Juan Antonio Morales-Lladosa

Positioning systems in Minkowski space-time: Bifurcation problem and observational data

In the framework of relativistic positioning systems in Minkowski space-time, the determination of the inertial coordinates of a user involves the {\em bifurcation problem} (which is the indeterminate location of a pair of different events receiving the same emission coordinates). To solve it, in addition to the user emission coordinates and the emitter positions in inertial coordinates, it may happen that the user needs to know {\em independently} the orientation of its emission coordinates. Assuming that the user may observe the relative positions of the four emitters on its celestial sphere, an observational rule to determine this orientation is presented. The bifurcation problem is thus solved by applying this observational rule, and consequently, {\em all} of the parameters in the general expression of the coordinate transformation from emission coordinates to inertial ones may be computed from the data received by the user of the relativistic positioning system.

Intrinsic vanishing of energy and momenta in a universe

We present a new approach to the question of properly defining energy and momenta for non asymptotically Minkowskian spaces in General Relativity, in the case where these energy and momenta are conserved. In order to do this, we first prove that there always exist some special Gauss coordinates for which the conserved linear and angular 3-momenta intrinsically vanish. This allows us to consider the case of creatable universes (the universes whose proper 4-momenta vanish) in a consistent way, which is the main interest of the paper. When applied to the Friedmann-Lemaître-Robertson-Walker case, perturbed or not, our formalism leads to previous results, according to most literature on the subject. Some future work that should be done is mentioned.

Painlevé–Gullstrand synchronizations in spherical symmetry

A Painleve–Gullstrand synchronization is a slicing of the spacetime by a family of flat space-like 3-surfaces. For spherically symmetric spacetimes, we show that a Painleve–Gullstrand synchronization only exists in the region where (dr)2 ≤ 1, r being the curvature radius of the isometry group orbits (2-spheres). This condition states that the Misner–Sharp gravitational energy of these 2-spheres is not negative and has an intrinsic meaning in terms of the norm of the mean extrinsic curvature vector. It also provides an algebraic inequality involving the Weyl curvature scalar and the Ricci eigenvalues. We prove that the energy and momentum densities associated with the Weinberg complex of a Painleve–Gullstrand slice vanish in these curvature coordinates, and we give a new interpretation of these slices by using semi-metric Newtonian connections. It is also outlined that, by solving the vacuum Einsteins equations in a coordinate system adapted to a Painleve–Gullstrand synchronization, the Schwarzschild solution is directly obtained in a whole coordinate domain that includes the horizon and both its interior and exterior regions.

Maximal slicings in spherical symmetry: Local existence and construction

We show that any spherically symmetric spacetime locally admits a maximal space-like slicing and we give a procedure allowing its construction. The designed construction procedure is based on purely geometrical arguments and, in practice, leads to solve a decoupled system of first-order quasi-linear partial differential equations. We have explicitly built up maximal foliations in Minkowski and Friedmann spacetimes. Our approach admits further generalizations and efficient computational implementation. As by-product, we suggest some applications of our work in the task of calibrating numerical relativity complex codes, usually written in Cartesian coordinates.

Intrinsic energy of Lemaı̂tre-Tolman-Bondi models and cosmological implications

Recently, some Lema{\^{\i}}tre-Tolman-Bondi metrics have been considered as models alternative to the dark energy within the Friedmann-Lema{\^{\i}}tre-Robertson-Walker universes. The vanishing of the intrinsic energy of these metrics is examined since such a vanishing, in the present case and in general, could be interpreted as a necessary condition to consider the possibility of the quantum creation of a metric. More specifically, this vanishing is examined in the particular case where the Lema{\^{\i}}tre-Tolman-Bondi metrics behave asymptotically like a Friedmann-Lema{\^{\i}}tre-Robertson-Walker universe. Finally, we deal with a particular model ruled out after being confronted with cosmic observations. In a minimal agreement with this negative result, leaving aside an unstable case, the value of the intrinsic energy of this particular model does not vanish and becomes in fact minus infinite.

On the uniqueness of the space-time energy in General Relativity: the illuminating case of the Schwarzschild metric

The case of asymptotic Minkowskian space-times is considered. A special class of asymptotic rectilinear coordinates at the spatial infinity, related to a specific system of free falling observers, is chosen. This choice is applied in particular to the Schwarzschild metric, obtaining a vanishing energy for this space-time. This result is compared with the result of some known theorems on the uniqueness of the energy of any asymptotic Minkowskian space, showing that there is no contradiction between both results, the differences becoming from the use of coordinates with different operational meanings. The suitability of Gauss coordinates when defining an intrinsic energy is considered and it is finally concluded that a Schwarzschild metric is a particular case of space-times with vanishing intrinsic 4-momenta.

From emission to inertial coordinates: an analytical approach

In a previous work [Class. Quantum Grav. 27 (2010) 065013] relativistic positioning systems in Minkowski space-time have been studied, and the transformation from emission to inertial coordinates have been obtained for an arbitrary configuration of the emitters. The formula giving this transformation applies in all the emission coordinate region and involves the orientation of the positioning system (the Jacobian sign of the map which gives the emission coordinates of an event). Nevertheless, there exists an inherent limitation on the applicability of this formula: only the users in the central region of a positioning system can obtain the orientation from the sole emission data. Here an observational method to determine the orientation of a relativistic positioning system is presented. In this procedure, a certain additional information allows any user to obtain its inertial coordinates, irrespectively of its location in the emission region of the positioning system.

Flat synchronizations in spherically symmetric space-times

It is well known that the Schwarzschild space-time admits a spacelike slicing by flat instants and that the metric is regular at the horizon in the associated adapted coordinates (Painleve-Gullstrand metric form). We consider this type of flat slicings in an arbitrary spherically symmetric space-time. The condition ensuring its existence is analyzed, and then, we prove that, for any spherically symmetric flat slicing, the densities of the Weinberg momenta vanish. Finally, we deduce the Schwarzschild solution in the extended Painleve-Gullstrand-Lemaitre metric form by considering the coordinate decomposition of the vacuum Einstein equations with respect to a flat spacelike slicing.

Geodesic family of spherical instantons and cosmic quantum creation

The Einstein field equations for any spherically symmetric metric and a geodesic perfect fluid source are cast in a canonical simple form, both for Lorentzian metrics and for instantons. Both kinds of metrics are explicitly written for the Lemaître–Tolman–Bondi family and for a general