Article

# Maxwell equations in matrix form, squaring procedure, separating the variables, and structure of electromagnetic solutions

06/2009; DOI:10.1007/s11587-010-0092-7
Source: arXiv

ABSTRACT The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell
electrodynamics in vacuum is investigated within the matrix formalism. The
matrix form of electrodynamics includes three real 4 \times 4 matrices. Within
the squaring procedure we construct four formal solutions of the Maxwell
equations on the base of scalar Klein -- Fock -- Gordon solutions. The problem
of separating physical electromagnetic waves in the linear space
\lambda_{0}\Psi^{0}+\lambda_{1}\Psi^{1}+\lambda_{2}\Psi^{2}+ lambda_{3}\Psi^{3}
is investigated, several particular cases, plane waves and cylindrical waves,
are considered in detail.

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##### Article:“True Transformations Relativity” and Electrodynamics
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ABSTRACT: Different approaches to special relativity (SR) are discussed. The first approach is an invariant approach, which we call the true transformations (TT) relativity. In this approach a physical quantity in the four-dimensional spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geometric quantity comprising both components and a basis (when some basis has been introduced). This invariant approach is compared with the usual covariant approach, which mainly deals with the basis components of tensors in a specific, i.e., Einstein's coordinatization of the chosen inertial frame of reference. The third approach is the usual noncovariant approach to SR in which some quantities are not tensor quantities, but rather quantities from 3+1 space and time, e.g., the synchronously determined spatial length. This formulation is called the apparent transformations (AT) relativity. It is shown that the principal difference between these approaches arises from the difference in the concept of sameness of a physical quantity for different observers. This difference is investigated considering the spacetime length in the TT relativity and spatial and temporal distances in the AT relativity. It is also found that the usual transformations of the three-vectors (3-vectors) of the electric and magnetic fields E and B are the AT. Furthermore it is proved that the Maxwell equations with the electromagnetic field tensor Fab and the usual Maxwell equations with E and B are not equivalent, and that the Maxwell equations with E and B do not remain unchanged in form when the Lorentz transformations of the ordinary derivative operators and the AT of E and B are used. The Maxwell equations with Fab are written in terms of the 4-vectors of the electric Ea and magnetic Ba fields. The covariant Majorana electromagnetic field 4-vector a is constructed by means of 4-vectors Ea and Ba and the covariant Majorana formulation of electrodynamics is presented. A Dirac like relativistic wave equation for the free photon is obtained.
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• ##### Article:An Invariant Formulation of Special Relativity, or the “True Transformations Relativity,” and Comparison with Experiments
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ABSTRACT: Different formulations of special relativity (SR) are briefly theoretically discussed. In the first formulation SR is understood as the theory of a four-dimensional (4D) spacetime with the pseudo-Euclidean geometry. It is an invariant formulation of SR, which we call the true transformations (TT) relativity. There a physical quantity in the 4D spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geometric quantity comprising both components and a basis (when some basis has been introduced). This invariant formulation is compared with the usual covariant formulation, which mainly deals with the basis components of tensors in a specific, i.e., Einstein''s coordinatization of the chosen inertial frame of reference. The third formulation is the usual noncovariant approach to SR in which some quantities are not tensor quantities, but rather quantities from 3+1 space and time, e.g., the synchronously determined spatial length. This formulation is called the apparent transformations (AT) relativity. Some of the well-known experiments: the muon experiment, the Michelson–Morley type experiments, the Kennedy–Thorndike type experiments, and the Ives–Stilwell type experiments are analyzed using the nonrelativistic theory and the mentioned different formulations of SR. It is shown that all the experiments (when they are complete from the TT relativity viewpoint) are in agreement with the TT relativity but not always with the AT relativity.
Foundations of Physics Letters 01/2002; 15(1):27-69.

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### Keywords

formal solutions

linear space

particular cases

physical electromagnetic waves

real 4 \times 4 matrices

scalar Klein