It is commonly accepted that the advance of the perihelion is "relativistic". To be relativistic, the speed of Mercury as it orbits the Sun must be a finite fraction of the speed of light. This fraction is 0.000158. In comparison, electronic relativistic corrections start becoming important when the relativistic speed reaches 0.007299, the value of the fine structure constant. Thus, we can safely consider the non-relativistic treatment as being sufficient to explain the anomalous advance of the perihelion of Mercury.
By replacing the square of the charge by the gravitational parameter, m=GM, we can define analogous quantities related to the fine-structure constant, etc. From general relativity, we know that the angular difference over one period of the motion amounts to 6 pi m/c^2 r. If we fix the radial coordinate at the semi-latus rectum, p=h^2/m, then this amounts to 6 pi (m/hc)^2. This looks relativistic enough, because it contains the speed of light, c, but it really isn't.
The gravitational Rydberg constant is
R=(m/h)^2/hc,
which is measured in inverse cm. To get a dimensionless ratio, we have to multiply it by a characteristic distance. In the non-relativistic limit, the semi-latus rectum is analogous to the Bohr radius. The product
Rp= m/hc,
the gravitational fine-structure constant. Squaring results in the expression for the advance of the perihelion, apart from a factor of 6 pi. It is the origin of this factor which explains the source of the advance.
Newtonian mechanics treats the Kepler problem as entirely planar. The semi-latus rectum, p, differs from the semi-major axis, a, by a factor 1-e^2, where e is the eccentricity. Classically, it is an arbitrary constant whose magnitude determines whether the orbit will be open (e>1) or closed (e<1). In terms of the polar coordinates, r, (theta), (phi), the last coordinate is cyclic, and performs rotational motion in the plane. Newtonian theory sets (theta)=pi/2, which forces the motion to be planar. But, if it is not planar, (theta), will perform a libration, or limiting motion that is symmetrical about pi/2, whose limits are given by
sin(theta)=cos i,
where i is the angle of inclination from the plane. Stationary conditions depend on two quantum conditions
J_1=nh and J_2=kh,
where n is the principal quantum number and k the subsidiary quantum number. J_2 is the total angular momentum, and J_3 is the angular momentum component along the polar axis. The total energy depends only on J_1; if J_2 were to vanish, it would correspond to pendulum orbits through the center of force.
The motion, therefore, consists of a libration in r between a(1-e) and a(1+e) together with a uniform rotation of the perihelion; the orbit takes the form of a rosette. The only quantity which is fixed is, a, the semi-major axis. The eccentricity, e, the semi-latus rectum, h, and the semi-minor axis b=(1-e^2)^(1/2) can assume any value consistent with a given value of a. The inverse of a corresponds to the average of the inverse of the radial coordinate.
If we follow the quantum argument, we have that for large values of k, the orbit is situated in an unhindered Coulomb field. For smaller values of k, the surrounding electrons make their presence felt. The potential energy may be expanded in powers of inverse r to give at lowest order
V=-m/r(1+ C(p/r)+...)
where we have taken the numerator in the second term to be the Bohr radius, p=h^2/m, and C is a small constant. The third order term was known to Newton to affect a rotation of the perihelion. However, in terms of the action, the angular momentum term now becomes
k^2 h^2 A
where A =(1-2c/k^2)^(1/2).
Following Born in his Mechanics of the Atom, the orbital equation is
r = p/[1+e cos A(theta-theta_0).
As r goes through one libration, the true anomaly, (theta), increases by 2 pi/A. The smaller the parameter, C, is the close we come to a true ellipse. For small values of C, the path is an ellipse whose perihelion rotates slowly with angular velocity, w C/k^2, where w is the mean motion of a point on the ellipse.
From Gerber's solution to the advance of the perihelion, we know that
(1-2C/k^2)^(1/2)= 1/(1+6m/c^2 r)^(1/2).
If we fix r at the Bohr radius, and expand the square roots we come out with
C/k^2=3 (m/ch)^2= 3(R D)^2,
which is proportional to the square of the gravitational fine structure constant. In the second equality R is the gravitational Rydberg constant, and the characteristic length chosen to make it a dimensionless constant is the impact parameter, D=h/c, or the Compton wavelength.
