Eddington on the Speed of gravity in General Relativity

Arthur Stanley Eddington was a very acco
mplished 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 vr/C, the denominator is equal to the present distance r, so the expression reduces to e/r in spite of the time propagation. The foregoing formula for the potential was found by Lienard and Wiehert." In other words, the electrical force always points directly to either charge regardless of their motion. There is therefore no applicable "retarded position", and therefore no force couple.  (See The Speed of Electric Fields  for an illustrated explanation.)

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:

https://en.wikipedia.org/wiki/Arthur_Eddington
https://www.britannica.com/biography/Arthur-Eddington  
https://en.wikibooks.org/wiki/Electrodynamics/Lienard-Wiechert_Potentials