Can GPS Test Gravity’s Speed of Propagation?

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 $Q=\left[{\left(c/c_g\right)}^2-1\right]$ . 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 $1/c^2$ 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 pulsars ³ 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.