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* E_{0}/ω e^{iω(t – z/c)},
*Φ* = 0.

This yields

(2)Hence we have

(3)
**D** = ε **E** = **i** ε E_{0} 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* E_{0}/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 C^{1} 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 C^{1}
on the whole space)

**Literature**

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