The true anomaly increases by
2 pi(1+ 3m^2/c^2 h^2);
the correction being proportional to the square of the gravitational fine-structure constant. This is comparable with the deflection of light by a massive object, which is (Moller, The Theory of Relativity, p. 345)
4 m/c^2 D=4(m/hc),
which is proportional to the gravitational fine-structure constant itself.
If we start with the gravitational analogue to the classical electron radius, m/c^2, and multiply it by the inverse of the gravitational fine-structure constant, hc/m, we come out with the Compton radius, h/c. Again, multiplying the latter by the inverse of the fine-structure constant results in the gravitational Bohr radius, h^2/m. A further multiplication of the latter by the fine-structure constant, gives h^3c/m^2, which is precisely the inverse of the Rydberg constant. To convert it into a frequency we multiply R by c. For electronic spectra this is 3.29 x 10^(15) 1/sec, which characterizes high frequency spectra in the X-ray region.
According to Sommerfeld, the formation of a rosette is relativistic; this can be found in his book Atomic Spectra, on page 467. Alternatively, according to Born, every multiply-periodic central motion takes the form of a rosette. In contrast to Newton's planar motion, the total angular momentum has a component in the direction of the polar axis. Quantum mechanically, this corresponds to the magnetic quantum number which differentiates orbitals withing a given subshell. Both the longitude of the node as well as the angular distance of the perihelion from the line of nodes remains constant.
However, Sommefeld's note 16 is enlightening because it establishes the true cause of the advance of the perihelion. The gravitational analog to his equation (9) is
{1+(m/hc)^2}u" + u = m/h^2 x const.,
where u=1/r and the prime stands for differentiation with respect to the azimuthal angle. If we compare this expression with Gerber's, we find
m/r = 6 h^2/r^2,
which requires a balance between gravitational and centrifugal energies. Sommerfeld dismisses his result of the change in azimuthal angle (true anomaly)
(m/hc)^2
by the fact that it is 6 times too small. The 6 appears in Gerber's orbital equation. The, according to Sommerfeld, is sufficient to reject his result in favor of general relativity. Yet, in deriving his result he uses Kepler's II and III laws. For rosette type of motion, these laws should not be strictly valid for the motion is not that of an ellipse.
Gerber's equation reads
(1+6m u/c^2) u'' + = m/h^2+ smaller order terms
If we evaluate r=1/u in the coefficient at the Bohr radius, we get the advance of the perihelion. Deflection of light usually entails taking h->infinity which would destroy the closed orbit. The angular change is
4m/c^2 D,
where D is the impact parameter. Setting it equal to h/c, the Compton wavelength, gives the angular change as the gravitational fine-structure constant. And this is precisely what Moller does in calculating the deflection of light. Note that his expression for the deflection of light following equation (45) on page 353, should have c in the denominator and not in the numerator.
The reason for letting h-> infinity would be to eliminate the constant term on the right-hand side of the orbital equation. From general relativity it follows
(u')^2 = 1/D^2- u^2+ 2m/c^2 u^3,
where D=h/c. Differentiating a second time
u"+u = 3m/c^2 u^2,
which is entirely equivalent to
(1+6m/c^2 u) u" +u =0
which Geber would have obtained.
However, it is not, for the approximations that go into evaluating Einstein's modified equation would lead to an angle of deflection of 4m/hc, instead of 3m/hc, found by Gerber. This has been sufficient reason for its rejection. Instead of finding an angle of 1.75", Gerber would have obtained 1.3125" for the deflection of light about the sun. The current measured deflection is 1.66"+/- 10%. It is not the numerical value that leads to an acceptance or rejection of the theory, but, rather, the physics involved. One cannot take Einstein's modified equation,
u" + u = m/h^2 +3 m/c^2 u^2,
and let h->infinity and come out with a result m/hc for the deflection of light.
It is obvious that we cannot continue to multiply m/c^2 by inverse characteristic lengths since the multiplication of the inverse classical radius would lead to a factor of unity. However, at the other end of the scale, multiplication of the inverse of the Rydberg radius,
h^3c/m^2
would lead to an angular difference of the third power of the fine-structure constant, something so minute that it couldn't be observed.