Positioning systems in Minkowski space-time: from emission to inertial coordinates

TL;DR

This paper derives the coordinate transformation between emission and inertial coordinates in Minkowski space-time for arbitrary emitter configurations, revealing dual domains and the role of emitter orientation.

Contribution

It provides explicit covariant transformations for emission to inertial coordinates in Minkowski space-time, including the concept of emitter orientation and domain distinctions.

Findings

Two distinct emission coordinate domains identified

Explicit covariant transformation formulas derived

Emitter orientation computable from emission coordinates

Abstract

The coordinate transformation between emission coordinates and inertial coordinates in Minkowski space-time is obtained for arbitrary configurations of the emitters. It appears that a positioning system always generates two different coordinate domains, namely, the front and the back emission coordinate domains. For both domains, the corresponding covariant expression of the transformation is explicitly given in terms of the emitter world-lines. This task requires the notion of orientation of an emitter configuration. The orientation is shown to be computable from the emission coordinates for the users of a `central' region of the front emission coordinate domain. Other space-time regions associated with the emission coordinates are also outlined.