The Speed of Gravity in General Relativity and Theoretical Interpretation of the Jovian Deflection Experiment
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According to Einstein, the notions of geodesic, parallel transport (affine connection), and curvature of space-time manifold have a pure geometric origin and do not correlate with any electromagnetic concepts. At the same time, curvature is generated by matter which is not affiliated with the spacetime geometric concepts. For this reason, the fundamental constant c entering the geometric and matter sectors of general theory of relativity have different conceptual meanings. Specifically, the letter c in the left side of the Einstein equations (geometric sector) entering the Christoffel symbols and its time derivatives is the ultimate speed of gravity characterizing the upper limit on the speed of its propagation as well as the maximal rate of change of time derivatives of the metric tensor, that is gravitational field. The letter c in the right side of the Einstein equations (matter sector) is the maximal speed of propagation of any other field rather than gravity. Einstein's general principle of relativity extends his principle of special relativity and equates numerical value of the ultimate speed of gravity to that of the speed of light in special theory of relativity but this general principle must be tested experimentally. To this end we work out the speed of gravity parameterization of the Einstein equations (cg-parameterization) to keep track of the time-dependent effects associated with the geometric sector of general relativity and to separate them from the time-dependent effects of the matter sector. Parameterized post-Newtonian (PPN) approximation of the Einstein equations is derived in order to explain the gravitational physics of the jovian deflection VLBI experiment conducted on September 8, 2002. The post-Newtonian series expansion in the cg-parameterized general relativity is with respect to a small parameter that is proportional to the ratio of the characteristic velocity of the bodies to the speed of propagation of the gravitational interaction cg. The Einstein equations are solved in terms of the Li´enard-Wiechert tensor potentials which are used for integrating the light-ray propagation equations. An exact analytic expression for the relativistic time delay in the propagation of a radio wave from the quasar to an observer is calculated under the assumption that the light-ray deflecting bodies move with constant velocities. A post-Newtonian expansion of the time delay proves that in general relativity the time delay is affected by the speed of gravity already to the first order in 1/cg beyond the leading (static) Shapiro term. We conclude that recent measurements of the propagation of the quasar's radio signal past Jupiter are directly sensitive to the time-dependent effect from the geometric sector of general relativity which is proportional to the speed of propagation of gravity cg but not the speed of light. It provides a first confirmative measurement of the fundamental speed c of the Einstein general principle of relativity for gravitational field. A comparative analysis of our formulation with the alternative interpretations of the experiment given by other authors is provided.