Electromagnetic Strings and Minimal Surfaces

In an extraordinary book published in 1914, and before he became a California immigrant, H. Bateman examined a number of different solution equivalence classes to the Electromagnetic Wave equations of Maxwell.

One class of solutions was generated by a complex vector field, M= E + icB, which satisfied the auxiliary constraint, M.M = 0. Such wave function solutions to Maxwell's equations were defined to be self conjugate fields, for both Poincare invariants of such elctromagnetic fields are zero.

That is: D.E-B.H = 0 and E.B=0.

In the modern language of Cartan both d(A^G) = 0 and d(A^F) = 0.

It is not clear that Bateman recognized that the constraint M.M=0 is sufficient to guarantee that the complex vector M generates a minimal surface [Osserman 1983]. This result had its basis in the S. Lie theorem, every holomorphic function generates a minimal surface.

Now suppose such minimal surface solutions have envelopes with 1-dimensional edges of regression, or self-intersections. Then on the propagating wave solutions there could be 1-dimensional defects or strings, and the strings or defects could themselves propagate. What are the propagation speeds of the strings? As these characteristic solution defects are not linear invariants, but are projective invariants, then such solutions to Maxwell's equations are NOT limited to the finite propagation speed imposed by the linear Lorentz equivalence class


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Last update 01/23/2009
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