0000000000102158
AUTHOR
Bartolomé Coll
Relativistic positioning systems: Perspectives and prospects
Relativistic positioning systems are interesting {\em technical objects} for applications around the Earth and in the Solar system. But above all else, they are basic {\em scientific objects} allowing developing relativity from its own concepts. Some past and future features of relativistic positioning systems, with special attention to the developments that they suggest for an {\em epistemic relativity} (relativistic experimental approach to physics), are analyzed. This includes {\em relativistic stereometry}, which, together with relativistic positioning systems, allows to introduce the general relativistic notion of (finite) {\em laboratory} (space-time region able to perform experiments…
Thermodynamic class II Szekeres-Szafron solutions. Singular models
A family of parabolic Szekeres-Szafron class II solutions in local thermal equilibrium is studied and their associated thermodynamics are obtained. The subfamily with the hydrodynamic behavior of a generic ideal gas (defined by the equation of state $p = k n \Theta$) results to be an inhomogeneous generalization of flat FLRW $\gamma$-law models. Three significative interpretations that follow on from the choice of three specific thermodynamic schemes are analyzed in depth. First, the generic ideal gas in local thermal equilibrium; this interpretation leads to an inhomogeneous temperature $\Theta$. Second, the thermodynamics with homogeneous temperature considered by Lima and Tiomno (CQG 6 1…
On the Leibniz bracket, the Schouten bracket and the Laplacian
International audience; 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.
Relativistic holonomic fluids
The notion of holonomic fluid in relativity is reconsidered. An intrinsic characterization of holonomic fluids, involving only the unit velocity, is given, showing that in spite of its dynamical appearance the notion of holonomic fluid is a kinematical notion. The relations between holonomic and thermodynamic perfect fluids are studied.
A hydrodynamic approach to the classical ideal gas
The necessary and sufficient condition for a conservative perfect fluid energy tensor to be the energetic evolution of a classical ideal gas is obtained. This condition forces the square of the speed of sound to have the form $c_s^2 = \frac{\gamma p}{\rho+p}$ in terms of the hydrodynamic quantities, energy density $\rho$ and pressure $p$, $\gamma$ being the (constant) adiabatic index. The {\em inverse problem} for this case is also solved, that is, the determination of all the fluids whose evolutions are represented by a conservative energy tensor endowed with the above expression of $c^2_s$, and it shows that these fluids are, and only are, those fulfilling a Poisson law. The relativistic …
On the geometry of Killing and conformal tensors
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…
Positioning in a flat two-dimensional space-time: the delay master equation
The basic theory on relativistic positioning systems in a two-dimensional space-time has been presented in two previous papers [Phys. Rev. D {\bf 73}, 084017 (2006); {\bf 74}, 104003 (2006)], where the possibility of making relativistic gravimetry with these systems has been analyzed by considering specific examples. Here we study generic relativistic positioning systems in the Minkowski plane. We analyze the information that can be obtained from the data received by a user of the positioning system. We show that the accelerations of the emitters and of the user along their trajectories are determined by the sole knowledge of the emitter positioning data and of the acceleration of only one …
On the relativistic compressibility conditions
The constraints imposed by the relativistic compressibility hypothesis on the square of the speed of sound in a medium are obtained. This result allows to obtain purely hydrodynamic conditions for the physical reality of a perfect energy tensor representing the energetic evolution of a perfect fluid in local thermal equilibrium. The results are applied to the paradigmatic case of the generic ideal gases. Then the physical reality of the ideal gas Stephani models is analyzed and the Rainich-like theory for ideal gas solutions is built.
Symmetric frames on Lorentzian spaces
Symmetric frames (those whose vectors are metrically indistinguishable) are studied both, from the algebraic and differential points of view. Symmetric frames which, in addition, remain indistinguishable for a given set of concomitants of the metric are analyzed, and the necessary and sufficient conditions for a space‐time to admit them are given. A new version of the cosmological principle then follows. Natural symmetric frames (induced by local charts) are also considered, and the space‐times admitting them are obtained.
