The Relativity of Kepler's Laws

Everyone thinks of Kepler's laws as being static with the period of orbital motion enters into his third and not time increments. That is, it is the period that is proportional to the area to the power $3/2$, and not the time increments themselves. Yet, time does enter into his law, and there are two different time depending on which focus is used as a reference.

Optical Basis of Kepler's Equation

Kepler's equation gives the orbit of a particle on an ellipse in terms of the time $t$, or better in terms of the mean anomaly, 

$$M=t\sqrt{\frac{GM}{a^3}}=E-\varepsilon\sin E,$$

where, $a$ is the radius that inscribes the ellipse, as shown in the figure below, $E$, is the central angle called the eccentric anomaly, and $varepsilon$ is the eccentricity of the ellipse.

From Newtonian Gravitation to Maxwell's Fish-Eye to Cassini Ovals

It is known [Needham, Am. Math. Monthly 100 (1993) 119-137] that there are only three pairs of integer exponents for dual force laws

$$[1,-2]\hspace{25pt}[-5,5]\hspace{25pt}[-4,-7].$$

 $[\tilde{a},a]$ stands for  pair of exponents in the force laws

$$\frac{d^2z}{dt^2}=-z|z|^{\tilde{a}-1},$$

and

$$\frac{d^2w}{d\tau^2}=-w|w|^{a-1},$$

where $z$ and $w$ are related by the conformal transform,

$$w=z^{\alpha},$$

Snell's Law for Rotational Motion

It has been burned into our minds that the index of refraction must be the inverse of the velocity in units where $c=1$. This has many ramifications like Snell's law here the ratio of the sine of the angle with respect to the normal and the velocity of the ray is constant. And in nonhomogeneous, but spherically symmetric, media where the index of refraction varies with the radial coordinate, the frequency of the wave remains constant making the velocity equal to the wavelength. 

Solving the orbital equation,

Optical Theory of Gravitation: What Newton Didn't Do

Following the publication of the first edition of Principia, Newton set himself to correct and revise it. Newton proposed three distinct methods for producing the solution to the equation of motion of Kepler's problem. In the first edition, curvature played a minor role, but all this changed in the new edition where curvature was elevated to play a major role in determining the law of force.

Incompatibility of the Schwarzschild solution and Newton's Inverse Square Law

Newton proved, save claims to the contrary, that under an inverse square law force every orbit is a conic section. And the reverse is also true that every orbit which is a conic section corresponds to and inverse square law for the force. Newton showed this by using uniform circular motion in which the radius of curvature is constant.

The radius of the Schwarzschild metric is not constant, but, rather

$$\kappa=-\frac{\mathcal{R}}{r^3}\left(1-\frac{3\mathcal{R}}{4r}\right),$$

Time Dilation and Space Contraction from Keplerian Orbits

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

Newton was well-aware of the fact that an inverse cubic orbit implies a bound orbit reaches the origin "by an infinite number of spiral revolutions..." This he first mentioned to Hooke in a letter addressed to him on the 13th of December 1679. General relativity, basing itself on the outer Schwarzschild solution, claims that "naked" singularities, like the inverse cubic law,  don't exist. But, has anyone ever asked to what central force law the Schwarzschild metric corresponds?

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,

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