A Relativistic Framework for Positioning Systems
Pascual-Sánchez, J.-F.
Universidad de Valladolid

Practically all experiments in General Relativity are done in a classical Newtonian conceptual framework. In this talk I will focus on the essential differences between a Newtonian plus relativistic corrections framework, as that of the current Global Navigation Satellite Systems (GNSS), and a fully relativistic framework which would be desirable to implement in the future Galileo system due to its theoretical and practical advantages.

The Newtonian conceptual framework uses a 3-dim spatial reference system and a time reference. In this framework, the “relativistic effects” are added with the same status as any non-desired perturbation (gravitational influence of other planets, effects of the atmosphere, ...). This is made with corrections, coming from general relativity when compared with classical mechanics, in weak gravitational fields and with small velocities (post-newtonian formalism).

Typically, this is what is done nowadays in all the current GNSS, where the primary relativistic effects, i.e., some post-newtonian corrections of second order 1/c^2, coming from both Special and General Relativity must be taken into account. In general, the satellites of the GNSS are affected by Relativity in three different ways: in the equations of motion, in the signal propagation and in the beat rate of the satellite clocks. In the first part of the talk I will review the main clock effects at second order, because they are the only measurable ones in the present GNSS due to the present accuracy of nanoseconds of the satellite clocks.

At present, the above approach is perfectly justified from a practical and numerical point of view. However, if the time resolutions are increased (more accurate clocks), it will be necessary in the future to consider other effects of order 1/c^2 and also it will be necessary to consider other effects of third (as the Shapiro delay) or fourth order (as the Lense-Thirring effect).

In this situation, it can be wondered if it would not be more convenient to change the framework to an exact formulation in General Relativity. This would imply to abandon the classical post-newtonian framework. Obviously, this is a jump with many implications and difficulties of many different kinds: from technical to sociological. However, a project is aimed to develop a theory of positioning systems in the framework of general relativity. This project, which is still in a state of theoretical construction, is called SYPOR (a French acronym for SYstéme de POsitionnement Relativiste) and it has been initiated by Coll and collaborators several years ago.

This project is based in the following mathematical result which is almost unknown in spite of its fundamental importance in the construction of positioning systems. In the 4-dim Newtonian spacetime there exists 4, and only 4, causal classes of reference frames, whereas in the relativistic 4-dim Lorentzian spacetime, due to the freedom introduced by the finite propagation of light, there exists 199, and only 199, causal classes of reference frames. A causal class is defined by a spacetime frame, a dual coframe and the 2-dim surfaces generated by the vectors of the frame.

Only a causal class, among the 199 Lorentzian ones, is privileged to construct a generic (valid for a wide class of spacetimes), gravity free (the previous knowledge of the gravitational field is not necessary) and immediate (non retarded) positioning system, this is the causal class of the emission coordinates. Emission coordinates are a class of spacetime coordinates defined and generated by four emitters (satellites) broadcasting their proper time by means of radio signals. The emission coordinates are covariant (frame independent) and completely independent of any observer or user. Any observer can measure the values of the emission coordinates which can give his position, the components of his 4-velocity and can obtain the metric of the spacetime (the gravitational field) acting on the constellation.

The study of emission coordinates is aimed to develop an exact theory of positioning systems, based on the framework and concepts of General Relativity, as opposed to introducing “relativistic effects” in a classical Newtonian framework, which is the customary approach yet now used in the GNSS. Just now important results have been obtained by Coll and collaborators in 2 spacetime dimensions, and some very interesting properties have been already obtained for 3 and 4 spacetime dimensions.