Global Navigation Satellite Systems (GNSS), such as Galileo and GPS, have to consider distortions caused by Earth on space and time in its vicinity (space-time curvature) and the effects of relative motions between the spacecrafts and the user (relativistic inertial effects). For the ground user, these relativistic effects can lead to corrections as large as twelve kilometres after one day. There are two ways of including relativity in the description of GNSS: one way is to keep the Newtonian concept of absolute time and space, and add a number of relativistic corrections to the level of the accuracy desired. This approach is used by present GNSS. An alternative, and more consistent, approach is to abandon the concept of absolute space and time and describe a GNSS directly in general relativity, i.e. to define a Relativistic Positioning System (RPS) with the use of emission coordinates (i.e. to use as four coordinates proper times of four satellites at the moment of emission of their signals).

In collaboration with the Advanced Concepts Team from ESA, we modeled an RPS in the idealized case of Schwarzschild geometry. Results showed that in this case an RPS is feasible, stable and accurate:  it is possible to highly accurately determine the receiver's position in emission coordinates and transform them to Schwarzschild coordinates. Furthermore, if satellites exchange their proper times via inter-satellite links, they can themselves determine their orbits and their orbital elements. Such a system of satellites would be independent from terrestrial reference frames and would constitute an Autonomous Basis of Coordinates (ABC).

We further investigated whether an RPS and ABC concepts are feasible, stable and accurate also in a more realistic case, where Earth's gravitational field is not spherically symmetric. We modeled an RPS in the space-time metric which includes all relevant gravitational perturbations: Earth's multipoles (up to the 6th), solid and ocean tides, gravitational influence of the Moon, the Sun, Venus and Jupiter, and the frame-dragging effect due to Earth's rotation (Kerr effect): We built a mathematical scheme for including all relevant gravitational perturbations to the background Schwarzschild metric in the weak-field limit with the linear perturbation theory. Using the perturbed metric, we calculated satellite orbits in the perturbed space-time we and used them to model relativistic positioning in gravitationally perturbed space-time. Next, we simulated inter-satellite links and built a model of the ABC system. It followed, that by using solely inter-satellite links and information on emission coordinates over several orbital periods it is possible to refine the initial values of orbital elements and thus construct an ABC. Last but not least, we have shown that, in principle, the inter-satellite links offer a new, independent way of probing space-time in the vicinity of Earth and measuring the gravitational influence of not only Earth and its multipoles and tides, but also of other celestial bodies. We believe that RPS GNSS has potential to measure gravitational perturbations, which can be beneficial in various scientific applications.


A. Gomboc, U. Kostić, M. Horvat, S. Carloni, P. Delva: Relativistic Positioning Systems and Gravitational Perturbations

U. Kostić, M. Horvat, S. Carloni, P. Delva, A. Gomboc: An Autonomous Reference Frame for Relativistic GNSS

U. Kostić, M. Horvat, A. Gomboc: Relativistic Positioning System in Perturbed Space-time

ESA PECS Relativistic GNSS Final Report