Propagation Speed of Gravity and the Relativistic Time Delay

We calculate the delay in the propagation of a light signal past a massive body that moves with speed v, under the assumption that the speed of propagation of the gravitational interaction cg differs from that of light. Using the post-Newtonian approximation, we consider an expansion in powers of v/c beyond the leading "Shapiro" time-delay effect, while working to first order only in Gm/c2, and show that the altered propagation speed of the gravitational signal has no effect whatsoever on the time delay to first order in v/c beyond the leading term, although it will have an effect to second and higher order. We show that the only other possible effects of an altered speed cg at this order arise from a modification of the parameterized post-Newtonian coefficient α1 of the metric from the value 0 predicted by general relativity. Current solar-system measurements already provide tight bounds on such a modification. We conclude that recent measurements of the propagation of radio signals past Jupiter are sensitive to α1 but are not directly sensitive to the speed of propagation of gravity.