Emission vs Fermi coordinates: applications to relativistic positioning systems

TL;DR

This paper develops a relativistic positioning system using emission coordinates applicable in general spacetimes, extending previous flat spacetime results, and offers an alternative to GPS that inherently accounts for relativistic effects.

Contribution

It introduces a 4D relativistic positioning system based on emission coordinates for general spacetimes, including Schwarzschild, with explicit mappings using the world function.

Findings

Coordinates are explicitly constructed for Schwarzschild spacetime.

The system provides a relativistic alternative to GPS.

Emission coordinates are extended from flat to curved spacetimes.

Abstract

A 4-dimensional relativistic positioning system for a general spacetime is constructed by using the so called "emission coordinates". The results apply in a small region around the world line of an accelerated observer carrying a Fermi triad, as described by the Fermi metric. In the case of a Schwarzschild spacetime modeling the gravitational field around the Earth and an observer at rest at a fixed spacetime point, these coordinates realize a relativistic positioning system alternative to the current GPS system. The latter is indeed essentially conceived as Newtonian, so that it necessarily needs taking into account at least the most important relativistic effects through Post-Newtonian corrections to work properly. Previous results concerning emission coordinates in flat spacetime are thus extended to this more general situation. Furthermore, the mapping between spacetime coordinates and emission coordinates is completely determined by means of the world function, which in the case of a Fermi metric can be explicitly obtained.