Relativistic location algorithm in curved spacetime

TL;DR

This paper presents a practical numerical method for relativistic positioning in curved spacetimes, integrating gravitational and atmospheric effects, and demonstrates its efficiency and high accuracy for terrestrial navigation scenarios.

Contribution

It introduces a novel algorithm and implementation for relativistic location in generic spacetimes, combining gravitational and atmospheric effects within a unified framework.

Findings

Achieves position determination in under 1 second on a desktop computer.

Submillimeter accuracy in vacuum with Kerr metric.

Submeter accuracy including atmospheric effects.

Abstract

In this article, we describe and numerically implement a method for relativistic location in slightly curved but otherwise generic spacetimes. For terrestrial positioning in the context of Global Navigation Satellite Systems, our algorithm incorporates gravitational as well as tropospheric and ionospheric effects modeled by the Gordon metric. The algorithm is implemented in the \textsc{squirrel.jl} code, which employs a quasi-Newton Broyden algorithm in conjunction with automatic differentiation of numerical geodesics. Our work provides a practical solution to the relativistic location problem in a generic spacetime and consolidates relativistic and atmospheric effects in a single framework. Though optimization is not our primary focus, our implementation is already fast enough for practical use, establishing a position from five emission points in $< 1~{\rm s}$ on a desktop computer for reasonably simple spacetime geometries. In vacuum, our implementation can achieve submillimeter accuracy considering the Kerr metric with terrestrial parameters and submeter accuracy including tropospheric and ionospheric effects.