L1 The Astrophysical Journal, 556:L1–L5, 2001 July 20  2001. The American Astronomical Society. All rights reserved. Printed in U.S.A. TESTING THE RELATIVISTIC EFFECT OF THE PROPAGATION OF GRAVITY BY VERY LONG BASELINE INTERFEROMETRY Sergei M. Kopeikin Department of Physics and Astronomy, University of Missouri at Columbia, 322 Physics Building, Columbia, MO 65211; kopeikins@missouri.edu Received 2001 May 17; accepted 2001 June 7; published 2001 July 10 ABSTRACT It is shown that the finite speed of gravity affects very long baseline interferometric observations of quasars during the time of their line-of-sight close angular encounter with Jupiter. The next such event will take place in 2002 September 8. The present Letter suggests a new experimental test of general relativity in which the effect of the propagation of gravity can be directly measured by very long baseline interferometry as an excess time delay in addition to the logarithmic Shapiro time delay. Subject headings: gravitation — quasars: individual (QSO J0842+1835) — relativity — techniques: interferometric 1. INTRODUCTION Experimental verifications of the basic principles underlying Einstein’s general relativity theory are important for fundamental physics. Numerous tests of general relativity in the solar system (Will 1993) and in the binary pulsar PSR 1913+16 (Taylor 1994) confirm its validity up to a precision of 1% or slightly better. It is worth emphasizing that all of the solar system tests of general relativity have relied on the Schwarzschild solution and have nothing to say about the effects of retardation associated with the propagation of gravity. Einstein’s theory of general relativity predicts that if the second time derivative of the quadrupole moment of a gravi- tating system (e.g., a binary pulsar) is not zero, the system emits gravitational waves that travel outward at the speed of light. Indirect evidence for the existence of gravitational waves consists of observations of the in-spiraling orbits of binary pulsars (Taylor 1994). Observations of binary pulsar in-spiral for PSR 1913+16 agree at the level of ∼1% with predictions based on the emission of energy by the binary in the form of (quadrupole) gravitational radiation (Kopeikin 1985; Scha ¨fer 1985; Damour 1987). This also provides indirect evidence that gravity waves must travel at the speed of light (Damour 1987; Will 1993). However, no one has yet directly detected gravi- tational waves, let alone measured their speed. It is the purpose of this Letter to point out that observing the propagation of light through the gravitational field of the solar system can serve as a tool for measuring the effects as- sociated with the finite speed of the propagation of gravity. This is based on our previous papers (Kopeikin 1990, 1997; Kopeikin et al. 1999; Kopeikin & Scha ¨fer 1999), where we have developed a post-Minkowskian approach for solving the problem of the propagation of light through time-dependent gravitational fields in the geometric optics approximation. This approach is based on making use of the retarded Lie ´nard- Wiechert–type solutions of the linearized Einstein equations and allows one to find a smooth analytic representation of the light-ray trajectory for arbitrary locations of the source of light and observer without imposing any restrictions on the motion of the light-ray–deflecting bodies. We find that electromagnetic signals interact with gravitating bodies only through the re- tarded gravitational field of the bodies. That is, if a light- ray–deflecting body moves with respect to a chosen coordinate system, the temporal variation in its gravity field must take time to reach the electromagnetic signal in order to perturb its trajectory. This observation constitutes the main idea of the proposed very long baseline interferometry (VLBI) test of the propagation of gravity elaborated in the following sections of the present Letter. In 1964, I. I. Shapiro suggested that the gravitational de- flection of light by the Sun—one of the three classical effects of general relativity analyzed by Einstein—could be measured more accurately at radio wavelengths with interferometric tech- niques than at visible wavelengths with the available optical techniques (Shapiro 1964). His idea led to stringent experi- mental limitations on the parameter g of the parameterized post- Newtonian (PPN) formalism (Shapiro 1967), thereby strongly restricting the number of viable theories of gravity. In the case of a static, spherically symmetric field, our post-Minkowskian approach gives the same result as predicted by Shapiro. Fur- thermore, we are able to calculate additional corrections to the Shapiro time delay related to the nonstationarity of the grav- itational field of the solar system and to associate them with the finite speed of the propagation of gravity. The largest con- tribution to the nonstationarity of the gravitational field of the solar system comes about via the orbital motions of the most massive planets—Jupiter and Saturn. Therefore, it is reasonable to undertake an attempt to detect the effect of the propagation of gravity by observing very accurately the deflection of light rays from a background source of light (quasar) caused by the motion of Jupiter or Saturn. This is the essence of our proposed VLBI test of general relativity discussed in the present Letter. In § 2, we consider the basic formula for relativistic time delay in time-dependent gravitational fields. In § 3, we outline the basic principles of the measurement of the effect of the propagation of gravity in radio interferometric experiments. Section 4 is dedicated to the description of the proposed VLBI experiment and gives numerical estimates for the Shapiro time delay in the field of Jupiter and for the effect of the propagation of gravity. Finally, in § 5, we discuss the proposed VLBI experiment. 2. RELATIVISTIC TIME DELAY IN TIME-DEPENDENT GRAVITATIONAL FIELDS Let us assume that the gravitational field is generated only by the solar system’s bodies and that there is a global four- dimensional coordinate system with its origin at the bary- (t, x)