Ever since Weber wrote down his equation that united Coulomb and Ampere forces, its generalization to include gravity was sought after by the likes of Tisserand and other astronomers. The problem consists in the delineation of two forms of motion of parallel and longitudinal masses. This has relevancy to electrodynamics where charges can attract and repel one another, but not to masses which only know attraction.
Consider Weber's logic for the construction of his force [cf. Hecht, "The significance of the 1845 Gauss-Weber correspondence" for greater detail]. Weber begins by generalizing Coulomb's law to where the charged particles are in relative motion. Although he uses the incorrect hypothesis of Gustav Fechner, in which a current consists of opposite flow of pairs of positive and negative charges. If the pairs of charges flow in the same direction, this will result in repulsion, whereas in the opposite case, attraction will result. These conclusions were supposedly the deductions of Weber. It certainly will not be the first time in the history of science that an incorrect hypothesis will lead to correct conclusions! Just look at Maxwell's vortex theory of the electrodynamic field.
Weber writes dr/dt for the relative motion of pairs of charges. Since this applies to the case where the pairs are approach or receding from one another, the relevant term in the force law will contain only the square of the relative velocity. So Weber writes his longitudinal force as
F = ee'/r^2 [1-a^2(dr/dt)^2] (1)
where the sign of the coefficient is unimportant, and a must be determined empirically.
Weber then considers the pairs of charges to be following parallel to one another. Again Ampere's theory describes attraction when the pairs of elements are flowing in the same direction, and repulsion when they are flowing in opposite directions. As the pairs approach one another, their relative velocity is negative, zero at the moment their paths cross, and positive when they recede from one another. Thus, a change in the relative velocity is an acceleration and this leads him to introduce an acceleration term into his force law, that, unlike the relative force term will lead to an overall increase in the force
F=ee'/r^2[1-a^2(dr/dt)^2+b d^2 r/dt^2], (2)
where b is another constant that needs to be determined.
To relate the coefficients, Weber introduces the condition
r d^2r/dt^2 = (dr/dt)^2. (3)
He says it comes from the realization that particles traveling in the parallel directions do not travel along the same straight line so that upon considering their distance of separation, condition (3) results. Be that as it may, it reduces to (2) to
F = ee'/r^2[1-(a^2-b/r)(dr/dt)^2]. (4)
The ratio, a^2/b/r, represents the ratio of the forces between the longitudinal and parallel elements, which Ampere had determined experimentally to be 1/2. Thus, the force is
F = ee'/r^2[1-a^2(dr/dt)^2+ 2a^2 r d^2r/dt^2].
In consideration of the 4 configurations that exist between the pairs of charges, Weber divides the constant "a" by 4.
Rather, if we multiply its square by 3, we come out with a force
F=m/r^2[1-3 a^2(dr/dt)^2+ 6 a^2 r d^2 r/dt^2]. (5)
Converting it into the equation of a closed ellipse by setting u=1/r, we get
u" + u =F/h^2 u^2= m[1-3a^2 u'^2-6 a^2 u u"]. (6)
Upon rearranging (6) becomes
(1+6 a^2m)u" + u = m(1-3a^2u'^2), (7)
which is precisely Gerber's equation if we set a=1/c. In fact, Gerber used his result
angular change in the line of apsides = 6 pi m a^2/ p, per period (8)
to determine that the speed of gravity was c.
However, (7) has nothing to do with the geodesic equation written down by Einstein
u" + u = m/h^2+ 3m/c^2 u^2. (9)
The non-Newtonian terms in (7) are quite different. Yet, by some miracle, it happens that the term 3m u^2/c^2 in Einstein's equation and the term -6m/c^2 u u" in Gerber's equation both result in a first order precession given by (8). This clearly shows that the dominating accelerative term in (7) when it comes to creating a first-order precession is a fluke. The fact that A=1+6m/c^2 u>1 results in a cumulative effect of having the angle (phi) increase beyond 2 pi in orther that the ellipse close upon itself. This creates a steady precession.
But, it has nothing to do with the precession that is derived from Einstein's equation. By successive approximation one comes out with [cf Danby, Celestial Mechanics, pp 66-67]
u=(1/p)[1 + e cos(phi) +3m/c^2 e/p^2 (phi) sin (phi). (10)
Neglecting the square of (m/c^2), the equation of the trajectory can be written as
u = (1/p)[1 + e cos [(1-3m/c^2 p)(phi)], (11)
where e is the eccentricity and p the semi-latus rectum of the ellipse. Just as we require
(1+6 a^2m u)>1 for a cumultative effect, so we have (1-3m/c^2p)<1 in (11).
The upshot is that although both give the correct numerical result to first-order, it is only (11) that is consisten with the physical picture of a heavenly body following a geodesic path. Inductive terms as they appear in (7) are completely foreign to gravitational phenomena. The proof of the pudding would be forthcoming if the non-Newtonian terms could some how be amplified to produce a finite difference between the two.
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