In 2002, Sergei Kopeikin suggested that measurement of the deflection of light from a quasar by the planet Jupiter could be used to measure the speed of the gravitational interaction. He argued that, since Jupiter is moving relative to the solar system, and since gravity propagates with a finite speed, the gravitational field experienced by the light ray should be affected by gravity's speed, since the field experienced here now depends on the location of the source a short time earlier, depending on how fast gravity propagates. According to his calculations, there should be a small correction to the normal general relativistic formula for the deflection, which depends on the velocity of Jupiter and on the velocity of gravity (technically, it's an extra term in the ``Shapiro'' delay in arrival of waves at a radio telescope). On September 8, 2002, Jupiter passed almost in front of a quasar, and, in collaboration with Ed Fomalont of the National Radio Astronomy Observatory, precise measurements were made of the Shapiro delay, with picosecond timing accuracy. Kopeikin and Fomalont argued that the results were in accord with the prediction of GR for this tiny effect, with a precision of about 20 per cent. This would be an interesting new confirmation of GR, albeit at modest accuracy.

The question is:
Does this tell us anything about the speed of propagation of gravity?
The consensus among relativists is NO!

Papers by Kopeikin claiming this tests the speed of gravity

• Testing the relativistic effect of the propagation of gravity by very long baseline interferometry,  S. Kopeikin, Astrophys. J. 556 (2001) L1-L5 (gr-qc/0105060)
• General relativistic model for experimental measurement of the speed of propagation of gravity by VLBI, S. Kopeikin and E. Fomalont, Proceedings of the 6th European VLBI Network Symposium, Ros, E., Porcas, R.W., Zensus, J.A. (eds.), June 25th - 28th, 2002, Bonn, Germany, p. 49 (gr-qc/0206022)
• The Post-Newtonian Treatment of the VLBI Experiment on September 8, 2002, S. Kopeikin, Phys. Lett. A312 (2003) 147 (gr-qc/0212121)
• The Measurement of the Light Deflection from Jupiter: Experimental Results, E. B. Fomalont, S. M. Kopeikin, Astrophys. J. 598 (2003) 704 (astro-ph/0302294)
• The Measurement of the Light Deflection from Jupiter: Theoretical Interpretation, S. Kopeikin, (astro-ph/0302462)
• Speed of Gravity in General Relativity and Theoretical Interpretation of the Jovian Deflection Experiment, S. Kopeikin, Class. Quantum Gravit. 21 (2004) 3251 (gr-qc/0310059)
• On the Speed of Gravity and Relativistic v/c Corrections to the Shapiro Time Delay, S. Kopeikin and E. B. Fomalont, Phys. Lett. A 355 (2006), 163 (online version) (gr-qc/0310065)
• Aberration and the Speed of Gravity in the Jovian Deflection Experiment, S. Kopeikin and E. B. Fomalont, Foundations of Phys. DOI: 10.1007/s10701-006-9059-7 (2006) (online version) (astro-ph/0311063)
• Comments on "On the speed of gravity and the Jupiter/Quasar measurement" by S. Samuel, S. Kopeikin, Int. J. Mod. Phys. D 15 (2006), 273 (online version)(gr-qc/0501001)
• Gravitomagnetism and the Speed of Gravity, S. Kopeikin, Int. J. Mod. Phys. D 15 (2006), 305 (gr-qc/0507001)
• Comment on 'Model-dependence of Shapiro time delay and the "speed of gravity/speed of light" controversy', S. Kopeikin, Class. Quantum Gravit. 22 (2005) 5181 (gr-qc/0510048)
• Gravitomagnetism and the Aberration of Gravity in the Gravitational Light-Ray Deflection Experiments, S. Kopeikin and E. B. Fomalont (gr-qc/0510077)
• Note on the relationship between the speed of light and gravity in the bi-metric theory of gravity, S. Kopeikin (gr-qc/0512168
• Radio interferometric tests of general relativity, E. B. Fomalont and S. Kopeikin, Proc. IAU Symposium No. 248, eds Jin et al. (2007), p 383.

Papers by authors claiming the measurement is NOT sensitive to the speed of gravity

• The Light-cone Effect on the Shapiro Time Delay, H. Asada, Astrophys. J. 574 (2002) L69 (astro-ph/0206266)
• Propagation Speed of Gravity and the Relativistic Time Delay, C. M. Will, Astrophys. J. 590 (2003) 683 (astro-ph/0301145)
• On the Speed of Gravity and the v/c Corrections to the Shapiro Time Delay, S. Samuel, Phys. Rev. Lett. 90 (2003) 231101 (astro-ph/0304006)
• The speed of gravity has not been measured from time delays, J. Faber (astro-ph/0303346)
• Comments on "Measuring the Gravity Speed by VLBI", H. Asada,Proc. of "Physical Cosmology", the XVth Rencontres de Blois, 15-20 June 2003 (astro-ph/0308343)
• Model-Dependence of Shapiro Time Delay and the "Speed of Gravity/Speed of Light" Controversy, S. Carlip, Class.Quantum Gravit. 21 (2004) 3803(gr-qc/0403060)
• Speed of gravity and gravitomagnetism, J.-F. Pascual-Sánchez, Int.J.Mod.Phys. D 13 (2004) 2345 (gr-qc/0405123)
• On the Speed of Gravity and the Jupiter/Quasar Measurement, S. Samuuel, Int. J. Mod. Phys. D 13 (2004) 1753 (astro-ph/0412401)

Other papers commenting on the effect

• Aberration by gravitational lenses in motion, S. Frittelli, MNRAS 344, L85 (2003) (astro-ph/0311189)
• Micro-arcsecond light bending by Jupiter, M. T. Crosta and F. Mignard, Class. Quantum.Gravit. 23 (2006) 4853 (astro-ph/0512359)
• Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers, S. Kopeikin and V. Makarov, Phys. Rev. D 75 (2007), 062002 (astro-ph/0611358)

The idea seems natural. What's wrong with it?

• In all calculations of the effects of finite propagation speed (electrodynamics, gravitational waves), for slowly moving sources, the first non-trivial effects appear at order (v/c)^2, NOT at first order (v/c), which is what Kopeikin claimed
• If you're working to only the first order in v/c, then all that counts is the uniform motion of the planet, Jupiter (its acceleration about the sun contributes a higher-order, unmeasurably small effect). But if that is the case, then the principle of relativity says you can view things from the rest frame of Jupiter. But in this frame, Jupiter's gravitational field is static, and the speed of propagation of gravity is irrelevant.
• A detailed calculation of the effect was done by Clifford Will (here), in a model of gravity in which the speed of gravity can be different from that of light, and showed explicitly that, at first order in (v/c), the effect depends on the speed of light, not the speed of gravity, in line with intuition.
• Stuart Samuel (here) has also argued that the experiment was not even sensitive to the general relativistic (v/c) correction, if one interprets the formulae properly.

How can we really measure the speed of propagation of gravity?

• If we could measure the effects on the Shapiro delay to order (v/c)^2, then we could test the speed of gravity. But these effects would be at the thousandths of a picosecond level, hopelessly small.
• The real way to measure the speed of gravity is to detect and study gravitational waves. By comparing the arrival of a gravitational-wave signal with that of an electromagnetic signal from an astrophysical source, one could compare the speed of gravity to that of light to parts in 10^(17).