Relativity, GPS and Gravitation
My colleague Bartolomé Coll and myself have been working quite hardly, in the past years, on the fundamentals of a GPS system. Our conclusion is that a paradigmatic shift is needed in the way a GPS system is operated. As a result, one could obtain a more precise system.Many relativistic corrections are applied to the GPS system. Neil Ashby presented in Physics Today (May 2002) a good account of how these relativistic corrections are applied, why, and which are their orders of magnitude. (Click here for an html version of Ashby's paper; click here for a PDF version of the html document) (a new 2003 paper here). Unfortunately, Ashby seems to propose that relativity is only a correction to be applied to Galilean physics. Coll and me believe that there is a fully relativistic way to understand a GPS system, and this leads to a new way of operating it.
To be short, in today's GPS, a system of three spatial (geocentric) coordinates and one time coordinate is used. These coordinates have many drawbacks:
- orbits are referenced through a terrestrial system, and the problems with ionosphere, with the poorly defined geocenter, etc., come in;
- the "universal time" being used forces the operators to struggle with the synchronization of clocks (is there anything less relativistic that the goal of synchronizing a system of clocks in relative motion)?
Our work (essentially Coll's work; see a serious discussion of this topic on his web page) has demonstrated, using fundamental principles, that there is only one coordinate system in space-time that is physically implementable, fully relativistic and immediate: if four clocks broadcast their (proper) time signals, any observer in space-time simultaneously receives, at any point of his space-time trajectory, four times, one sent by each clock. These four times are the only "good" space-time coordinates (let us call them light-coordinates).Now, if each clock listens to the three others, it knows its own trajectory (in these light-coordinates). We have demonstrated that, if instead of broadcasting its proper time, each clock broadcasts its trajectory (in these light coordinates), then, any observer not only knows its coordinates, but is also able to compute the components of the space-time metric in these coordinates (i.e., she/he is able to use the coordinates to build a local clock and a local meter). And no a priori knowledge of the possible acceleration of the four clocks is necessary.
This theorem is only true in flat space-time. In the actual gravitational field of the Earth, more than four satellites (clocks) are needed, the redundancies being used to model the gravity field (i.e., the space-time metric) itself. In fact, we believe that our system would be the ultimate space gravimeter.
The agency in charge of running a GPS system should forget about the Earth, its geocenter, and any terrestrial receiver: the satellite system must be referenced internally (the orbits have to be defined using the light-coordinates only). This will achieve maximum accuracy for the orbits and the primary reference system. The problem of attaching these light-coordinates to some terrestrial coordinate system, is an attachment problem, that should not interfere with the problem of defining the primary coordinate system itself.
The day a GPS system would be run this way, we will considerably simplify the theory, increase the accuracy of the positioning system, and provide accurate models of the Earth's gravity field (the geoid and so on).
Please click here to download a working draft explaining some of our ideas.