Positioning systems in Minkowski space-time: from emission to inertial coordinates
The coordinate transformation between emission coordinates and inertial coordinates in Minkowski space-time is obtained for arbitrary configurations of the emitters. It appears that a positioning system always generates two different coordinate domains, namely, the front and the back emission coordinate domains. For both domains, the corresponding covariant expression of the transformation is explicitly given in terms of the emitter world-lines. This task requires the notion of orientation of an emitter configuration. The orientation is shown to be computable from the emission coordinates for the users of a `central' region of the front emission coordinate domain. Other space-time regions a…
Newtonian and relativistic emission coordinates
Emission coordinates are those generated by positioning systems. Positioning systems are physical systems constituted by four emitters broadcasting their respective times by means of sound or light signals. We analyze the incidence of the space-time causal structure on the construction of emission coordinates. The Newtonian case of four emitters at rest is analyzed and contrasted with the corresponding situation in special relativity.
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…
Intrinsic characterization of space‐time symmetric tensors
This paper essentially deals with the classification of a symmetric tensor on a four‐dimensional Lorentzian space. A method is given to find the algebraic type of such a tensor. A system of concomitants of the tensor is constructed, which allows one to know the causal character of the eigenspace corresponding to a given eigenvalue, and to obtain covariantly their eigenvectors. Some algebraic as well as differential applications are considered.
Local thermal equilibrium and ideal gas Stephani universes
The Stephani universes that can be interpreted as an ideal gas evolving in local thermal equilibrium are determined. Five classes of thermodynamic schemes are admissible, which give rise to five classes of regular models and three classes of singular models. No Stephani universes exist representing an exact solution to a classical ideal gas (one for which the internal energy is proportional to the temperature). But some Stephani universes may approximate a classical ideal gas at first order in the temperature: all of them are obtained. Finally, some features about the physical behavior of the models are pointed out.
Positioning with stationary emitters in a two-dimensional space-time
The basic elements of the relativistic positioning systems in a two-dimensional space-time have been introduced in a previous work [Phys. Rev. D {\bf 73}, 084017 (2006)] where geodesic positioning systems, constituted by two geodesic emitters, have been considered in a flat space-time. Here, we want to show in what precise senses positioning systems allow to make {\em relativistic gravimetry}. For this purpose, we consider stationary positioning systems, constituted by two uniformly accelerated emitters separated by a constant distance, in two different situations: absence of gravitational field (Minkowski plane) and presence of a gravitational mass (Schwarzschild plane). The physical coord…
Relativistic Positioning Systems in Flat Space-Time: The Location Problem
The location problem in relativistic positioning is considered in flat space-time. When two formal solutions are possible for a user (receiver) of the system, its true location may be obtained from a standard set of emission data extended with an observational rule. The covariant expression giving the location of the user in inertial coordinates is decomposed with respect to an inertial observer.
On the thermodynamics of inhomogeneous perfect fluid mixtures
It is shown that inhomogeneous Szekeres and Stephani universes exist corresponding to non-dissipative binary mixtures of perfect fluids in local thermal equilibrium. This result contradicts a recent statement by Z\'arate and Quevedo (2004 Class. Quantum Grav. {\bf 21} 197, {\it Preprint} gr-qc/0310087), which affirms that the only Szekeres and Stephani universes compatible with these fluids are the homogeneous Friedmann-Robertson-Walker models. Thus, contrarily to their conclusion, their thermodynamic scheme do not gives new indications of incompatibility between thermodynamics and relativity. Two of the points that have generated this error are commented.
Ideal Gas Stephani Universes
The Stephani Universes that can be interpreted as an ideal gas evolving in local thermal equilibrium are determined, and the method to obtain the associated thermodynamic schemes is given
Two-dimensional approach to relativistic positioning systems
A relativistic positioning system is a physical realization of a coordinate system consisting in four clocks in arbitrary motion broadcasting their proper times. The basic elements of the relativistic positioning systems are presented in the two-dimensional case. This simplified approach allows to explain and to analyze the properties and interest of these new systems. The positioning system defined by geodesic emitters in flat metric is developed in detail. The information that the data generated by a relativistic positioning system give on the space-time metric interval is analyzed, and the interest of these results in gravimetry is pointed out.
