Relativistic positioning systems: the emission coordinates

International audienceThis paper introduces some general properties of the gravitational metric and the natural basis of vectors and covectors in four-dimensional emission coordinates. Emission coordinates are a class of spacetime coordinates defined and generated by four 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 two parameters (scalar functions) dependent exclusively on the trajectory of the emitters, hence independent of the broadcasted time, and four parameters, one for each emitter, scaling linearly with the time broadcasted by the corresponding satellite, hence independent of the others