Dynamics and Geometry from the Index of Refraction

In what was a most beautiful idea, Luneburg was able to connect the index of refraction for a Coulomb potential with Maxwell's fish-eye whose index of refraction is the line element for a sphere. From Fermat's integral

$$\int n(r)\sqrt{1+\varphi^{\prime\;2}}dr= \mbox{extremum},$$

where the prime stands for differentiation with respect to $r$, it follows that rays will follow geodesics that are given by equation of the orbit

$$\varphi=\int\frac{Ldr}{r\sqrt{r^2n^2(r)-L^2}},$$

where $L$ is the constant appearing in the first  integral for $\dot{\varphi}$.       

The index of refraction in Fermat's integral, $n(r)$, a function of the radial coordinate $r$ in a medium of spherical symmetry, has the form of a conformal factor in the line element,

$$ds^2=n^2(r)\left(dx^2+dy^2\right).$$

What Luneburg realized that the index of refraction for Maxwell's fisheye,

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

gave the line element for a sphere,

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

Now the beauty of his argument is that the index of refraction for the fisheye is related to the index of refraction for the Coulomb potential,

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

where $C$ is an arbitrary constant, by a Legendre transform. 

The eikonal equation is

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

where the prime stands for differentiation with respect to the argument, in this case, $r$. We can define the Legendre transform of Hamilton's principal function $S(r)$, as

$$R(p)=pr-S(r),$$

where a new coordinate, $p$ is introduced by

$$p=S^{\prime}(r).$$

Likewise, the inverse function $R(p)$, can be used to derive $S$ as

$$S(r)=pr-R,$$

where

$$r=R^{\prime}(p).$$

In terms of the coordinate $p$, the eilonal becomes

$$p^2=C+R^{\prime\;-1}.$$

Rearranging and squaring gives

$$R^{\prime\;2}=\left(\frac{1}{p^2-C}\right)^2=\left(\frac{1}{1+p^2}\right)^2=n^2(p),$$

which is the index of refraction for Maxwell's fish-eye when we set $C=-1$. We can forget about thinking about $p$ as momentum and $r$ as a radial coordinate, and consider them both as coordinates on par with one another. 

Now, we can introduce the two expressions for the indices of refraction into the equation of the orbit and consider the geometrical figures that are obtain in the projected plane and on the surface of the sphere. In the former case we have

$$\varphi^{\prime}(r)=\frac{L}{r\sqrt{-r^2+r-L^2}},$$

which can easily be integrated to get the equation of an ellipse,

$$r=\frac{2L}{1+\epsilon\cos\varphi},$$

where the eccentricity is $\epsilon=\sqrt{1-4L^2}<1$. 

On the sphere, the geodesics will be given by

$$\varphi^{\prime}=\frac{\sqrt{1-4L^2}L(1+r^2)}{r\sqrt{r^2-L^2(1+r^2)^2}}.$$

The equation of light rays on the surface of the sphere is

$$r^2-r\frac{\sqrt{1-4L^2}}{L}\cos\varphi-1=0.$$ 

Following Luneburg, we transform to Cartesian coordinates in the $x,y$-plane, and obtain the equation of a set of circles,

$$\left(x-\frac{\sqrt{1-4L^2}}{2K}\right)^2+y^2=R^2,$$

of radius $R=1/2L$ that go through points $y=\pm 1$ on the $y$-axis. 

If we consider the extreme points on the $x$-axis on the unit circle with center at the origin, all rays that intersect at $x_0$ will also intersect at the second extreme point $x_1=-1/x_0$. The same holds true for any pair of diametrically opposite points with $x_0x_1=-1$ whose line goes through the origin. Transferring to the sphere the conclusion follows that for any sphere of radius $r_0$ there will be a conjugate, or inverse, sphere of radius $r_1=1/r_0$ which is a "perfect and undistorted optical image of the sphere of radius $r_0$." The image is, of course, inverted. As a quaint way of putting it, Hebe, the Greek goddess of youth would see the back of her head upside down in such a spherical mirror.

On the sphere, great circles intersect the equator at two opposite points, so that if one is the source the other is the sink, as shown in the figure.

 

diametrically opposite points on the sphere in which all rays intersect

 

All rays emanating from one point will coalse at the second point indicating perfect, undistorted imaging. The spherical mirror can be replaced by a pseudo sphere, and what was elliptic geometry becomes hyperbolic geometry. The interesting thing is that the index of refraction, $n^2$, becomes negative. Either the permittivity or the permeability is negative, but not both. The  single NIMs (Negative Index Materials reported in the literature have negative permittivities. We will develop this theme in the next blog.

The point that we want to stress is: There is no need for using any criterion for determining the metric coefficients. Everything follows once the index of refraction has been specified. And the specification comes from physical considerations, as opposed to the vanishing of the Ricci tensor in an "empty" universe. It also provides a criterion of which coefficients are acceptable on physical grounds. We will see that the inner solution to the Schwarzschild metric is acceptable whereas the outer solution is not. In fact, we will see that the inner solution corresponds to the Poincare' disc model of the hyperbolic plane.

When light orbits on a sphere, or pseudosphere, become projected onto the plane they become distorted required the introduction of an index of refraction, just like light passing from air into water. In the spherical case, the conjugate curve, or the antipode of the original curve, has the same optical length. It's optical image in the plane will also have the same optical length when the index of refraction is introduced.