Can GPS Test Gravity's Speed of Propagation?

I wonder if the system's achieved precision is sufficient for determining the fundamental constant of the gravitational action's propagation speed, and if it has been determined yet. One possible way to do that would be to analyze the eccentricity effect on the satellites' orbits from the tidal force of the Moon. On a geostationary satellite at a height of 36 000 km, the effect is ±2 km , and the axis of eccentricity precesses around Earth following the Moon's 28-day journey. The phase of that precession follows the Moon's orbit with a delay of 1 second, if one assumes a gravitational propagation speed equal to the speed of the electromagnetic interaction. Within 14 days, the eccentricity has rotated by 180 degrees. The challenge for researchers is to determine the precession to better than 1 second, the time the gravitational field needs to travel from the Moon to the satellite. Determining the 4-km eccentricity in 14 days to better than 1 second calls for a precision of 1 part in 10¹⁰.

Perhaps this analysis has already been done. Physicists may think it is trivial, because all predictions of special and general relativity have proven to be correct so far. However, I think this investigation would be a fundamental one, since the expansion speed of two different interactions, gravitational and electromagnetic, need not be identical.

It was good to see Neil Ashby's article about general relativity and the global positioning system. Let me add a few historical details.

In 1965, I landed a position with Aerospace Corp in El Segundo, California; I had completed a PhD in general relativity some years earlier. Aerospace Corp had become involved in developing what was eventually to become the GPS, and W. Begley of the tracking and radar department asked me to do a study of possible relativistic effects on clocks carried by satellites.

The project was classified, so all I was told was that the military had become very interested in setting up an ultraprecise navigation system. I was happy to help by writing a research report; a brief, unclassified version of it was later submitted for publication.¹

Many years later, pocket-sized GPS receivers hit the civilian market, and I began to realize the full implications of my research. I could finally tell my wife what I had been up to 30 years before!

It is remarkable that the GPS is presently the only practical application of Einstein's gravitation theory. I urge that the general public be made more aware of this very useful result of a very abstract physical theory.

1. W. J. Cocke, Phys. Rev. Lett. 16, 662 (1966).

Ashby replies: Dieter Proetel raises the very interesting question of whether the propagation speed of gravity, c_g, can be observed by accurate position measurements on satellites orbiting Earth. A clear answer cannot be given without also considering retardation effects from all important sources in the system, such as Earth, the Sun, and the Moon. Some terms in the approximate solutions of Einstein's field equations for the Solar System resemble the retarded Liénard-Wiechert potentials of electrodynamics. One can therefore obtain estimates of perturbations that are due to retardation by changing the speed c in such terms to a propagation speed c_g that is different from c. Electromagnetic waves that propagate with speed c are universally used, however, to make meaningful position and timing measurements in Earth's neighborhood. One way to approach the problem is to introduce normal Fermi coordinates, which are simple to interpret in terms of proper distances and proper times.

If the propagation speed for gravity is c_g, one finds, after transforming to normal Fermi coordinates, several small new orbital effects that are proportional to the quantity Q = [(c/c_g)² - 1]. Such a form for the orbital perturbations results from a combination of many relativistic effects: Lorentz contraction, resynchronization of local clocks, rescaling of lengths due to external potentials, relativistic precession of axes, and so on. The calculation is lengthy.¹

When c = c_g, Q vanishes. There are then no surviving retardation corrections to the relativistic equations of motion of a satellite as it orbits Earth, to the order 1/c² of the calculation. This finding is consistent with the analysis by Steven Carlip,² who points out that such cancellations occur as a result of velocity-dependent terms in general relativity. He also says that, for a uniformly moving source, the force is directed toward the instantaneous, rather than the retarded, position of the source. Similar effects occur in electrodynamics.

Even if c and c_g are unequal, the coefficients of Q are discouragingly small. Considering only the Earth-Moon-satellite system as point masses, the coefficients of Q that correspond to corrections to lunar tidal displacements of Earth-orbiting satellites are far smaller than a millimeter. Furthermore, observations of the orbital decay of binary pulsars³ imply Q < 0.02. It thus appears that retardation effects from the Moon's gravity field will be extremely small and difficult to detect. A more attractive possibility would be to look for retardation effects from the gravity field of Earth or the Sun on more rapidly moving satellites such as LAGEOS (Laser Geodynamic Satellite), for which the coefficients of Q are considerably larger--a few centimeters. Some years ago, I discussed this possibility with John Ries of the Texas Center for Space Research. He analyzed some LAGEOS data with retardation effects from Earth's and the Sun's gravity included, but found that such effects were too small to discern.

1. N. Ashby, B. Bertotti, Phys. Rev. D 34, 2246 (1986).
2. S. Carlip, Phys. Lett. A 267, 81 (2000).
3. J. H. Taylor, Rev. Mod. Phys. 66, 711 (1994).