Page:AbrahamMinkowski1.djvu/17

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

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

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

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 '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 ; 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 's theory, i.e., as being the contribution of the aether at the electric and magnetic excitation. In the theory of, the vectors