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,

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

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

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

Twisted Tales from General Relativity: The Deflection of Light

Supposedly one of the greatest predictions of twentieth century physics, was Einstein's prediction of the deflection of light from a massive star like the sun. His pre-relativistic calculation gave half the value that his post-relativistic calculation gave, and this was heralded as a great triumph for general relativity. Yet, there are so many loop holes in the calculation that it's amazing that no one has ever questioned it. 

Gravitational Waves: From Schwarzschild to Michelson

We know very well how a static gravitational field influences the propagation of light from Schwarzschild's solution of Einstein's field equations. But, do we know how light behaves in a passing gravitational wave? It is argued that gravitational waves are "ripples in spacetime", and their effect is to stretch and contract it so that spacetime is a new aether and not a vacuum in which light (and gravitational waves) propagate.

Electromagnetic Waves on Curved Spaces versus Gravitational Waves on Flat Spaces

Einstein developed a set of field equations in order to determine the metric coefficients in what would be otherwise the hyperbolic metric of special relativity. At least one metric, so derived, called the Schwarzschild metric can be written as a Hamilton-Jacobi equation (Landau and Lifshitz, Classical Theory of Fields, p. 306 4th edition). The Hamilton-Jacobi equation is equivalent to the eikonal equation of geometrical optics.

Why is a Black Hole Black?

Black holes started off as "frozen" stars until they were renamed black by Wheeler who also introduced the "no hair" theorem which has fallen from prominence together with the misconceived thermodynamics of black holes. The same Newtonian potential is at work in a black hole as in an ordinary star with the exception that its mass has surpassed 10 solar masses so that neither the electron pressure nor the neutron pressure will be sufficient to stop the star's collapse.

A Black Hole without Gravity

A "black" hole by definition is black to anyone observing it from the outside. But, is it "black" inside? The same gravitational potential exists whether "inside" or "outside". Then what distinguishes "inside" from "outside"? It was shown by Luneburg [Luneburg, The Mathematical Theory of Optics, p. 172.] that light rays acted upon by a Coulomb potential cannot penetrate into regions greater than a fourth of the inverse of the absolute value of the total energy.

How to Derive the Correct Equations of Motion for the Electro-and Gravito-Fields

In our last blog we noticed (at least) two fallacies in the derivation of the equations of motion of the gravitational field equations: 

I. The constraint $A\dot{t}$, where $A$ is any metric coefficient depending on $r$, and $\dot{t}$ is the derivative of time (only one time!) with respect to an affine parameter, $lambda$, cannot be introduced into the variational equations more than once, and

II. Since $t$ is a cyclic (ignorable, kinosthenic, or whatever you want to call it) variable the Lagrange equation reads

Why Gravitational Repulsion Doesn't Exist

There has been much discussion in the literature of anti-gravity machines based on gravitational repulsion that would be the source for propulsion of payloads in the foreseeable future. The possibility of gravitation repulsion comes from an analysis of the Schwarzschild metric by Droste and Hilbert which predict that "a large mass moving faster that $c/\surd 3$ could serve as a driver to accelerate a much smaller mass from rest to a good fraction of the speed of light" (Felber: http://arxiv.org/abs/0910/1084).