Relativistic Positioning Systems: The Emission Coordinates

TL;DR

This paper explores the properties of emission coordinates in relativistic positioning systems, developing a theoretical framework based on general relativity to characterize their geometric and causal structure.

Contribution

It introduces a comprehensive theory of emission coordinates, analyzing their metric properties, causal character, and parameter dependencies within a relativistic context.

Findings

Characterization of the causal nature of coordinate vectors and covectors

Derivation of conditions for the metric to be Lorentzian

Identification of parameters dependent on emitter trajectories and broadcasts

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.