Emission versus Fermi coordinates: applications to relativistic positioning systems

Abstract

Abstract. A 4-dimensional relativistic positioning system for a general spacetime is constructed by using the so called "emission coordinates." The results apply in a small region around the world line of an accelerated observer carrying a Fermi triad, as described by the Fermi metric. In the case of a Schwarzschild spacetime modeling the gravitational field around the Earth and an observer at rest at a fixed spacetime point, these coordinates realize a relativistic positioning system alternative to the current GPS system. The latter is indeed essentially conceived as Newtonian, so that it necessarily needs taking into account at least the most important relativistic effects through Post-Newtonian corrections to work properly. Previous results concerning emission coordinates in flat spacetime are thus extended to this more general situation. Furthermore, the mapping between spacetime coordinates and emission coordinates is completely determined by means of the world function, which in the case of a Fermi metric can be explicitly obtained.

Approaches to relativistic positioning around Earth and error estimations

“…Various concepts and techniques being useful to develop the 1-order RPS have been found in previous papers, among them, we may point out the definition and uses of the world function (Synge, 1931;Bahder, 2001;Bini et al, 2008;San Miguel, 2007) and the time transfer function, the form of this last function in the S-ST (Teyssandier and Le Poncin-Lafitte, 2008), and a method to find the user position coordinates by using the time transfer function (Čadež and Kostić, 2005;Čadež et al, 2010;Delva et al, 2011). Here, this last method is modified by using the analytical formula derived by Coll et al (2010) -instead of numerical iterations-to work with photons moving in M-ST The Earth's center is at rest in the asymptotic M-ST; hence, the S-ST may be considered as a perturbation of the asymptotic M-ST with a static metric g αβ = η αβ +s αβ , where η αβ is the Minkowski metric, and s αβ are perturbation terms depending on GM ⊕ /R, where R is the Schwarzschild radial coordinate.…”

Deviation and precession effects in the field of a weak gravitational wave

“…We take advantage of the high symmetrical character of the background spacetime as well as of the weak field approximation to derive the exact transformation between TT coordinates and Fermi coordinates, generalizing previous works for geodesic world lines [9][10][11]. Indeed, this can be achieved in closed form only for a few spacetimes around very special world lines [12,13], while most applications use a series expansion approximation from the very beginning [14][15][16][17][18][19].…”

Four Causal Classes of Newtonian Frames

“…This is the case, for example, of the Solar system synchronization, foliating the space-time by timelike instants. And more importantly, the case of the positioning systems, cutting any (history of an) extended object by four (histories of) electromagnetic pulses or sound waves [8][9][10][11][12].…”

“…This means that, for its development, it needs no other physical concepts than the ones contained in its specific foundations, or those that can be coherently deduced from them. But in practice, despite efforts made in this sense [3][4][5][6][7][8][9][10][11][12], to develop physical applications must, for the moment, resort to Newtonian concepts and post-Newtonian methods. This situation reduces Relativity theory, with few exceptions, to a role of corrective algorithm for Newtonian theory, relegating its best specific concepts to a simple historically admirable, but otherwise ineffective, method of setting the main equations of the theory, the Einstein equations.…”