The covariance of GPS coordinates and frames

TL;DR

This paper investigates the properties of GPS coordinates and frames within general relativity, demonstrating their covariance, observer independence, and practical measurability, enabling precise localization and gravitational tests.

Contribution

It introduces a covariant, observer-independent GPS coordinate system in general relativity, linking measurable quantities like redshifts to position, motion, and metric reconstruction.

Findings

GPS coordinates are covariant and observer-independent.

Observers can measure their position, velocity, and metric components using spectroscopic data.

The method applies to various spacetime models, including Minkowski and Friedmann-Lemaître.

Abstract

We explore, in the general relativistic context, the properties of the recently introduced GPS coordinates, as well as those of the associated frames and coframes. We show that they are covariant, and completely independent of any observer. We show that standard spectroscopic and astrometric observations allow any observer to measure (i) the values of the GPS coordinates at his position, (ii) the components of his [four-]velocity and (iii) the components of the metric in the GPS frame. This provides to this system an unique value both for conceptual discussion (no frame dependence) and for practical use (involved quantities are directly measurable): localisation, motion monitoring, astrometry, cosmography, tests of gravitation theories. We show explicitly, in the general relativistic context, how an observer may estimate its position and motion, and reconstruct the components of the metric. This arises from two main results: the extension of the velocity fields of the probes to the whole (curved) spacetime; and the identification of the components of the observer's velocity in the GPS frame with the (inversed) observed redshifts of the probes. Specific cases (non relativistic velocities; Minkowski and Friedmann-Lema\^{i}tre spacetimes; geodesic motions) are studied in details.