We would like to invert the program of general relativity. Instead of beginning with a theoretical framework, which we have shown to be shaky at the very least, let us start with the deflection of light by a massive body and the advance of the perihelion and try to infer the refractive index. This will give us the both the mechanism and potential by which the phenomena occur.

We have seen that that the generalized index of refraction,

$$n=\frac{2r^{\gamma-1}}{1+r^{2\gamma}},$$

pertains to Maxwell's fish-eye for $\gamma=1$, and the element of the line element,

$$ds^2=n^2\left(dx^2+dy^2\right)=4\frac{dx^2+dy^2}{(1-r^2)^2},$$

corresponds to the line element of a sphere. In the projection of the sphere onto the plane, it corresponds to an ellipse with an index of refraction

$$n^2=\frac{2}{r}-1,$$

which is a Coulomb potential with negative total energy $-1$.

For $\gamma=2$, we have the bipolar polar equation (in its most general form)

$$r^{2\gamma}-2a^2r^{\gamma}\cos(2\vartheta) +a^4(1-\epsilon^4)=0,$$

where $\epsilon=b/a$, $b^2=r_0r_1$, the square of the geometric average of distances from each of the focal points to a point on the curve, and $a$ is distance between the pole and the focal points. Cassini ovals are loci of points for which the geometric mean, $b$ of the distances to any two points, the foci, is constant. The field lines are the sum of two orthoradial $1/r$ fields.

If the geometric mean is replaced by the arithmetic mean, there results the ellipses of Kepler's orbits, while if it were replaced by the harmonic mean, there would result Cayley ovals. Instead of slicing a cone to get conic sections, we can slice a torus to get spiric sections, which were already known to Menaechmus some 150 BC. It has been hypothesized that that such generalizations may lead to a gravitational theory with a fourth-rank metric tensor [cf. references in J A Nieto and L A Beltran, arXiv:1406.0779]. It seems that the Einstein bug has caught on and one tries, no matter how hard, to generalize the second-rank metric tensor of general relativity. Happily, things are much more simple so that Nature does not have to go to such exaggerations.

As Luneburg has shown, the key is in the generalization of the polar equation of a straight line,

$$\vartheta=\Delta\int\frac{dr}{r\sqrt{r^2-\Delta^2}}=\sin^{-1}\frac{\Delta}{r}.$$

If we introduce the so-called turning parameter, $\rho\equiv rn$, an analogous integral can be written as

$$\Delta\int\frac{d\Omega(\rho)}{\sqrt{\rho^2-\Delta^2}}=\Psi(\Delta),$$

where $\Omega=\ln\rho$. To find $\Psi$, we avail ourselves of an Abel transform,

$$\Omega(\rho)=\ln\rho=\frac{2}{\pi}\int\frac{\Psi(\Delta)d\Delta}{\sqrt{\Delta^2-\rho^2}}.$$

Oddly, enough, the two integrals are related by the expression for the total angle of deflection,

$$\frac{\chi}{2}=\frac{\pi}{2}-\vartheta_{\infty},$$

where $\vartheta_{\infty}$ is the angle that a ray makes at its closest point to a massive body. Since $\vartheta_{\infty}=\cos^{-1}(\Delta)$ when the limits of the above integral is taken to be from $\Delta$ to $1$, the trigonometric expression,

$$\cos^{-1}(\Delta)=\frac{\pi}{2}-\sin^{-1}(\Delta),$$

gives $\chi/2=\sin^{-1}(\Delta)$. We can express this as the integral equivalence,

$$\frac{2}{\pi}\int_{\Delta}^{1}\frac{\cos^{-1}(\Delta)d\Delta}{\sqrt{\Delta^2-\rho^2}}=\int_{\Delta}^{1}\frac{d\Delta}{\sqrt{\Delta^2-\rho^2}}-\frac{2}{\pi}\int_{\Delta}^{1}\frac{\sin^{-1}(\Delta)d\Delta}{\sqrt{\Delta^2-\rho^2}}$$

$$\hspace{25pt}=-\ln\rho$$.

Now,

$$\int_{\Delta}^{1}\frac{d\Delta}{\sqrt{\Delta^2-\rho^2}}=\ln\left(\frac{1+\sqrt{1-\rho^2}}{\rho}\right),$$

we can infer that the second integral is

$$\frac{2}{\pi}\int_{\Delta}^1\frac{\sin^{-1}(\Delta)d\Delta}{\sqrt{\Delta^2-\rho^2}}=\ln(1+\sqrt{1-\rho^2}).$$

It is now easy to see that we can replace the arcsine by the total angle of deflection, $\chi$,

$$1+\sqrt{1-\rho^2}=\left(\rho^{-1}+\sqrt{\rho^{-2}-1}\right)^{1/\gamma},$$

where we have set $\chi=\pi/\gamma$, with $\gamma>0$.

The beauty of the above equation is that by setting it equal to $r$, the left-hand side gives

$$n^2=\frac{2}{r}-1,$$

while the right-hand side gives

$$n=\frac{2r^{\gamma-1}}{1+r^{2\gamma}}.$$

That is, the index of refraction in the projected plane gives all the indices of refraction on the sphere and its generalizations!

The former index of refraction is that of the Eaton lens at an angle of refraction of $\pi$. It's negative is the index of refraction in the Schwarzschild metric [cf. Moller, op. cit. p. 355 eqn (54)]. Since it requires $r>2$, it is not valid for discussing deflections! We will have more to say about this in a forthcoming blog.

It is important to observe that $\chi$ is the *negative* of what the general relativistic treatment of the deflection of a light ray gives [cf. eqn (52), page 354 in Moller, *The Theory of Relativity*]. If that angle is positive, our $\chi$ is negative meaning that the right ray will bend inward, and not outward, in accordance to Snell's law. We will have more to say about this in the next blog.