Page:AbrahamMinkowski1.djvu/17

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which according to (37) and (40b) is to be brought into the form

(44a) \psi=\frac{1}{2}\mathfrak{ED}+\frac{1}{2}\mathfrak{HB}+\mathfrak{qW}

In order to facilitate the comparison of our results with the approaches of Minkowski, we write

\begin{array}{ccccc}
c\mathfrak{g}_{x}=X_{t}, &  & c\mathfrak{g}_{y}=Y_{t}, &  & c\mathfrak{g}_{z}=Z_{t}\\
\mathfrak{S}_{x}=cT_{x}, &  & \mathfrak{S}_{y}=cT_{y}, &  & \mathfrak{S}_{z}=cT_{z},\\
ct=l, &  & \mathfrak{wK}+Q=c\mathfrak{K}_{t}, &  & \psi=T_{t}.\end{array}

Then the momentum equations (6) and the energy equation (7) read

\begin{cases}
\mathfrak{K}_{x}=\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}-\frac{\partial X_{t}}{\partial l},\\
\\\mathfrak{K}_{y}=\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}-\frac{\partial Y_{t}}{\partial l},\\
\\\mathfrak{K}_{z}=\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}-\frac{\partial Z_{t}}{\partial l}.\\
\\\mathfrak{K}_{t}=-\frac{\partial T_{x}}{\partial x}-\frac{\partial T_{y}}{\partial y}-\frac{\partial T_{z}}{\partial z}-\frac{\partial T_{t}}{\partial l}.\end{cases}

There, the relation exists according to (19a)

X_{x}+Y_{y}+Z_{z}+T_{t}=0

Now, relation (40) means

X_{t}=T_{x},\ Y_{t}=T_{y},\ Z_{t}=T_{z}.

Together with (6a), these relations contain a remarkable symmetry property, which cannot be found in Minkowski's approach. Regarding the behavior under Lorentz transformations, the 10 quantities

\begin{array}{c}
X_{x},\ Y_{y},\ Z_{z},\ -T_{t},\ X_{y}=Y_{x},\ Y_{z}=Z_{y},\\
Z_{x}=X_{z},\ -X_{t}=-T_{x},\ -Y_{t}=-T_{y},\ -Z_{t}=-T_{z},\end{array}

transform as the squares and products of coordinates xyz and of the light-path l. Accordingly, this "space-time-tensor" satisfies the "principle of relativity" in the sense of Minkowski; Also the ponderomotive forces, which we are going to calculate in § 12, thus satisfies the relativity principle.


§ 10. The relation between the theories of Lorentz and Minkowski.


We have emphasized the illustrative meaning of vectors \mathfrak{E,H} in Lorentz's theory, i.e., as being the contribution of the aether at the electric and magnetic excitation. In the theory of Minkowski, the vectors