In my book, A New Perspective on Relativity, Ch. 7, I draw attention to the fact that the gravitational field acts like a medium to which we can assign a definite index of refraction. Passing from the geometrical optic limit to the wave domain, we can explain things like the orbits as stationary standing waves that are quantized, the perihelion shift as perturbation in the radial wave function in terms of Laguerre polynomials, the deflection of light as a limiting form of the index of refraction of the Eaton lens and so on.

There now seems to be a growing body of evidence that solar orbits are governed by a radial square law in the principal quantum number[cf. references in H Arp, Seeing Red. p. 220.] It has been known for quite some time that the Bode law, where the radii of planets scale as 2^n, with n integer, fails miserably for the outer planets of Neptune and Pluto. Rather, the mean planetary radii using a Bohr condition fare much better for all the planets from Mercury to Pluto, as seen in the diagram taken from Arp's book Seeing Red.

The light rays in a medium of index of refraction n=n(r) are plane curves that are the same for a particle moving in a field of central force. Since the angular momentum is conserved, and points in the direction normal to the x,y-plane we can limit our investigation to rays that lie in the plane. From geometrical optics we have

S_x^2 +S_y^2 =n^2(r)

Transforming to polar coordinates, and noting that dg/dh=a, an arbitary constant, we have

S_r^2+(1/r^2)S_(theta)^2=n^2(r), (a)

or

dS/dr=S_r=+/- {n^2-h^2/r^2}^(1/2). (b)

Now the only assumption that we need is to specify the index of refraction. We take it to be a generalization of the Eaton lens,

n^2= 1/(1-2GM/r) - E. (c)

We take the arbitrary constant, E, to be positive which ensures that the light rays in this medium are identical to the paths of particles that move in a gravitational field, -GM/r. We could, and have, solved (b) directly for the orbits, but it now appears more enlightening to notice that (a) is the geometrical limit of a wave equation. That is we set

R=R_0 exp(iS/K),

where R_0 and K are constant, and introducingi it into (a) we get

R_rr +(1/K^2){1/(1-2GM/r c^2) - E -h^2/r^2}R=0, (d)

by equating S_rr/h=0. Now the type of approximations we make on (c) will lead to different phenomena. Supposing GM/r c^2<<1, results in

R_rr-(1/K^2){h^2/r^2 -(2GM/r c^2-E)}R=0 .

Introducing

R=U exp(-E^1/2 r)

gives

U_rr-2E^1/2 U_r+ (2GM/r c^2K^2-h^2/r^2)U =0,

which is closely allied to the Laguerre d.e.

A non-elegant, but almost always used method is to solve it in terms of a power series, in r^(N+s), where N will turn out to be the principal quantum number. The series will be broken off a N in the coefficients of the power series provided

2(N+s-1)E^1/2=2GM/c^2K.

It is clear that K must have units of action, and since we are not interested in quantizing angular momentum, we find

E_N=(GM/c^2K N)^2. (e)

This makes the energy levels proportional to 1/N^2, just like in the Bohr atom. And since E_N=v_N^2, the invariant v_N x N=const., in terms of the velocity.

These quantized conditions are supposedly applicable to the solar system, thereby replacing Bode's law [Rubcic & Rubcic and Agnese & Festa, as referenced in Arp.]

The advance of the perihelion is obtain by multiplying and dividing through by (1-2GM/c^2) in (c) and omitting the second term in the denominator. The advanced of the perihelion is interpreted as a decrease in the radius of the orbit which is expressed in terms of Laguerre polynomials, analogous to the Stark shift in the hydrogen atom. The deflection of light by a massive body is obtained by setting E=0 in (d).

The decrease in the radius of the orbit that would accompany the slow rotation of the line of apsides can be determined as a perturbation effect, but, also from the condition that R satisfies equation (d), viz.,

S_rr=0.

The condition is

2GM r=h^2(1-2GM/r c^2)^2.

Solving for r, by neglecting the square term in GM, yields for the positive root

r_+=h^2/2GM{1+1-8(GM)^2/h^2c^2}

or

r_+ =h^2/GM - 4GM/c^2=p{1-(2GM/ch)^2}. (f)

The second term is the diminuition in the radius of the orbit.

The period of the orbit is

P=2 pi a (a/GM)^1/2, (g)

where a is the semi-major axis. The energy equation for an elliptical orbit is

v^2=GM(2/r-1/a),

which can be taken as a first-order approximation to our expression for the index of refraction, (c), where E-1=GM/a. By replacing a in (g), and retaining only linear terms in GM,

we find the relative change in the period as

(P-P_0)/P_0= - 6(GM/c h)^2,

where the unperturbed period, P_0, is given by (g).

We can understand this by realizing that a planet will move inward toward the sun by perturbations, i.e., the sun's oblateness, perturbations by other planets, etc. Its potential well become more steep meaning that the planet will complete an orbit having a shorter period. Consider the equation of the orbit,

u" +u =GM/h^2 + n (GM/c^2) u^2, (h)

where u=1/r, the primes mean differentiation with respect to the angle phi, and n is an integer. Inverting (f) gives

u_+= GM/h^2 + 4(GM)^3/h^4c^2,

approximately. Writing

u=u_+ w,

and introducing it into (h), and retaining only linear terms in w, we get

w" + (1-8(GM/c h)^2)w=0,

if n=4. The solution to this harmonic oscillator equation is

w= A cos(k phi-phi_0),

where

k= {1-8(GM/ch)^2}^1/2,

and A and phi_0 are constants. This means the planet will complete one oscillation when

k phi =2 pi.

Solving for phi,

phi= 2 pi/k =2 pi {1+(2GM/ch)^2},

where the second equality is only approximate.

According to GR, we would need only 3/4 of the angular difference. But, then again, the sum of the inverse radii in GR's expression is not constant so that the claimed coincide with the missing 44.98 arcsec is completely fortitious.

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