Projections from the Sphere and Pseudosphere and their Indices of Refraction

Based on the eikonal equation of geometrical optics and the Legendre transform, Luneburg was able to associate an elliptical orbit created by a Coulomb field with the line element of a sphere. This meant that to any sphere of a given radius, $r_0$, there belongs a conjugate sphere of radius $r_1=1/r_0$, which represents a perfect undistorted optical image of the original sphere. Its image, however, is inverted, and its magnification is the ratio of the radii, $r_1/r_0$.

The line element,

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

is that of a sphere of radius $r$. The ellipse in the plane is related to the closed curves on the sphere through a stereographic projection, as shown in the figure below.

stereographic projection

Stereographic projection, as is well known from Mercator's maps of the world, preserves angles but distorts lengths, taking circles into circles or lines.

The index of refraction of the sphere,

$$n^2=\frac{1}{(1+r^2)^2},$$

is thus related to the index of refraction,

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

in the plane which represents a Coulomb field of negative total energy $-C$. Consequently, the index of refraction, which represents that found by Maxwell in 1854, determines both the geometry (elliptic) and the dynamics (motion in a Coulomb field). The Coulomb field can be replaced by the Newtonian field, and everything carries through. So this might be a model of gravitational phenomena, like the advance of the perihelion of the planets.

The paths of light rays can be obtained by a conformal mapping of a sphere onto a plane, with the curves on the sphere representing great circles with intersect at antipodal points. All light rays that emanate from one of these points will coalesce at the other, inverse, point, and thus represent a perfect, undistorted image of the former. This was Maxwell's discovery. We can generalize this still further by considering an index of refraction given by

$$n=\frac{r^{\gamma-1}}{1+r^{2\gamma}},\hspace{10pt}\gamma\ge 1,$$

which reduces to Maxwell's fish-eye for $\gamma=1$. Light rays in such a medium are still determined by the equation

$$\varphi-\varphi_0=\int_{r_0}^r\frac{\Delta dr}{r\sqrt{r^2n^2-\Delta^2}},$$

where $\Delta$ represents the distance of closed approach, and $r_0$ and $\phi_0$ determine the origin of the ray.  Given the above index of  refraction, the paths of the light rays in such a medium will be given by

$$r^2+r^{2\gamma}A\cos\varphi+B=0,$$

where $A$ and $B$ are arbitrary constants. 

The form of the equation for the path of rays is extremely interesting for it can represent Cassini's ovals, which he introduced as an alternative to Kepler's ellipses. Cassin's ovals are the loci of points on the plane for which the geometric mean of distances $r$ and $r'$ from each of the foci to the same point is a constant. We obtain it here as a distortion of projecting the curve on the sphere to the plane, requiring the introducing of an index of refraction in a spherically symmetric, but inhomogeneous medium, like that give above.

 

However, when it comes to light deflection by a massive body, the orbits are no longer elliptical, but, rather, hyperbolic. This means a change in the geometry. The coefficient $1-\mathcal{R}/r$ in Schwarzschild's outer solution is not a candidate, but his coefficient for the inner solution, $1-\alpha r^2$ is, where $\alpha is a constant. Instead of the line element of a sphere, we now have

$$ds^2=\frac{dx^2+dy^2}{(1-\alpha r^2)^2}.$$

The line element was first written down by Riemann in 1954, while Beltrami referred to the transformation as "stereograph", who must have been onto its interpretation. Apparently, Riemann was cognisant of its conformal property, but missed out on the connection with hyperbolic geometry. Poincare, using complex analysis, was aware of it all, so, as Stillwell claims, Beltrami is the "missing link" between Riemann and Poincare.

The projection from the hyperboloid, a model of a pseudosphere, onto the plane is shown below.

projection from hyperboloid

Like stereographic projection, it is conformal, but geodesics are no longer straight lines. The index of refraction in the disc is

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

which is the negative of the stereographic projection since $C>0$. We have here a classical scattering problem  with $c^2$ replacing the kinetic energy, and $\Delta=L/c$ as the impact parameter. The equation of the trajectory is

$$\varphi-\pi=\int_{r_0}^r\frac{\Delta dr}{r\sqrt{r^2-2\mathcal{R}r-\Delta^2}}.$$

The hyperbola is given by

$$r=\frac{\Delta}{1+\epsilon\cos\varphi},$$

with an eccentricity given by

$$\epsilon=\sqrt{1+\Delta^2/\mathcal{R}^2}.$$

As, is well-known, the scattering angle is

$$\cot(\theta/2)=\sqrt{\epsilon^2-1},$$

which for small angles gives

$$\theta=2\mathcal{R}/\Delta.$$

The scattering configuration is shown below.

Rutherford scattering

Factors of $2$ in the expression for the angle of deflection, $\theta$, are red-herrings. Without the Newtonian potential you have no scattering at all. The general relativistic expression cannot be consider as a "perturbation": perturbation of what? one may ask, a straight line? It is misleading to justify a theory on numerical coincidences between predictions and observation because with the advancement of technology those "observations" may change. 

A good case in point is the perihelion of Mercury. As Vankov [arXiv:1008.181] contends that instead of try "to fit" Einstein's value to the "anomaly gap", attention should be turned to devising more rigorous methods of analyzing empirical data that remain at Le Verrier's level, where the real precision of the current ephemerides is still largely unknown.

Poincare's conformal metric determines the potential acting in the projection in the plane for $\alpha=8\pi/3(c\tau_N)^2$, where $\tau_N$ is the Newtonian free-fall time. It is a Coulomb type potential, but, unlike stereographic projection, the total energy, $C$ must be positive. Thus, it would be analogous to a negative index of refraction. A "gap" is just that, a gap, and the assumption is that all effects, the precession of the equinoxes and the perturbations by the planets are additive.