Thank @KyleKanos for a suggestion, I am rephrasing this question to hopefully make it more clear.
According to this publication in Physics Letters, 2000: Aberration and the Speed of Gravity:
It is well known that if a charged source moves at a constant velocity, the electric field experienced by a test particle points toward the source’s “instantaneous” position rather than its retarded position.
The paper then extends this result to gravity:
aberration in general relativity is almost exactly canceled by velocity-dependent interactions
More specifically:
the gravitational acceleration is directed toward the retarded position of the source quadratically extrapolated toward its “instantaneous” position, up to small nonlinear terms and corrections of higher order in velocities.
The author immediately clarifies:
Does [this] imply that gravity propagates instantaneously? As in the case of electromagnetism, it clearly does not.
Following the logic in the paper, the Earth is attracted to the instantaneous position of the Sun 8 minutes or 4 Sun's diameters ahead of its observed retarded position in the sky.
The paper also explains that this conclusion is precise only to the radiative term. In case of the Earth rotating around the Sun, this correction is very small, because the gravitational radiation emitted by the Earth does not significantly change its orbit. (The situation, of course, would be very different in a strong gravity of a neutron star or black hole.)
My question is on the meaning of "toward the instantaneous position" in differential geometry. For example, in the Schwarzschild spacetime, does the gravitational acceleration of the Earth point in the direction of the spacelike geodesic between the Earth and the Sun? If not, then what is the mathematical definition of the direction "toward the instantaneous position" described in the paper?
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