Beyond the Confines of Relativity: The Breakdown of Symmetry in Past and Future

There is a deep connection between the formula for aberration,

$$\cos\omega=\frac{\cos\omega'-\beta}{1-\beta\cos\omega'},$$

and the Mobius automorphism that swaps the relative state of motion, $\beta$, with one at rest, $0$,

$$\mathcal{M}(\cos\omega')=\frac{\cos\omega'-\beta}{\beta\cos\omega'-1},$$

except for the negative sign. If $\cos\omega'=\beta$, then $\mathcal{M}(\beta)=0$, while if $\omega'$ is a right-angle, then $\mathcal{M}(0)=\beta$. But, there is more to this than what meets the eye.

The Limited Scope of Special Relativity

In his first paper on special relativity, Einstein wrote (in our notation)

$$\cos\omega=\frac{\cos\omega'-\beta}{1-\beta\cos\omega'},$$

for "the law of aberration in its most general form. If $\omega'=\pi/2$, the equation takes the simple form

$$\cos\omega=-\beta$$. In the annotated version of this paper, the angle $\omega$ is formed from "light source-observer" was changed to "direction of motion."

Can You Fit the Entire Universe into a Nutshell?

An old math joke tells you how to catch a tiger: Build a cage in the interior of a perimeter and perform inversion.

Not only does Maxwell's constant $b^2=1/\epsilon_0\mu_0$ not coincide with the velocity of light, but different definitions of the speed of light turn out to be inverses of one another.

One definition used in the Kennedy-Thorndike (KT) interferometer uses a metric

$$c^2dt^2(1-\beta^2)=ds^2\pm2v\cos\omega dt,$$

Will the Real Speed of Light Please Make Itself Known

Never have two numbers been so similar, and, yet, so far apart. James Clerk-Maxwell—-not without understandable reservation—-proposed the equivalence of the speed of light, $c$, and his constant, $b$, which he found as the ratio of two completely and entirely stationary forces, the Coulomb force, $F_E$, and the magnetic force, F_M$, from the Biot-Savart law. 

Indistinguishability Between Electrodynamic and Gravitational Waves

According to Dennis Coyne, "The excitement surrounding gravitational wave astrophysical observation stems from the significant differences between electromagnetic waves and gravitational waves...." According to the conventionally accepted view that "a gravitational wave is a propagating distortion of spacetime which alternately produces out of phase elongations and contractions of space along two axes perpendicular to the propagation direction" would, indeed, make it different from electromagnetic wave propagation which is normal to the electric and magnetic fie

From Kepler's Second to Newton's Third: From Optics to Electrodynamics and Beyond

The conservation of angular momentum is always assumed in systems in which torques to not operate. Yet, non-central forces which depend on angles and velocities seem to challenge the conservation of energy. Angle dependent forces have been known since the time of Ampere, and velocity dependent force which invalidate Newton's third law of action-and-reaction are also well known since the work of Grassmann, and a half a decade later by Lorentz. 

It Takes Two to Tango: The Decline of Observer-Physics

The velocity $v$ seems innocuous enough. But, it has been the seat of many a misunderstanding. In fact, Einstein introduced his theory of special relativity to emphasize the fact that the ultimate velocity is the speed of light, $c$. The position of a body moving from point $x=0$ at $t=0$ along the $x$-axis is simply $x=vt$. That is, we measure $x$ and $t$ and obtain the velocity as $v=x/t$.

Does Gravity Need a Maxwellian Foundation?

The Ampere-Neumann-Weber theory of electrodynamics describes the mutual interaction of current elements or between moving charges that would account for phenomena like that of induction and radiation. Ampere's force is action-at-a-distance and is the antithesis of Maxwell's circuit equations that describe the propagation of electrodynamic fields. In order to apply the latter to material systems one needs an expression for the force that these fields exert on matter.

Weber's Electromagnetic Theory As an Optical Gravitational Theory

Weber's electrodynamics fell out of favor in the later half of the nineteenth century thanks to Maxwell's field theory. The distinction between the two is that the former predicts a pondermotive force between current elements whereas the later gives an electrodynamic force. The former produces a longitudinal force along the radius vector connecting the two elements while the latter produces a transverse force that acts normal to the direction of propagation.