Page:AbrahamMinkowski1.djvu/14

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The first two terms are contributions of the aether in the sense of Lorentz's theory, the two latter ones are to be seen as contributions of polarized matter to the electromagnetic momentum density.

We begin to calculate the energy current. With respect to (32) and (31), the expression

(34) \mathfrak{W=[DB]-[EH]=[EM]+[PH]+[PM]}

holds in Lorentz's theory for the vector \mathfrak{W} introduced at the end of § 5. According to (31) and (30), we have

\begin{array}{l}
\mathfrak{E'-[qB]=E-[qM],}\\
\mathfrak{H'+[qD]=H+[qP],}\end{array}

so that equation (21) assumes the form

\frac{\mathfrak{S}}{c}\mathfrak{=\left[E-[qM],\ H+[wP]\right]-q(qW)}

Now, since it is to be set according to (34)

\mathfrak{q(qW)=\left[[qE][qM]\right]+\left[[qP][qH]\right]+\left[[qP][qM]\right]}

then eventually if follows as the value of the energy current

(35) \frac{\mathfrak{S}}{c}\mathfrak{=[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}

The first term can be interpreted as a contribution of the aether, the second one as the contribution of the electrically polarized matter at the energy current, as G. Nordström[1] has shown in a recently published paper, which is remarkable also in other respects; the third term which is added at the motion of magnetically polarized matter, corresponds in such a way to the second one, as it is required by the symmetry of electric and magnetic vectors assumed at this place.


§ 9. Theory of H. Minkowski.


In this theory, the following relations between the electromagnetic vectors hold

(36) \begin{cases}
\mathfrak{D}=\epsilon\mathfrak{E}'-[\mathfrak{qH}],\\
\mathfrak{B}=\mu\mathfrak{H}'+[\mathfrak{qE}];\end{cases}
(37) \begin{cases}
\mathfrak{E'}=\mathfrak{E}+[\mathfrak{qB}],\\
\mathfrak{H}'=\mathfrak{H}-[\mathfrak{qD}].\end{cases}

Also here, besides the vector pairs contained in the main equations, a new vector pair is added, which mediates the relation between them.

  1. G. Nordström, Die Energiegleichung für das elektromagnetische Feld bewegter Körper (Dissertation, Helsingfors 1908).