Relativistic location algorithm in curved spacetime

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 $<1\mathrm{s}$ 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.

19 pages, 16 figures, 1 table

References in corpus (7)

• Relativistic Positioning Systems
• The covariance of GPS coordinates and frames
• Positioning in a flat two-dimensional space-time: the delay master equation
• Pulsars as celestial beacons to detect the motion of the Earth
• FANTASY: User-Friendly Symplectic Geodesic Integrator for Arbitrary Metrics with Automatic Differentiation
• Geometric approach to the definition of emission coordinates
• Coordinates and frames from the causal point of view