The covariance of GPS coordinates and frames

Abstract

We explore, in the general relativistic context, the properties of the recently introduced global positioning system (GPS) coordinates, as well as those of the associated frames and coframes that they define. 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 4-velocity and (iii) the components of the metric in the GPS frame. This provides this system with a unique value both for conceptual discussion (no frame dependence) and for practical use (involved quantities are directly measurable): localization, motion monitoring, astrometry, cosmography and tests of gravitation theories. We show explicitly, in the general relativistic context, how an observer may estimate his 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–Lematre spacetimes, geodesic motions) are studied in detail.