Mathematical explanations of physical phenomena may not always be unique. Never has it been truer than in the explanation of gravitational phenomena.
Einstein built into his general relativity that particles move along geodesic paths. This excludes acceleration, period. But then how could Paul Gerber derive the first-order precession from a retarded potential that involved a potential with a "double-dose" of retardation and Einstein's modification of Keplerian motion both give the same answer--to first order, of course?
The discussion is even more poignant when one considers gravity at propagating at a finite speed. Indeed, this was the intention of Gerber, to use his formula for the advance of the perihelion as a means of determining the propagation speed of gravity. The value that coincided with the observational one was, of course, the speed of light.
By expanding his potential
V= m/r (1-dr/dt/c)^2,
in powers of the relative velocity, he cam out with the equation of motion
d^2 r/dt^2- r(d(phi)/dt^2 = F =-m/r^2{1-3(dr/dt)^2+6r(d^2 r/dt^2), (3)
which, among other things contains accelerations. Helmholtz argued that such a term would violate the conservation of energy. But, such a force can be derived from the Weber potential,
W=m/r X {1-(dr/dt)^2/c^2}, (4)
and, hence, is conservative.
As Schrodinger pointed out if (4) were slightly modified to read
W=m/r X {1-3 (dr/dt)^2/c^2}, ( 5)
it would give Gerber's equation (3). The essential point is that if we look upon (4), or (5), as the square of the index of refraction, n^2, the
F = grad n^2,
would be Newton's second law. But, it would violate the assumption that the paths are geodesic! since it contains a velocity term.
In terms of the inverse radial coordinate, u=1/r, Einstein's modification of Kepler's equations reads
u" + u = 1/p + 3m/c^2 u^2 (6)
where p is the semi-latus rectum, and the primes denote the derivative with respect to the azimuthal angle, (phi). In contrast, Gerber's equation (3) would give
u" + u = (1/p){1- 3h u'^2 +6m u u"} (7)
which just happens to give the same result because 3 m u^2 acts the same as 6m u u" to first order. To the list we could add the purely radial equation for the Schwarzschild metric,
u"+u=(1/p){1-(2m/c^2)u+2(V/c)^2}-3m u'^2/(1-2m u/c^2), (8)
which contains extraneous terms not found in Einstein's modified equation (6), and is missing the last term. The term that would come closest to it, is to be found in the Schwarzschild transverse velocity squared, V^2, which would give a term 2(m/c^2)u^2 on the right-hand side of (8), which is one less than in (6). So the same criticism that can be lodged against the derivation of the Gerber equation applies, also, to radial Schwarzschild equation (8).
The Schwarzschild radial equation (8) is derivable from the geodesic equation
d^2 x^a/dp^2 + G^a_(bc)dx^b/dp dx^c/dp =0 (9)
where p is an affine parameter that is related to coordinate time through
dt/dp=1/(1-2m/c^2 r).
G^a_(bc) are the connection coefficients. All this is spelled out in Weinberg's book, Gravitation and Cosmology.
But, what is not spelled out is why go through all the bother when a first integral exists:
u'2 +u^2(1-2m/c^2u) = 1-E(1-2m/c^2 u), (10)
where, according to Weinberg, E is the total energy: >0 for material particles and 0 for photons [cf eqn (8.4.13)in Weinberg]. Clearly taking the derivative of (10) with respect to the azimuthal angle gives
u"+u -3(m/c^2)u = (1/p)(E/c^2), (11)
which encompasses both the perihelion shift and the deflection of light by a massive body where E=0.
So why do business with (8)? which has a long a glorious history [cf McGrudder, "Gravitational repulsion in the Schwarzschild field", and all the references therein that contain illustrious names like David Hilbert]. In fact, it leads to nonsensical results like gravitational repulsion for relative velocities greater than the square root of 3. This is clearly on account of the second-order equation (8) when viewed as a force like that of the Weber force. This I will treat in the next posting.
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