The advance of the perihelion of Mercury has always been the guinea pig to test out theories of gravitation. Rather, we turn the process on its head, and use the advance of the perihelion to determine the properties of gravity when matter is in motion.
To motivate our argument, we start with Weber's law:
F_W=ee'/r^2{1-a^2(dr/dt)^2+(b/r)(d^2r/dt^2)}
for two charges e and e', where the coefficients a and b are to be determined through experiment. It is commonly acknowledged that Weber's force is a synthesis of Coulomb's static field and Ampere's magnetostatic law. This would be true in the absence of the acceleration term in the force law.
Weber divided Ampere's law into longitudinal elements and parallel elements. Weber realized that in the parallel case, the relative velocity of the particles was zero, but, nevertheless, had a relative acceleration. For this reason he added the acceleration term in the above formula. So, he went beyond Ampere's magnetostatic law. But, in order to determine the coefficients in the above formula, he had to reduce it to the magnetostatic case. Weber claimed in his 1846 paper (Sec. 19) that since particles in the parallel current elements do not move in the same straight line, it follows that
d^2r/dt^2=(1/r)(dr/dt)^2.
This is, indeed, cryptic, and nowhere have I found a physical justification of doing so. Yet, it claims that
dr/dt/r=const.
and it is the condition for the vanishing of the electromotive force, Eq (34) in Sec. 860 in his second volume on Electricity and Magnetism. This is nothing else than the condition for the constancy in time of his vector potential, whose time rate of change generalizes Ampere's law and makes it one of Maxwell's circuital equations that gives rise to the propagation of electromagnetic waves. In other words, it is the condition for zero electromagnetic induction..
This, the mysterious condition reduces Weber's force to the sum of Coulomb's force and Ampere's law. It was then up to experiment to show that the ratio of the two coefficients in the reduced law is the ratio of longitudinal to parallel circuit elements of the force for which Ampere determined was equal to the absolute value 1/2. This meant that the longitudinal component is only half as big as the parallel one; moveover, the former decreases electrostatic attraction of opposite charges, while the latter increases it. Having derived this, Weber went back to his full equation,
F_W=ee'r^2{1-a^2(dr/dt)^2+2a^2r(d^2r/dt^2)},
and subsequently showed that the inverse of his coefficient, a^2, is "that relative velocity which electrical masses e and e' have and must retain, if they are not to act on each other at all". This was obviously connected to the speed of light that he and Kohlrausch determined in 1856.
In Newton's case, the condition for the reduction of Weber's equation to an inertial one is
r(dr/dt)=const.
When the conservation of angular momentum is introduced, h=r^2dw/dt, it becomes r'/r=z=constant, where the prime stands for the differentiation with respect to the angle w.
Now, if we introduce this condition into Weber's force, we get
F=2GM/r^2{1-(3/c^2)(dr/dt)^2},
showing that accelerating the acceleration of masses in the parallel case always decreases gravitational attraction where GM replaces ee' in the electrodynamic case. It says that the gravitational field is twice as strong in the transverse plane of a mass than along its axis of propagation. Both fields are attractive, and their magnitudes will show up in the discrepancy of the advance of the perihelion of Mercury, as we shall now see.
When Weber's force is considered the external force in Newton's law,
d^2r/dt^2-h^2/r^3=-2GM/r^2{1+3r(d^2r/dt^2)/c^2},
and replacing r by 1/u, we come out with the equation of the trajectory as
u"(1+6GM u/c^2)+u=2GM/h^2,
where the primes stand for differentiation with respect to the angle, w.
It will be appreciated that the equation of the trajectory is exactly that which was derived by Paul Gerber in 1898, using retarded potentials, to lowest order in 1/c^2. The solution to the equation of motion is
u(w)=1/p{1+e cos(w/A^{1/2})}
where the coefficient A is related to the coefficient in the equation of motion. Since this is always greater than 1, the angle w must increase more than 2pi in order for the system to return to its original position after a revolution. And for orbital motion, the mean value of u or r can be expressed in terms of the orbital constant, p, the semi-latus rectum of the elliptical orbit, namely, u=1/p. This is obtained by setting the derivative u"=0 in the equation of motion. The result,
w/{1+6(GM/hc)^2=2pi,
shows that the angular difference is 6pi(GM/hc)^2, after a single revolution, and this coincides exactly with what general relativity gives. For greater detail see: Seeing Gravity, Sec 5.3.1. But, the new point shown here is that masses in parallel motion decrease the gravitational force twice as much as those in longitudinal motion. In addition, the condition that reduces it to the inertial case is that the aberration angle, alpha=cot^{-1}z, in Newton's inverse square law is a constant of the motion, where z=r'/r.
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