Special Relativity and Diagonal Transformations

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Abstract
We discuss the form of the special Lorentz transformation, and the corresponding transformation of the electromagnetic field, in which the transformation matrix is diagonal. We derive the diagonal form of the special Lorentz transformation directly, in a simple way, and show that it is sometimes more convenient to apply than the algebraically equivalent conventional form of the transformation. The convenience is especially evident in deriving the linear Doppler effect, and the relativistic addition of more than two parallel velocities. By writing Maxwell's equations in terms of linear combinations of coordinates which have simple transformation properties, we arrive at the transformation equations of the Maxwell fields in a diagonal form, as well as at the plane wave solutions, in a natural manner. The derivations and applications described above should be of use in a course on relativity because of their simplicity and directness.

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