking, the matching hypersurface is restricted to be tangent to the orbits of a desired local group of symmetries admitted at both sides of the matching and thus admitted by the whole matched spacetime. This general definition is shown to lead to conditions on the properties of the preserved groups. First, the algebraic type of the preserved group must be kept at both sides of the matching hypersurface. Secondly, the orthogonal transivity of two-dimensional conformal (in particular isometry) groups is shown to be preserved (in a way made precise below) on the matching hypersurface. This result has in particular direct implications on the studies of axially symmetric isolated bodies in equilibrium in General Relativity, by making up the first condition that determines the suitability of convective interiors to be matched to vacuum exteriors. The definition and most of the results presented in this paper do not depend on the dimension of the manifolds involved nor the signature of the metric, and their applicability to other situations and other higher dimensional theories is manifest.Comment: LaTeX, 19 page

Dynamical laws of superenergy in General Relativity

The Bel and Bel-Robinson tensors were introduced nearly fifty years ago in an attempt to generalize to gravitation the energy-momentum tensor of electromagnetism. This generalization was successful from the mathematical point of view because these tensors share mathematical properties which are remarkably similar to those of the energy-momentum tensor of electromagnetism. However, the physical role of these tensors in General Relativity has remained obscure and no interpretation has achieved wide acceptance. In principle, they cannot represent {\em energy} and the term {\em superenergy} has been coined for the hypothetical physical magnitude lying behind them. In this work we try to shed light on the true physical meaning of {\em superenergy} by following the same procedure which enables us to give an interpretation of the electromagnetic energy. This procedure consists in performing an orthogonal splitting of the Bel and Bel-Robinson tensors and analysing the different parts resulting from the splitting. In the electromagnetic case such splitting gives rise to the electromagnetic {\em energy density}, the Poynting vector and the electromagnetic stress tensor, each of them having a precise physical interpretation which is deduced from the {\em dynamical laws} of electromagnetism (Poynting theorem). The full orthogonal splitting of the Bel and Bel-Robinson tensors is more complex but, as expected, similarities with electromagnetism are present. Also the covariant divergence of the Bel tensor is analogous to the covariant divergence of the electromagnetic energy-momentum tensor and the orthogonal splitting of the former is found. The ensuing {\em equations} are to the superenergy what the Poynting theorem is to electromagnetism. See paper for full abstract.Comment: 27 pages, no figures. Typos corrected, section 9 suppressed and more acknowledgments added. To appear in Classical and Quantum Gravit

Kerr-Schild Symmetries

We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any n-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixed. The properties of these Lie algebras are briefly analyzed and we show that they are generically finite-dimensional but that they may have infinite dimension in some relevant situations. The most general vector fields of the above type are explicitly constructed for the following cases: any two-dimensional metric, the general spherically symmetric metric and deformation direction, and the flat metric with parallel or cylindrical deformation directions.Comment: 15 pages, no figures, LaTe

Review on exact and perturbative deformations of the Einstein-Straus model : uniqueness and rigidity results

The Einstein-Straus model consists of a Schwarzschild spherical vacuole in a Friedman-Lema^ tre-Robertson-Walker (FLRW) dust spacetime (with or without ). It constitutes the most widely accepted model to answer the question of the in uence of large scale (cosmological) dynamics on local systems. The conclusion drawn by the model is that there is no in uence from the cosmic background, since the spher- ical vacuole is static. Spherical generalizations to other interior matter models are commonly used in the construction of lumpy inhomogeneous cosmological models. On the other hand, the model has proven to be reluctant to admit non-spherical generalizations. In this review, we summarize the known uniqueness results for this model. These seem to indicate that the only reasonable and realistic non- spherical deformations of the Einstein-Straus model require perturbing the FLRW background. We review results about linear perturbations of the Einstein-Straus model, where the perturbations in the vacuole are assumed to be stationary and axially symmetric so as to describe regions (voids in particular) in which the matter has reached an equilibrium regime.M.M. acknowledges financial support under the projects FIS2012-30926 (MICINN) and P09-FQM-4496 (J. Andalucia-FEDER). F. M. thanks the warm hospitality from Instituto de Fisica, UERJ, Rio de Janeiro, Brasil, projects PTDC/MAT/108921/2008 and CERN/FP/123609/2011 from Fundacao para a Ciencia e a Tecnologia (FCT), as well as CMAT, Univ. Minho, for support through FEDER funds Programa Operacional Factores de Competitividade (COMPETE) and Portuguese Funds from FCT within the project PEst-C/MAT/UI0013/2011. R. V. thanks the kind hospitality from the Universidad de Salamanca, where parts of this work have been produced, and financial support from project IT592-13 of the Basque Government, and FIS2010-15492 from the MICINN

From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit

Petrou types D and II perfect-fluid solutions in generalized Kerr-Schild form.

Petrov types D and II perfect-fluid solutions are obtained starting from conformally flat perfect-fluid metrics and by using a generalized KerrSchild ansatz. Most of the Petrov type D metrics obtained have the property that the velocity of the fluid does not lie in the two-space defined by the principal null directions of the Weyl tensor. The properties of the perfect-fluid sources are studied. Finally, a detailed analysis of a new class of spherically symmetric static perfect-fluid metrics is given