Mapping Cartesian Coordinates into Emission Coordinates: some Toy Models

TL;DR

This paper explores how to explicitly map Cartesian coordinates to emission coordinates in relativistic positioning systems within Minkowski and slightly inhomogeneous space-times, providing models in 1+1 and 1+3 dimensions.

Contribution

It presents explicit mappings between Cartesian and emission coordinates for known emitter world-lines in relativistic space-times, including inhomogeneous models.

Findings

Explicit maps derived for Minkowski space-time

Maps extended to inhomogeneous space-times with small gravitational fields

Models developed in both 1+1 and 1+3 dimensions

Abstract

After briefly reviewing the relativistic approach to positioning systems based on the introduction of the emission coordinates, we show how explicit maps can be obtained between the Cartesian coordinates and the emission coordinates, for suitably chosen set of emitters, whose world-lines are supposed to be known by the users. We consider Minkowski space-time and the space-time where a small inhomogeineity is introduced (i.e. a small "gravitational" field), both in 1+1 and 1+3 dimensions.