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Lorentz invariance of Maxwell electromagnetic equations
is demonstrated in two complementary ways: first, we give
a pedestrian review with three-vector equations, and we
then express Maxwell equations in a four-vector matrix form
(the Maxwell-Lorentz matrix) which demonstrates the intimate
connection of Maxwell equations with the Lorentz group.
Each Maxwell-Lorentz matrix component is the product of
three matrices: a derivative matrix, a 4 x 4 Lorentz group
generator matrix, and an electromagnetic field matrix. We
obtain rotary Lorentz transformations of the electromagnetic
field matrix from Lorentz equation matrices. We then transform
the derivative and electromagnetic matrices and obtain an
explicit matrix demonstration of Lorentz invariance of Maxwell
equations. To obtain this result, we express all transformation
matrices in exponential form to facilitate the application
of simple Lorentz group algebra. The pedestrian approach
illustrates what the Lorentz group matrix approach actually
accomplishes and helps one to gain some appreciation of
group theory methods.
A complete collection of equations adequate for relativistic transformation of Maxwell
equations is given by Lorrain et al. (1970) and Lorrain and Corson (1970). We adopt their
convention that the primed system is moving with velocity v with respect to the unprimed
system. In several books that treat relativity with group theory (Tung, 1985; Ryder, 1996; and Carmeli, 2000), the equations show that their primed coordinate system moves with
velocity – v with respect to the unprimed system (inspite of any illustrations to the contrary).
For simplicity, we consider the respective axes of the primed and unprimed systems to be
aligned, and we consider the velocity v to be parallel to the x, y, or z-axis (i.e., an x, y, or
z-boost). |
