Relativistic positioning: four-dimensional numerical approach in Minkowski space-time

TL;DR

This paper develops a numerical method for relativistic positioning in Minkowski space-time, analyzing satellite constellations and emission coordinates, and addressing the bifurcation problem in GPS-like systems.

Contribution

It introduces a four-dimensional numerical approach to relativistic positioning, applying it to satellite constellations and analyzing localization issues.

Findings

Detailed analysis of emission coordinate regions.

Implementation of a numerical code for event localization.

Discussion of the bifurcation problem in satellite positioning.

Abstract

We simulate the satellite constellations of two Global Navigation Satellite Systems: Galileo (EU) and GPS (USA). Satellite motions are described in the Schwarzschild space-time produced by an idealized spherically symmetric non rotating Earth. The trajectories are then circumferences centered at the same point as Earth. Photon motions are described in Minkowski space-time, where there is a well known relation, Coll, Ferrando & Morales-Lladosa (2010), between the emission and inertial coordinates of any event. Here, this relation is implemented in a numerical code, which is tested and applied. The first application is a detailed numerical four-dimensional analysis of the so-called emission coordinate region and co-region. In a second application, a GPS (Galileo) satellite is considered as the receiver and its emission coordinates are given by four Galileo (GPS) satellites. The bifurcation problem (double localization) in the positioning of the receiver satellite is then pointed out and discussed in detail.