Arthur Stanley Eddington was a very accomplished and brilliant British physicist, mathematician, and astronomer. Shortly after Einstein's theories of Special and General Relativity were published, Eddington authored two books on the subject: Space, Time and Gravitation (1920) and The Mathematical Theory of Relativity (1923). The presentation below concerns only what these two books have to say about how fast the force of gravity is propagated.

Eddington points out that it was long thought that the speed of gravitational force was far higher than that of light. The historical argument was that if gravity was propagated at the speed of light, there would be a "force couple" that develops between the present position of Jupiter and the so-called "retarded position" of Jupiter. This would result in an alteration of Jupiter's orbit around the Sun which would be easily observed over time. But no such alteration has been observed. Hence, something is wrong with the argument.

Eddington's belief was that the gravitational force acts much like that of electrical forces between charges. As the footnote referenced on the page below explains "To the first order of v

Eddington still assumes, strangely, that gravitational force is propagated at the speed of light, and likewise for forces between electric charges. A. P. French commits the same error as shown in The Speed of Electric Fields . Evidently, even brilliant minds found instantaneous action-at-a-distance ("non-locality") too inconceivable to take seriously, whether in the 1920s or the 1960s.

Space, Time and Gravitation An Outline of the General Relativity Theory,

A. S. Eddington (1920; reprint 2013) p.94

Eddington's
second book (below) shows that the speed of gravity in General
Relativity depends on the choice of a coordinate system. In Einstein's
General Relativity "the
coordinates used were purposely introduced in order to obtain the
simplification which results from representing the propagation as
occurring with the speed of light." But the choice is arbitrary and
"The argument thus follows a vicious circle."

Eddington
says that "the speed of gravitation is quite definite". But
in
the 1920s no actual laboratory experiments had been done to measure the
speed. Hence, the speed was still an open question. Einstein
arbitrarily chose the speed of light to simplify his theory.

The Mathematical Theory of Relativity
,
A. S. Eddington (1923; reprint 2017 by Forgotten Books) p130-131

The
take-away from
all this is that there is really nothing in Einstein's General
Relativity that "proves" or "requires" the speed of gravity to be the
same as the speed of light. Indeed, recent investigations indicate that
the speed of gravity is very much faster than the speed of light and is
probably instantaneous ("non-local").

Links:

Links:

https://www.britannica.com/biography/Arthur-Eddington

https://en.wikibooks.org/wiki/Electrodynamics/Lienard-Wiechert_Potentials

https://en.wikibooks.org/wiki/Electrodynamics/Lienard-Wiechert_Potentials