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.Comment: 11 pages, 5 figures. v2: a brief description of the principal bibliography has been adde

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 associated with the emission coordinates are also outlined.Comment: 20 pages; 1 figur

Relativistic Positioning Systems: The Emission Coordinates

This paper introduces some general properties of the gravitational metric and the natural basis of vectors and covectors in 4-dimensional emission coordinates. Emission coordinates are a class of space-time coordinates defined and generated by 4 emitters (satellites) broadcasting their proper time by means of electromagnetic signals. They are a constitutive ingredient of the simplest conceivable relativistic positioning systems. Their study is aimed to develop a theory of these positioning systems, based on the framework and concepts of general relativity, as opposed to introducing `relativistic effects' in a classical framework. In particular, we characterize the causal character of the coordinate vectors, covectors and 2-planes, which are of an unusual type. We obtain the inequality conditions for the contravariant metric to be Lorentzian, and the non-trivial and unexpected identities satisfied by the angles formed by each pair of natural vectors. We also prove that the metric can be naturally split in such a way that there appear 2 parameters (scalar functions) dependent exclusively on the trajectory of the emitters, hence independent of the time broadcast, and 4 parameters, one for each emitter, scaling linearly with the time broadcast by the corresponding satellite, hence independent of the others.Comment: 13 pages, 3 figures. Only format changed for a new submission. Submitted to Class. Quantum Gra

Painlev\'e-Gullstrand synchronizations in spherical symmetry

A Painlev\'e-Gullstrand synchronization is a slicing of the space-time by a family of flat spacelike 3-surfaces. For spherically symmetric space-times, we show that a Painlev\'e-Gullstrand synchronization only exists in the region where $(dr)^2 \leq 1$, $r$ being the curvature radius of the isometry group orbits ($2$-spheres). This condition says 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 Painlev\'e-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 Einstein's equations in a coordinate system adapted to a Painlev\'e-Gullstrand synchronization, the Schwarzschild solution is directly obtained in a whole coordinate domain that includes the horizon and both its interior and exterior regions.Comment: 16 page