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Why the Relativistic Explanation of the Advance of the Perihelion of Mercury is Incorrect

Updated: Oct 31, 2019

A triumphant feat was attributed to Einstein's General Theory of Relativity in that it hit on the nail the missing 43 arcseconds per century that Newtonian theory could not explain. Prior to Einstein's explanation various authors tried to account for the advance in a variety of ways. Not least of all was Hall's attempt to show that a minute modification of the exponent in Newton's law would do the trick.


Hall's hopes were seeming dashed by Ernest Brown's 1903 refinement of the Hill=Brown lunar theory which was of sufficient precision to rule out Hall's hypothesis by empirically constraining the absolute departure d in the modified force law 1/r*(2+d) from the exact inverse square law to be less than 4x10^(-8). That is, if the value of the motion of the moon's perigee and node are correct, the greatest difference between theory and observation is 3 tenths of an arcsecond making the exponent d <4x10^{-8). Such a value would be insufficient to explain the outstanding deviation in the motion of the perihelion of Mercury. According to unanimous opinion, the true explanation had to await Einstein's relativistic explanation of the anomalous precession found from Schwarzschild's metric.


Yet, what seems to have gone unnoticed that Schawarschild's solution also entails a modification of the Newtonian law of attraction! That is a new force law, of the inverse fourth power, is superimposed upon the Newtonian law of attraction. Now, superimposing one force law upon another is not new--it was known to Newton himself. By adding an attractive centrifugal force term to the already existing one, Newton was able to obtain a new orbit that resembled the old one except that the new orbit is continuously rotating. The apsidal angles are increased by a factor of 1/k, where k is the ratio of the old to new angular momenta.


So when an inverse cubic law is superimposed upon an inverse square law, the stability of the orbit is not changed---it remains inverse square. Not so in Einstein;s case. Here an inverse fourth law is superposed on an inverse square law, and the orbit will remain stable so long as the radial distance is greater than sqrt(3)h/c, where h is the angular moment and c is the speed of light in vacuum. This condition is testimony to the fact that the law is no longer inverse square because if it were, the second term in the force law

f(r)=GM/r^2{1+3h^2/c^2r^2}

would have no effect upon the stability of the orbit, just like when h^2/r^3 is added onto the force law in Newton's case. His force law reads

f(r)=GM/r^2{1+p/r},

where p is the semi-latus rectum. In fact, the coefficient of the second term is immaterial since the inverse cubic will have absolutely no affect on the shape or stability of the orbit. In other words, Newton's force satisfies the stability condition

3f+rf'>0

unconditionally, where the prime stands for differentiation with respect to r. In contrast, Einstein;s force requires

r^2>3h^2/c^2,

and, consequently, the superimposed force law is not an inverse square law. This is the exactly the same criticism that was lodged against Hall!


But, Einstein's case is a little more complicated because it has the angular momentum in the potential. Considering orbits with constant energy, E, but variable angular momentum, h, The expression for the angular momentum is

h^2={2r^2E+2GMr-r^2r*^3}/(1-2GM/c^2r).

The condition that h have a maximum will determine E as a function of r_0, obtained from the stationary condition. If we vary h with respect to r and r' and demand that r*=0, we will limit ourselves to a circular orbit.


From the stationary condition with respect to r,

E=-GM/2r_0{1-GM/c^2r),

to terms of first order in GM/c^2. The condition that h be a maximum is

E(1-4GM/c^2r)+ (GM/cr)^2 <0,

which is not the condition that would be obtained from the inverse square law, viz. E<0! This clearly shows that Einstein is open to the same criticism as lodged against Hall.


There is no disputing the numerical coincidence of Einstein's result and the 43 arcseconds per century. However if we set this equal to the ratio of the new to old angular momenta, we find that the angular momentum of the superimposed field must be

h'=(GM/p)^(1/2) 3GM/c^2.



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