Gerhard W. Bruhn, Darmstadt University of Technology
Let us generate a plane wave solution of the Maxwell-equations by using the potentials
(1) A = – i i E0/ω eiω(t – z/c), Φ = 0.
This yields(2) B = curl A = j E0/c eiω(t – z/c) and E = – At = i E0 eiω(t – z/c).
Hence we have
(3) D = ε E = i ε E0 eiω(t – z/c).
Applying Kiehn's definitions of topological transversality we obtain the plane wave (2),(3) to be not transverse electric (since A·D ≠ 0) but transverse magnetic (since A·B = 0).
But now we come to the critical point: Instead of using (1) we could start as well with the potentials
(1') A = j – i i E0/ω eiω(t – z/c), Φ = 0,
which yields the same wave (2),(3) to be NOT transverse magnetic contradicting our former result, since A·B ≠ 0 now.
But no physicist is able to decide by measurements, whether the choice (1) of the vector potential A or the choice (1') or any other additive grad χ (χ time-independent in C1 on the whole space) to (1) would give a correct description of the plane wave. One cannot decide by means of measurements whether a given em-wave is topologically transverse or not.
Hence Kiehn's concept of topological transversality is not well-defined.
Remark: The addition of grad χ to (1) is equivalent to the addition of the exact form dχ to Kiehn's fundamental 1-form A. (χ time-independent in C1 on the whole space)
R.W. Kiehn: Electromagnetic Waves in the Vacuum with Torsion and Spin