Positioning with stationary emitters in a two-dimensional space-time

TL;DR

This paper explores how stationary relativistic positioning systems can be used for gravimetry in two-dimensional space-time, demonstrating that additional information allows distinguishing gravitational fields and measuring their properties.

Contribution

It introduces a method to differentiate between flat and curved space-times using stationary emitters and emission coordinates, enabling gravimetric measurements within relativistic positioning systems.

Findings

Emitters' trajectories are identical in both flat and curved space-times in emission coordinates.

Additional information allows distinguishing space-time curvature and measuring gravitational mass.

The approach enables complete dynamical and gravimetric descriptions using relativistic signals.

Abstract

The basic elements of the relativistic positioning systems in a two-dimensional space-time have been introduced in a previous work [Phys. Rev. D {\bf 73}, 084017 (2006)] where geodesic positioning systems, constituted by two geodesic emitters, have been considered in a flat space-time. Here, we want to show in what precise senses positioning systems allow to make {\em relativistic gravimetry}. For this purpose, we consider stationary positioning systems, constituted by two uniformly accelerated emitters separated by a constant distance, in two different situations: absence of gravitational field (Minkowski plane) and presence of a gravitational mass (Schwarzschild plane). The physical coordinate system constituted by the electromagnetic signals broadcasting the proper time of the emitters are the so called {\em emission coordinates}, and we show that, in such emission coordinates, the trajectories of the emitters in both situations, absence and presence of a gravitational field, are identical. The interesting point is that, in spite of this fact, particular additional information on the system or on the user allows not only to distinguish both space-times, but also to complete the dynamical description of emitters and user and even to measure the mass of the gravitational field. The precise information under which these dynamical and gravimetric results may be obtained is carefully pointed out.