On Newtonian frames
In Newtonian space-time there exist four, and only four, causal classes of frames. Natural frames allow to extend this result to coordinate systems, so that coordinate systems may be also locally classified in four causal classes. These causal classes admit simple geometric descriptions and physical interpretations. For example, one can generate representatives of the four causal classes by means of the {\em linear synchronization group}. Of particular interest is the {\em local Solar time synchronization}, which reveals the limits of the frequent use of the concept of `causally oriented oordinate', such as that of `time-like coordinate'. Classical {\em positioning systems}, based in sound …
Sur les repères symmetriques lorentziens
Résumé.- Définition et propriétés des repères symétriques. Charactérisation des variétés admettant des repères symètriques naturels. Abstract.- Definition and properties of the symmetric frames. Characterization of the manifolds admitting holonomic symmetric frames.
Thermodynamic class II Szekeres–Szafron solutions. Regular models
In a recent paper (Coll {\em et al} 2019 {\it Class. Quantum Grav.} {\bf 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 ($p = \tilde{k} n \Theta$), and we analyze in depth two notable interpretations that follow on from the choice of two specific thermodyna…
Emission and null coordinates: geometrical properties and physical construction
A Relativistic Positioning System is defined by four clocks (emitters) broadcasting their proper time. Then, every event reached by the signals is naturally labeled by these four times which are the emission coordinates of this event. The coordinate hypersurfaces of the emission coordinates are the future light cones based on the emitter trajectories. For this reason the emission coordinates have been also named null coordinates or light coordinates. Nevertheless, other coordinate systems used in different relativistic contexts have the own right to be named null or light coordinates. Here we analyze when one can say that a coordinate is a null coordinate and when one can say that a coordin…
Comment to the reply to "On the thermodynamics of inhomogeneous perfect fluid mixtures"
In spite of a recent reply by Quevedo and Z\'arate (gr-qc/0403096), their assertion that their thermodynamic scheme for a perfect fluid binary mixture is incompatible with Szekeres and Stephani families of universes, except those of FRW ones, remains wrong.
Relativistic perfect fluids in local thermal equilibrium
Every evolution of a fluid is uniquely described by an energy tensor. But the converse is not true: an energy tensor may describe the evolution of different fluids. The problem of determining them is called here the {\em inverse problem}. This problem may admit unphysical or non-deterministic solutions. This paper is devoted to solve the inverse problem for perfect energy tensors in the class of perfect fluids evolving in local thermal equilibrium (l.t.e.). The starting point is a previous result (Coll and Ferrando in J Math Phys 30: 2918-2922, 1989) showing that thermodynamic fluids evolving in l.t.e. admit a purely hydrodynamic characterization. This characterization allows solving this i…
Comments on space–time signature
In terms of three signs associated to two vectors and to a 2-plane, a formula for the signature of any four-dimensional metric is given. In the process, a simple expression for the sign of the Lorentzian metric signature is obtained. The rela- tionship between these results and those already known are commented upon.
199 Causal Classes of Space-Time Frames
It is shown that from the causal point of view Minkowskian space-time admits 199, and only 199, different classes of frames.
Epistemic Relativity: An Experimental Approach to Physics
The recent concept of relativistic positioning system (RPS) has opened the possibility of making Relativity the general standard frame in which to state any physical problem, theoretical or experimental. Because the velocity of propagation of the information is finite, epistemic relativity proposes to integrate the physicist as a truly component of every physical problem, taking into account explicitly what information, when and where, the physicist is able to know. This leads naturally to the concept of relativistic stereometric system (RSS), allowing to measure the intrinsic properties of physical systems. Together, RPSs and RSSs complete the notion of laboratory in general relativity, al…