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# Every Which Way But Win: Relativistic Correction to the Anomalous Advance of Perihelion of Mercury

No matter what one adds to the soup, it always tastes the same. The same can be applied to the anomalous advance of the perihelion of Mercury. Gerber's exact expression for the advance of the perihelion was labelled a freak because he could not produce the correct numerical value for the deflection of light by a massive body. Yet, when one mixes a stability criterion for a circular orbit, it is claimed that the exact expression for the shift is obtained eventhough the force term that is used violates the claimed condition for the stability of a circular orbit!

Moreover, the derivation, which can be found in R. Fitzpatrick, Newtonian Dynamics (2011), contains an ad hoc mixture of non-relativistic and relativistic expressions. Be that as it may, it should have come as a surprise that the supposely "exact" expression contrains the radius of the stabile circular orbit, and not the semi-latus rectum, which differs from the former by the eccentricity which happens to be non-negligible for Mercury, 0.2056!

The claim that is made is that because "the apsidal angle is an irrational fraction of 2 pi it means that non-circular orbits never close on themselves." However, the entire proof is based on circular orbits, so non-circular orbits never enter the discussion. And the apsidal angle of a circular orbit is an oxymoron. Furthermore, the circular orbit stability theorem, upon which the celebrated Betrand theorem rests [cf. Seeing Gravity] claims that all orbits for which the index n>3 in the force law, f=c/r^n, where c is a constant, are unstable. Yet, Einstein's correction to the Newtonian correction involves a force component with n=4. Would that mean that the advance of the perihelion is unstable?

The expression for the apsidal angle is claimed to be

pi/(3-n)^1/2

where the index n>3 would lead to an imaginary angle and violate the condition of stabilty

3 F(a)/a+ F'(a) >0,

where the prime denotes differentiation with respect to the radius, r, and evaluated at the constant radius of the orbit, a. Yet, the force proposed by Einstein is

F= -GM(1/r^2 + 3h^2/c^2 r^4),

where the angular momentum, h, satisfies the non-relativistic Newtonian law, h^2=GM r, eventhough there is a relativistic correction involving the square of the speed of light, c. Even worse is the finding of General Relativity (Eq. (18) of Chapter XII of Moller's Theory of Relativity) that the angular momentum per unit mass is not conserved, but differs from one by a term -2GM/c^2r, i.e.,

h/(1-2GM/c^2r) = angular momentum per unit mass.

Neglect of the second term in the denominator, which is done, is equivalent to throwing the baby out with the bath water since all relativistic corrections to Newtonian laws are of the same order. The claim that this expression "cannot in general be interpreted as angular momentum, since the notion of a "radius vector" occurring in the definition of angular momentum has an unambiguous meaning only in Euclidean space." (Moller, p. 350) Who affims something like that must be completely ignorant of non-Euclidean geometry.

To this list we may add the error committed by Einstein in his derivation. He finds three roots to the stationary orbital equation. The two larger roots represent the apsides of the orbit. The third, smaller root, is a function of the other two, yet is varied independently of the other two which are then assumed to be constant! Einstein was aware of this inconsistency, and later tried to remedy it. Since he was unable to, which do you throw out: the correct result or the incorrect proof?

If such elementary errors and blatant contradictions can be foun in the derivation of the apsidal angle, in Bertrand's theorem, in the absence of conservation laws, what guarantees us the supposed experimental verifications of General Relativity including the evanescence of gravitational waves?