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A relativistic positioning system is a set of four emitters broadcasting their proper times by means of light signals. The four emitter times received at an event constitute the emission coordinates of the event. The covariant quantities associated with relativistic positioning systems are analysed relative to an observer in Minkowski space-time by splitting them in their relative space-like and time-like components. The location of a user in inertial coordinates from a standard set of emission data (emitted times and satellite trajectories) is solved in the underlying 3+1 formalism. The analytical location solution obtained by Kleusberg for the GPS system is recovered and interpreted in a Minkowskian context. [ABSTRACT FROM AUTHOR]

The necessary and sufficient conditions are obtained for a unit time-like vector field u to be the unit velocity of a divergence-free perfect fluid energy tensor. This plainly kinematic description of a conservative perfect fluid requires considering eighteen classes defined by differential concomitants of u. For each of these classes, we get the additional constraints that label the flow of a conservative energy tensor, and we obtain the pairs of functions { ρ , p } , energy density and pressure, which complete a solution to the conservation equations. [ABSTRACT FROM AUTHOR]

The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group of isometries of dimension r acting on s-dimensional orbits are obtained. These conditions are Intrinsic, Deductive, Explicit and ALgorithmic and they offer an IDEAL labeling that improves previously known invariant studies. [ABSTRACT FROM AUTHOR]

The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a transitive group of isometries are obtained. These conditions are intrinsic, deductive, explicit and algorithmic, and they offer an IDEAL labeling of these geometries. It is shown that the transitive action of the group naturally falls into an unfolding of some of the ten types in the Bianchi–Behr classification. Explicit conditions, depending on the Ricci tensor, are obtained that characterize all these types. [ABSTRACT FROM AUTHOR]

In a recent paper (Coll et al 2019 Class. Quantum Grav. 36 175004) we have studied a family of Szekeres–Szafron solutions of class II in local thermal equilibrium (singular models). In this paper we deal with a similar study for all other class II Szekeres–Szafron solutions without symmetries. These models in local thermal equilibrium (regular models) are analyzed and their associated thermodynamic schemes are obtained. In particular, we focus on the subfamily of solutions which are compatible with the generic ideal gas equation of state (), and we analyze in depth two notable interpretations that follow on from the choice of two specific thermodynamic schemes: firstly, as a generic ideal gas in local thermal equilibrium and, secondly, as a model having the homogeneous temperature of the FLRW limit. The models above are shown to fulfill the general necessary macroscopic requirements for physical reality (positivity of matter density, internal energy and temperature, energy conditions and compressibility conditions) in wide domains of the spacetime. [ABSTRACT FROM AUTHOR]

We extend our recent analysis (Ferrando J J and S'áez J A 2012 Class. Quantum Grav. 29 075012) on the Bel radiative gravitational fields to the super-energy radiative gravitational fields defined by García-Parrado (2008 Class. Quantum Grav. 25 015006). We give an intrinsic characterization of the new radiative fields and consider some distinguished classes of both radiative and nonradiative fields. Several super-energy inequalities are improved. [ABSTRACT FROM AUTHOR]

We give the necessary and sufficient conditions for a 3-metric to be the adapted spatial metric of a static vacuum solution. This work accomplishes for the degenerate cases the already known study for the regular ones (Bartnik and Tod 2006 Class. Quantum Grav. 23 569-71). [ABSTRACT FROM AUTHOR]

We analyze the symmetries and other invariant qualities of the D-metrics (type D aligned Einstein-Maxwell solutions with cosmological constant whose Debever null principal directions determine shear-free geodesic null congruences). We recover some properties and deduce new ones about their isometry group and about their quadratic first integrals of the geodesic equation, and we analyze when these invariant symmetries characterize the family of metrics. We show that the subfamily of the Kerr-NUT solutions are those admitting a Papapetrou field aligned with the Weyl tensor. [ABSTRACT FROM AUTHOR]

We show that Petrov type I vacuum solutions admitting a Killing vector whose Papapetrou field is aligned with a principal bivector of the Weyl tensor are the Kasner and Taub metrics, their counterpart with timelike orbits and their associated windmill-like solutions, as well as the Petrov homogeneous vacuum solution. We recover all these metrics by using an integration method based on an invariant classification which allows us to characterize every solution. In this way we obtain an intrinsic and explicit algorithm to identify them. [ABSTRACT FROM AUTHOR]

The second order Killing and conformal tensors are analyzed in terms of their spectral decomposition, and some properties of the eigenvalues and the eigenspaces are shown. When the tensor is of type I with only two different eigenvalues, the condition to be a Killing or a conformal tensor is characterized in terms of its underlying almost-product structure. A canonical expression for the metrics admitting these kinds of symmetries is also presented. The space–time cases 1+3 and 2+2 are analyzed in more detail. Starting from this approach to Killing and conformal tensors a geometric interpretation of some results on quadratic first integrals of the geodesic equation in vacuum Petrov-Bel type D solutions is offered. A generalization of these results to a wider family of type D space–times is also obtained. [ABSTRACT FROM AUTHOR]

The Leibniz bracket of an operator on a (graded) algebra is defined and some of its properties are studied. A basic theorem relating the Leibniz bracket of the commutator of two operators to the Leibniz bracket of them is obtained. Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie algebra structure. In particular, those algebras generated by the Leibniz bracket of the divergence and the Laplacian operators on the exterior algebra are considered, and the expression of the Laplacian for the product of two functions is generalized for arbitrary exterior forms. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We give a classification of the type D space–times based on the invariant differential properties of the Weyl principal structure. Our classification is established using tensorial invariants of the Weyl tensor and, consequently, besides its intrinsic nature, it is valid for the whole set of the type D metrics and it applies on both, vacuum and nonvacuum solutions. We consider the Cotton-zero type D metrics and we study the classes that are compatible with this condition. The subfamily of space–times with constant argument of the Weyl eigenvalue is analyzed in more detail by offering a canonical expression for the metric tensor and by giving a generalization of some results about the nonexistence of purely magnetic solutions. The usefulness of these results is illustrated in characterizing and classifying a family of Einstein–Maxwell solutions. Our approach permits us to give intrinsic and explicit conditions that label every metric, obtaining in this way an operational algorithm to detect them. In particular a characterization of the Reissner–Nordström metric is accomplished. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The conditions for a unit vector field to be the velocity of a relativistic barotropic perfect fluid are given. These conditions induce an eightfold classification of such fluids; for every class, the admissible barotropic variables are found. Some special cases, in particular polytropic fluids, are analyzed separately. [ABSTRACT FROM AUTHOR]

The conditions for a relativistic perfect fluid to admit a thermodynamic scheme are considered, and the necessary and sufficient requirements for a metric to define a thermodynamic perfect fluid space-time are given. [ABSTRACT FROM AUTHOR]

The relations on the electromagnetic field obtained by Teukolsky and Press for type D vacuum space-times are considered; these are four second-order equations in two complex components of the field with respect to a principal null tetrad. A rigorous geometric interpretation of these relations is given, showing the essential role played by the Maxwellian character of the basic null tetrad. It appears that, generically, the Teukolsky–Press relations are incomplete. Once completed, their generalizations to the general Maxwell equations (with source term) with respect to non-necessarily Maxwellian tetrads on arbitrary space-times are given. [ABSTRACT FROM AUTHOR]