Page:AbrahamMinkowski1.djvu/17

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