R.W. Kiehn's Transversality Concept in the
Case of Plane Waves

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 e^{iω(t – z/c)},        Φ = 0.

This yields

Hence we have

(3)                   D = ε E = i ε E₀ e^{iω(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 e^{iω(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)

Literature

R.W. Kiehn: Electromagnetic Waves in the Vacuum with Torsion and Spin