Introducing relativity in global navigation satellite systems

Abstract

Today, the Global Navigation Satellite Systems, used as global positioning systems, are the GPS and the GLONASS. They are based on a Newtonian model and hence they are only operative when several relativistic effects are taken into account. The most important relativistic effects (to order 1/c 2 ) are: the Einstein gravitational blue shift effect of the satellite clock frequency (Equivalence Principle of General Relativity) and the Doppler red shift of second order, due to the motion of the satellite (Special Relativity). On the other hand, in a few years the Galileo system will be built, copying the GPS system unless an alternative project is designed. In this work, it will be also shown that the SYPOR project, using fully relativistic concepts, is an alternative to a mere copy of the GPS system. According to this project, the Galileo system would be exact and there would be no need for relativistic corrections.

Relativistic Positioning System in perturbed spacetime

“…> 1 and rewrite this equation using the variable u = 1/x on the domain of interest u ∈ [0, 1]:(1 − u)u 2 : h 0 + 2u(1 − u) 9 h 0 + [2u − n(n + 1)]h 0 = 0 , (A.28)The point u = 0 is a regular singular point, so it can be solved with Frobenius method in a similar way as equation (A.11). The solution for h 0 is h 0 (r) = α nm r −n P (2) n r s r + β nm r n+1 R(2) …”

Relativistic versus Newtonian Frames

“…This is the causal class {s s s s; S S S S S S; l l l l} of the Coll homogeneous coordinate system [2,4,5]. In this causal class the null emission coordinates of the Coll positioning system are included.…”

“…The basic idea of this project, that was conceived by Coll in [6] and also exposed in [2,3], is the following one: A satellite constellation provided with clocks that interchange their proper time among them (interlinks) and with Earth receivers, is a fully relativistic autonomous or autolocated positioning system. Note that, nowadays, this procedure of proper time auto navigation can be technically fulfilled.…”

“…Among the relativistic effects on the rate of the satellite clocks with a time accuracy of nanoseconds and 10 −12 of frequency accuracy the most important ones (to first post-Newtonian order 2 1 c ) are: the Einstein effect or gravitational frequency blue shift of the atomic clocks of the satellites (Equivalence principle of General Relativity) with respect to Earth bound clocks due to their position in the Earth gravitational field, time dilation or Doppler shift of second order due to the speed of the satellites (Special Relativity) and the kinematical Sagnac effect due to the rotation of the Earth (Special Relativity), see [1][2][3] for reviews. If they were not corrected by imposing an offset, the GNSS would not be operative after few minutes.…”

Emission versus Fermi coordinates: applications to relativistic positioning systems

“…Although clock speeds are small in comparison with the speed of light and gravitational fields are weak near the Earth, they give rise to significant relativistic effects. The most important ones affecting the rate of clocks (to the order 1/c 2 , which is just the order of approximation used in GPS) are first and second order Doppler frequency shifts of clocks due to their relative motion, gravitational frequency shifts due to the Earth's mass, and the Sagnac effect due to the Earth's rotation [1,2,9]. If these corrections were not made, the system would not be operational after a few minutes.…”