Geometric approach to the definition of emission coordinates

TL;DR

This paper introduces a geometric method to define emission coordinates in flat spacetime, using properties of geodesic triangles, and provides an analytic solution applicable in various configurations.

Contribution

It presents a novel geometric approach to emission coordinates in flat spacetime, extending from 2D to 4D, with an explicit analytic solution for receiver positioning.

Findings

Coordinates can be defined using geodesic triangle properties.

Analytic solution for receiver coordinates in flat spacetime.

Method applicable even with redundant emitters.

Abstract

We investigate a relativistic positioning system where the coordinates of the users are determined by the proper times broadcasted by clocks in motion in spacetime: these are the so-called emission coordinates. In particular, we focus on emission coordinates in flat spacetime: in this case, we show that these coordinates can be defined using an approach based on simple geometrical properties of geodesic triangles. We analyse the 2-dimensional case and then we show how the whole procedure can be applied to 4 dimensions, also in terms of ordinary three-dimensional spheres. An analytic solution for the coordinates of the receiver is obtained. The solution remains valid at almost any place, in particular when redundancy in the number of emitters can be exploited.