Positioning in a flat two-dimensional space-time: the delay master equation

TL;DR

This paper explores how relativistic positioning systems in a two-dimensional Minkowski space-time can determine the accelerations of emitters and users using minimal data, highlighting the role of the master delay equation and echo intervals.

Contribution

It demonstrates that accelerations can be deduced from emitter data and one emitter's acceleration during echo intervals, advancing understanding of relativistic positioning.

Findings

Accelerations are determined by emitter data and one emitter's acceleration.

The master delay equation relates data to acceleration during echo intervals.

The method allows deriving dynamics from specific received data sets.

Abstract

The basic theory on relativistic positioning systems in a two-dimensional space-time has been presented in two previous papers [Phys. Rev. D {\bf 73}, 084017 (2006); {\bf 74}, 104003 (2006)], where the possibility of making relativistic gravimetry with these systems has been analyzed by considering specific examples. Here we study generic relativistic positioning systems in the Minkowski plane. We analyze the information that can be obtained from the data received by a user of the positioning system. We show that the accelerations of the emitters and of the user along their trajectories are determined by the sole knowledge of the emitter positioning data and of the acceleration of only one of the emitters. Moreover, as a consequence of the so called master delay equation, the knowledge of this acceleration is only required during an echo interval, i.e., the interval between the emission time of a signal by an emitter and its reception time after being reflected by the other emitter. We illustrate these results with the obtention of the dynamics of the emitters and of the user from specific sets of data received by the user.