## Relativistic Global Navigation Satellite System

For successful use, GNSS must take into consideration relativistic effects - that is they must take into consideration the warping of space and time as described by Einstein's field equations. This is currently achieved by keeping the Newtonian concept of absolute space and time, and adding a number of relativistic corrections to the desired accuracy. A more natural approach may be to describe a GNSS directly in general relativity, in this manner "all required corrections" are already included in the calculation.

The natural evolution of GNSS is to become more and more accurate and autonomous, with the help of very stable clocks and cross links capabilities. If we employ a relativistic GNSS in future systems, it will allow for:

- A fully relativistic model allowing for a much higher precision in location than current GNSS's.
- An Autonomous Basis of Coordinates (ABC) whereby the satellites can determine their own orbits, orbital parameters and constants of motion
- The use of the GNSS satellites to measure the gravitational field in which they exist by refining the gravitational parameters.

If relativistic coordinates are used to account for General Relativity effects, a relativistic GNSS can be modeled to locate users to a higher accuracy than the current GNSS. With the addition of inter-satellite cross links, it is also possible to model a system where the satellites exchange their proper time; this allows them in turn to determine their own orbital parameters and constants of motion. By tracking their orbits the spacetime geometry acting on the satellite constellation can also be inferred solving an inverse problem [1].

In order to simulate the set of data delivered by a GNSS one has to solve the time transfer problem between the satellite. In a preliminary study, the ACT studied three methods of solving the time transfer problem; using the world function and numerically solving for the null coordinates; solving the time transfer equation analytically in terms of elliptic functions; and using expansions of the time transfer function [2]. By considering a very simplistic system in Schwarzschild coordinates, we were able to validate and compare the efficiency of all three methods.

The ACT started an Ariadna Project in collaboration with the University of Ljubljana to study the feasibility of doing relativistic gravimetry. The study was so successful, that it was immediately followed by an Ariadna continuation study and eventually a 3 year PECS (Plan for Cooperating States) project.

The aim of the first Ariadna was to simulate the data generated by a GNSS in an ideal framework, the Schwarzschild geometry, and to assess the influence of non gravitational perturbations (clock noise, ...) on the relativistic coordinates. This study proceeded with the analytical elliptic function solution of the time transfer equation to obtain solutions for light-like and time-like geodesics to model a system of satellites. It successfully illustrated the use of null coordinates (see figure) to determine the positions of both satellites and user as well as a small analysis of non-gravitational sources of noise [3].

The Ariadna continuation study's goal was to build on the RPS system already modeled in Schwarzschild geometry by developing the ABC system; this allows satellites to set up their own coordinate system independent of the terrestrial system and effectively track their own movement. This is realised by using the Hamilton formalism and defining a new approach to the regular coordinate system; allowing inter-satellite links to continuously refine the orbital information, thus improving the Hamiltonian and constants of motion for the satellites with the use of null coordinates [4].

The PECS project combined the work of the two previous Ariadna's but replaced the Schwarzschild geometry of spacetime with a more realistic case, where the Earth's gravitational field is not spherically symmetric. It, thus, modeled a relativistic positioning system in a spacetime metric that includes all relevant gravitational perturbations; Earth's multipoles (up to 6th); solid and ocean tide; gravitational influence of the Moon, Sun, Venus and Jupiter; and the frame dragging effect due to the Earth's rotation. In this more complicated spacetime, we showed the successful simulation of the relativistic GNSS as well as the ABC system with refinement of some gravitational parameters [5].

To date, we have found the a relativistic GNSS with gravitational perturbations is feasible, highly accurate and stable. With the addition of inter-satellite links, we have also successfully illustrated how the ABC system can constitute an accurate and stable coordinate system, independent of the Earth based system. We are now looking at moving this project from purely theoretical simulations to a more realistic system that would investigate the system engineering aspects of a relativist GNSS.

### References

[1] Tarantola, A., Klimes L., Pozo, J.M., and Coll, B., "Gravimetry, Relativity, and the Global Navigation Satellite Systems", Int. School on Relativistic Coordinates, Reference and Positioning Systems held at Salamanca (Spain), 2005, arXiv:gr-qc/09053798[2] Delva, P. and Olympio, J., "Mapping the Spacetime Metric with GNSS: a preliminary study", 2nd International Colloquium, Scientific and Fundamental Aspects of the Galileo Programme, COSPAR colloquium, Padua, Italy, 2009.

[3] Čadež, A., Kostić, U. and Delva, P., "Mapping the Spacetime Metric with a Global Navigation Satellite System", Ariadna Final Report ID 09/1301, 2010

[4] Čadež, A., Kostić, U., Delva, P. and Carloni, S., "Mapping the Spacetime Metric with a Global Navigation Satellite System - extension of study: Recovering of orbital constants using inter-satellites links", Ariadna Final Report ID 09/1301 CCN, 2011

[5] Gomboc, A., Horvat, M. and Kostić, U., "Relativistic GNSS", The PECS Project Final Report, Contract NO. 4000103741/11/NL/KML, 2014