Page:Grundgleichungen (Minkowski).djvu/41



These quantities are all real. In the theory for bodies at rest, the combinations ($$X_{x},\ X_{y},\ X_{z},\ Y_{x},\ Y_{y},\ Y_{z},\ Z_{x},\ Z_{y},\ Z_{z}$$ are known as 's Stresses", $$T_{x},\ T_{y},\ T_{z}$$ are known as the Poynting's Vector, $$T_{t}$$ as the electromagnetic energy-density, and L as the Langrangian function.

On the other hand, by multiplying the alternating matrices of f and F, we obtain

and hence, we can put

where by L, we mean L-times the unit matrix, i.e. the matrix with elements

$$\left|Le_{hk}\right|\ \left(\begin{array}{c} e_{hh}=1,\ e_{hk}=0,\ h\gtrless k\\ h,k=1,2,3,4\end{array}\right)$$

Since here $$SL = LS$$, we deduce that,

$$F^{*}f^{*}fF = (-S-L)(S-L) = -SS + L^{2}$$,

and find, since $$f^{*}f = Det^{\frac{1}{2}}f,\ F^{*}F = Det\frac{1}{2}F$$, we arrive at the interesting conclusion

i.e. the product of the matrix S into itself can be expressed as the multiple of a unit matrix — a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal