Positioning systems in Minkowski space-time: Bifurcation problem and observational data

TL;DR

This paper addresses the bifurcation problem in relativistic Minkowski space-time positioning systems by proposing an observational rule that allows users to determine their orientation and accurately compute inertial coordinates from emission data.

Contribution

It introduces an observational method for users to resolve the bifurcation problem and determine all parameters needed for coordinate transformation in relativistic positioning systems.

Findings

The observational rule effectively solves the bifurcation problem.

All transformation parameters can be derived from received data.

The method enhances the accuracy of relativistic positioning systems.

Abstract

In the framework of relativistic positioning systems in Minkowski space-time, the determination of the inertial coordinates of a user involves the {\em bifurcation problem} (which is the indeterminate location of a pair of different events receiving the same emission coordinates). To solve it, in addition to the user emission coordinates and the emitter positions in inertial coordinates, it may happen that the user needs to know {\em independently} the orientation of its emission coordinates. Assuming that the user may observe the relative positions of the four emitters on its celestial sphere, an observational rule to determine this orientation is presented. The bifurcation problem is thus solved by applying this observational rule, and consequently, {\em all} of the parameters in the general expression of the coordinate transformation from emission coordinates to inertial ones may be computed from the data received by the user of the relativistic positioning system.