'Relativistic' Phenomena in a Non-Relativistic Arena

We have discussed repulsive gravity within the context of the conformal equivalence between the squares of Hookean ellipses and the orbits in a gravitational field. In particular attractive oscillators corresponds to attractive gravity while a negative Hookean law is equivalent to repulsive gravity. Here, we show that oscillators and gravitational forces lead to non-Euclidean geometries. This has been discussed before by Milnor, "On the geometry of the Kepler problem" in Not. Am Math Soc90 (1983) 353-365.

A Simple Derivation of the Angle of Deflection by Transmuting Elliptic into Hyperbolic Orbits

We have seen that the conformal transformation,

$$z=w^2,$$

converts Newton's nonlinear law of gravity into a linear equation of a harmonic oscillator. Here, we will show that it converts elliptical orbits into hyperbolic ones, and in so doing derive the expression for the angle of deflection of light  when it grazes a massive body like the sun. 

'Naked' Singularities Exist: the Schwarzschild Metric Doesn't Apply

Moller, in his Theory of Relativity dissects Einstein's expression for the deflection of light by a massive body into the sum of a light ray passing through a medium with a nonuniform index of refraction,

$$n_S^2=\frac{c^2}{1-\mathcal{R}/r},$$

and a second contribution coming from the non-Euclidean nature of the metric,

$$\frac{\dot{r}^2}{1-\mathcal{R}/r}+r^2\dot{\varphi}^2=c^2.$$

Observational Gravity

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,

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

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.