The reason for the advance of the perihelion had been in plane sight, yet it was deep-sixed and contorted approximation made to solving approximately the equation of the trajectory of the Schwarzschild field. That is, the advance of the perihelion of Mercury is caused by the lack of conservation of angular momentum in the laboratory frame. Transferring to Levi-Civita's fictitious time, the angular momentum is conserved!

The relation between Levi-Civita's fictitious time and coordinate time is

ds/dt=B,

where B=1-2GM/rc^2, the coefficient in the Schwarzschild metric,

d tau^2=B dt^2-dr^2/B+r^2 d phi^2,

in the plane theta=pi/2. The relation between the local time tau and t would then be the square root of B, and not B itself. The concept of a local time in GR is illusory, and so is the metric. It was recognized that the angular momentum in coordinate time,

r^2(d phi/dt)/B=h, (*)

was not conserved. Yet, it is conserved in the Levi-Civita frame,

r^2(d phi/ds)=h. (**)

Yet, (**) was chosen as the approximation since (*) "cannot in general be interpreted as angular momentum, since the notion of 'radius vector' occurring in the definition of the angular momentum has an unambiguous meaning only in Euclidean space." This is definitely untrue.

Using (**) in place of (*), and basing the analysis on a cubic Weierstrass integral for the equation of the orbit, Einstein assumes that the two main roots are unaffected by the small, third root. Yet, he considers their sum as a fixed constant, inversely related to the perihelion shift. If the other two roots representing the apsides are unaffected by the presence of the third root, there will be no effect. What his solution implies is that they are shifted in magnitude by the advance of the perihelion. His subsequent attempts at giving a more convincing proof is testimony to his unhappiness with the path taken to derive a result that had already been given by Paul Gerber shortly before the turn of the century.

In the previous blog we have found the arc length in velocity space to be given by the hyperbolic distance:

int B'dt/B=(1/2)ln{[1+Bw/2^(1/2)][1-Bw/2^(1/2)]},

where the argument is the cross-ratio and w is the inverse velocity in Euclidean space. The prime stands for differentiation with respect to r. Introducing the fictitious time and the conservation of angular momentum, dt->ds->d phi, give:

u_h\int(d phi/B^2)~u_h\int(1+4k/c^2 r)d phi,

u_h=2k/h, is the radius of the velocity circle, inversely proportional to the angular momentum, where K is the gravitational constant, GM. Performing the integration over a period of the motion, we get a positive increase in the orbital angle by an amount

8 pi k/(c^2 L),

where L is the semi-latus rectum, a(1-e^2). The configuration space is increase is 3/4 as large. However, the denominator is often quoted as just a, the semi-major axis, which reduces the quoted value by 1-e^2=0.9559, as in the book by Moller, The Theory of Relativity.

It is indeed strange that over the years many derivations have been presented, but no matter what their differences, the same final result is obtained. That is too close for coincidence. And all make the approximation that angular momentum is conserved in coordinate time, which, needless to say at this point, obliterates the effect. On the observational side, there is still the uncertainty of the ephemerides, which has hardly improved since Le Verrier's time. What astronomers have done is the fit what the ephemerides should be so that the 43 arcseconds/century should be verified. This can hardly be considered as an observational confirmation of general relativity, least of all the approximations to solve the equation of the orbit in configuration space after the effect has been cancelled by assuming that there is angular momentum conservation in laboratory time!

All this raises the more important question: If there is no local time in General Relativity, what is the meaning of the indefinite metric that Einstein generalized from the Special theory?

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