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Planetary Orbits in Constant and Non-Constant Curvature Planes

In the Foward to Peter Bergmann's book on The Introduction to the General Theory of Relativity, Einstein wrote "It is true that the theory of relativity, particular the general theory, has played a rather modest role in the correlation of empirical facts so far...." Undoubtedly, this is due to the minuteness of the effects that it has predicted. Yet, even these effects have been given questionable derivations, as Einstein and Bergmann have admitted on page 211 by providing yet a still additional proof of the advance of the perihelion using a Fourier series approach. The "proof" could not have gone astray since the exact expression for the shift was known since the work of Gerber who derived it from a totally different approach that involved a "double dose" of retardedness in the gravitational potential, opened the demonstation to criticism.


Einstein's development generalizes the indefinite metric of light propagation by allowing the (gravitational) coefficients to vary both in space and time. These are supposedly determined by solving the Einstein field equations which connect the Einstein tensor with the energy-stress tensor. The Einstein tensor is a variation of the Ricci tensor that was necessary in order that its covariant divergence vanish, being an expression of energy-momentum conservation.


The perihelion advance was simpler in that only the gravitational potentials were dependent upon space, and the Einstein conditions were solved "in vacuum" albeit a central mass popped up as an arbitrary constant of integration--an apparent contradiction!


Here, we will show that the "same type" of correction is required to straighten out paths on a surface of constant positive curvature so that the same orbital equation is obtained as in the Euclidean plane--with the only difference that instead of the constant total energy appearing in the integral, there is now the difference between twice the total energy and the square of the angular momentum per unit mass.


Schrodinger tells us that the correct form of the Coulomb (Newton) potential (corresponding to 1/r in flat space) is cot(theta), where theta like the of two angles necessary to define a spherical harmonic varies from 0 to pi, and corresponds to the distance from the origin divided by the radius of curvature. Since cot(theta) corresponds to 1/r,

r = tan(theta), (1)

and in the hyperbolic case, r= tanh(theta), which is a Lobachevskian straight line. The justification of (1) follows from the fact that the potential U, satisfies the Laplace-Beltrami equation

[1/sin^2(theta)] d/d(theta)[sin^2(theta)dU/d(theta)] =0, (2)

which is easily seen to be satisfied by

U=-m cot(theta) + const. (3)

where m is the gravitational parameter.


The metric is well-known to be

dt^2= d^2(theta) + sin^2(theta) d^2(phi). (4)

Since phi is an ignorable coordinate, a first integral should exist. Consider

integral [1+ sin^2(theta) (phi)'^ 2]d(theta) = extremum.

Taking the variation with respect to (phi)', where the prime indicates differentiation with respect to (theta), we get the angular momentum integral,

h= sin^2(theta) d(phi)/dt (5)

provided

(d(theta)/dt)^2 + sin^2(theta) (d(phi)/dt)^2 =1,

which follows from (4).


However, since (1) holds, we would have expected to see an angular momentum of the form

h= r^2 d(phi)/dt] = tan^2(theta) d(phi)/dt, (6)

than that of (5). The conversion of (6) into (5) requires the division of (6) by sec^2(theta), and this is precisely the square of the "relativistic index of refraction in the orbital equation

(dr/d(phi))^2= (r/h)^2[n_1^2 r^2- h^2n_2^2] (7)

where

n_1^2 =2(E+U(theta))=2(E+m cot(theta)) (8)

E is the total energy, and

n_2= sec(theta)=1/cos(theta). (9)


The orbital equation (7) thus becomes

csc^2(theta) d(theta)/d(phi) =+/- (1/h^2){2(E+m cot(theta))-h^2/sin^2(theta)}^(1/2), (10)

where special attention should be paid to the last term, which has the proper form for the angular momentum. Transforming to u=1/r=cot(theta), gives an energy integral

(h^2/2){u'^2+u^2} -m u = E-h^2/2, (11)

which differs from its flat version by the appearance of the last term on the right hand side instead of the energy itself. Nonetheless, the second order orbital equation,

u" + u =m/h^2=1/p, (12)

where p is the semi-latus rectum, is identical to its flat version. So the second index of refraction, (9), was necessary to bring about the coincidence of the orbital equation of a positive constant curvature space and flat space.


For an ellipse, (11) must be less than zero. This takes the pressure off of E<0 since the angular momentum aids the fact that (11) can be negative even with positive total enery, E.


Under the transform (rho)=sin (theta), or d(rho)=cos(theta)d(theta), the metric (4) becomes

dt^2=d^2(rho)/(1-rho^2) + (rho)^2 d(phi)^2, (15)

where the denominator in the first term on the right hand side is the square of the inverse of the second index of refraction, (9). In fact, if x and y are any two vectors, their inner product (x,y) defines the elliptic metric [ cf. H Busemann, The Geometry of Geodesics, p. 375]

theta = Arc cos (x,y) (16)

if x and y are of unit lengths. The surface area is proportional to sin^2(theta), and the length of the circumference of a great circle on a sphere is proportional to sin(theta). For hyperbolic geometry of constant curvature, the substitution (theta)->i (theta), where i is the imaginary number so that for a hyperbolic metric,

theta= Arc cosh (x,y).


Let G= sin(r/R), and take the derivative with respect to r to give G'=(1/R) cos(theta). A second derivative yields

-G"/G= 1/R^2,

the constant curvature of a sphere with radius, R. This is equal to the sectional curvature

(1-G'^2)/G^2= 1/R^2,

and it makes no sense to look for an average of the two as one would to form the eigenvalue of the Ricci tensor in a given direction. In the relativistic case,

G'={1-2m/c^2/G}^1/2. (17)

Surprisingly enough, this can be considered the imaginary of

{2(E+m/G)}^1/2 (18)

for an energy, E=-(1/2)c^2, in which the magnitudes of (17) and (18) differ by a mere scale factor, c, the speed of light. Imaginary indices of refraction are significant of reflection.


Now, the sectional curvatures are proportional to the densities

G"/G = m/G^3 c^2 (19)

and

(1-G'^2)/G^2 = 2m/G^3 c^2. (20)

when referred to the principal axes of a spheroid, it is seen to be a prolate ellipsoid, a>b=c,

where a is proportional to (20) and b and c to (10).


We thus have found a general procedure for considering "relativistic" corrections to flat plane orbits by specifying the second index of refraction, n_2. A criterion for the validity of such an index of refraction is that its sectional curvatures must correspond to physical quantities, like the square of the inverse of the radius of curvature, or a constant density.


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