Relativistic Electrodynamics: Various Postulate and Ratiocination Frameworks

Abstract

Abstract-Presently various models consistent with Einstein's Special Relativity theory are explored. Some of these models have been introduced previously, but additional models are possible, as shown here. The topsy-turvy model changes the order of postulates and conclusions of Einstein's original theory. Another model is given in the spectral domain, with the relativistic Doppler Effect formulas replacing the Lorentz transformation. In this model a new principle tantamount to the constancy of the speed of light in vacuum is stated and analyzed, dubbed as the constancy of light slowness in vacuum. Because the slowness is derived in the spectral domain from the Doppler Effect formulas, this result is not trivially semantic. It is shown that potentials and equations of continuity can replace the Maxwell Equations used by Einstein for his "Principle of Relativity" in electrodynamics. It is also shown that defining convection currents and assuming the current-charge densities transformations can replace the Lorentz transformation. The list of feasible models representative rather than exhaustive, since parts of the models presented here can be combined to yield additional models. The two underlying elements of Einstein's original Special Relativity theory are always present: (1) the theory requires a kinematical element (e.g., the constancy of the speed of light in vacuum in Einstein's original model), and (2), a dynamical element (e.g., the form-invariance of the Maxwell Equations in all inertial systems of reference in Einsteins original model).

Non-Relativistic Scattering: Pulsating Interfaces

“…This so-called principle of the invariance of the phase [4,16], (41), becomes exact in Special-Relativity theory which involves inertial reference-frames moving at constant velocities. Therefore, taking off the amplitude effect from the waves in question, and avoiding the approximations of the velocity-dependent exponentials, yields the waves as measured by an observer at-rest with respect to medium {1 }, but expressed in terms of r T , t T coordinates.…”

“…Finally, the spatiotemporal transformation (2), (3), in the form (16) relevant to the present problem can be substituted, in order to derive the scattered wave in terms of the coordinates of an observer at-rest with respect to the initial medium {1 } at-rest.…”

Broadband Spatiotemporal Differential-Operator Representations for Velocity-Dependent Scattering

“…The wave-equation (4) is obtained from Maxwell's equations in sourceless domains (2), including the constitutive equations, e.g., (3), by repeated substitution reduction, or equivalently, by equating to zero the symbolic determinant of the system. Applying the wave operator F (∂ R ) to the integral (1) and equating to zero defines f (R) as a solution of (4). Now, interchanging order of integration and differentiation, and applying the derivatives to the exponential, yields…”

“…For more detail see also [3,4]. Accordingly if (2) is valid in an inertial reference-frame Γ, it is also valid in another arbitrary inertial reference-frame Γ in the form…”

“…Postulating the phase-invariance "principle" (which is actually superfluous if we already have declared K, R, to be Minkowski fourvectors [3,4]), we have…”