The Schwarzschild metric has been used to predict the discrepancy between the observed and calculated advance of the perihelion of Mercury and the deflection of light by a massive object. These are two classical confirmations of the validity of General Relativity. However, the Schwarzschild metric,
dr^2 +G^2(r)d phi^2,
where
G'(r)=[1-2m/c^2 G^2]^(1/2) (1)
has zero scalar curvature, where the prime stands for differentiation. That is, the scalar curvature,
R = -G"/G + [(n-2)/2](1-G^2)/G^2 = 0,
for n=3.
The construction of an ellipse as an orthographic projection of a great circle of a sphere given two points on it, identifies the major axis as the diameter of the sphere. The minor axis of the ellipse can be derived from the orthogonal projection of the great circle called the eccentric circle. The inscribed ellipse can be defined as the locus of points P on the ellipse such that the ratio PD/QD is constant, where PD is the vertical distance from the point on the ellipse to the major axis of the ellipse and QD is the distance from the eccentric circle to the same point on the major axis: the ratio represents the minor to the major axes of the ellipse.
Thus we must have a constant radius of curvature, 2/G^2, in order to get elliptical motion. Rather than the relativistic correction (1), containing c^2 in the denominator, we use the index of refraction of the Eaton lens
G' = [2m/G - E]^(1/2), (2)
where E represents the magnitude of the total energy, we get a scalar curvature
R = (1+E)/2G^2.
With respect to (2), the orbital equation is
phi = h integral dr/r[(G'r)^2-h^2]^(1/2), (3)
where h is the constant angular momentum. The solution to this equation is
r = p/[1-e sin phi],
where p= h^2/m, is the semi-latus rectum, and e is the eccentricity
e= [1-h^2E/m^2]^(1/2),
which will result in elliptical motion provided
m/h>E^(1/2).
Calling (1) and (2) the indices of refraction, n_(rel) and n_(norm), respectively, the advance of the perihelion of Mercury is given by the orbital equation:
phi = h integral dr/r[(r n_(norm))^2- (h n_(rel))^2]^(1/2), (4)
and the deflection of light by a massive body being given by the orbital equation with n_(norm)=1. In the latter case, h->h/c which is identified as the impact parameter, having dimensions of length instead of dimensions of action (energyxtime). This is where the relativistic correction (1) applies. By using "natural" units where G=c=1, one is allowed to get away with "murder". The appearance of c^2, or lack thereof, makes all the difference in the world. But, even without it, (2), with E=1, would be the imaginary of (1)! What you would need is E<1, and the whole expression (2) divided by c^2 to get the same result for the bending of light.
A crucial distinction between the orbital equations (3) and (4) is that only the former can be derived from the principle of least time by requiring that
n_(normal){dr^2+r^2 phi^2}^(1/2)
be an extremum. The condition is
n_(norm) r^2 phi'/[1+r^2 phi'^2]^(1/2) = h,
with h a constant and phi'=d phi/dr. Expression (1) will not result in any extremum condition for the path of least time.
To derive both these classic results we need not go beyond dimension n=3. In n=3, scalar curvature characterizes the curvature of the surface, whereas if we were to assume n>3, it would represent only one particular part of the Riemann curvature tensor that would be needed to characterize the manifold.
In regard to the orbital equation (4), the ratio of r/h modulates the magnitude of the two indices of refraction. In this case h must be considered as the impact parameter, or the least distance that the deflecting particle can approach the deflector of mass, m (in "natural" units) so that n_(norm)>n_(rel).
Kepler's transition from circles to ellipses was, indeed, a great step. Surely, it must have involved a projective argument, but scanning the literature, it is not entirely obvious (at least to me) where he used it. In fact, by the arguments of Kepler and Desargues, a projective line can be visualized as a great semi-circle which contains an antipodal pair of a great circle. So it was natural for them to consider a projective line as a closed curve, i.e., a circle. Yet, Kepler remained steadfast in his explanation of the distances to the planets in terms of the five regular polyhedra. The only problem was when more planets were discovered there were not enough polyhedra to go around.
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