Relativistic Positioning Systems: The Emission Coordinates

Abstract

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.

13 pages, 3 figures. Only format changed for a new submission. Submitted to Class. Quantum Grav

References in corpus (4)

• World function and time transfer: general post-Minkowskian expansions
• Two-dimensional approach to relativistic positioning systems
• Positioning with stationary emitters in a two-dimensional space-time
• The covariance of GPS coordinates and frames

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• Using global positioning systems to test extensions of General Relativity
• Mapping Cartesian Coordinates into Emission Coordinates: some Toy Models
• The Relativistic framework of Positioning systems
• Mapping the spacetime metric with GNSS: a preliminary study
• Equivalent Generic Forms for Metric Fields Yielded by Relativistic Positioning Systems
• Conformal proper times according to the Woodhouse causal axiomatics of relativistic spacetimes
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