Relativistic Positioning System in Perturbed Space-time

TL;DR

This paper develops a relativistic positioning system modeled in a perturbed space-time around Earth, demonstrating high accuracy and efficiency in determining user coordinates using emission signals and ray-tracing methods.

Contribution

It introduces a novel relativistic positioning system that accounts for multiple perturbations in Earth's space-time, achieving high-precision user localization.

Findings

Positioning in perturbed space-time is feasible.

Algorithms are highly accurate and efficient, with sub-10^{-26} relative errors.

Real-time coordinate determination is possible on standard laptops.

Abstract

We present a variant of a Global Navigation Satellite System called a Relativistic Positioning System (RPS), which is based on emission coordinates. We modelled the RPS dynamics in a space-time around Earth, described by a perturbed Schwarzschild metric, where we included the perturbations due to Earth multipoles (up to the 6th), the Moon, the Sun, Venus, Jupiter, solid tide, ocean tide, and Kerr rotation effect. The exchange of signals between the satellites and a user was calculated using a ray-tracing method in the Schwarzschild space-time. We find that positioning in a perturbed space-time is feasible and is highly accurate already with standard numerical procedures: the positioning algorithms used to transform between the emission and the Schwarzschild coordinates of the user are very accurate and time efficient -- on a laptop it takes 0.04 s to determine the user's spatial and time coordinates with a relative accuracy of $10^{-28}-10^{-26}$ and $10^{-32}-10^{-30}$, respectively.