Page:The Meaning of Relativity - Albert Einstein (1922).djvu/56

when we consider Maxwell's equations that these may be looked upon as tensor equations, provided we regard the electromagnetic field as a skew-symmetrical tensor. Further, it is clear that the skew-symmetrical tensor of the third rank (skew-symmetrical in all pairs of indices) has only four independent components, since there are only four combinations of three different indices.

We now turn to Maxwell's equations (19a), (19b), (20a). (20b), and introduce the notation:^[1]

$\left.{\begin{matrix}\phi _{23}&\phi _{31}&\phi _{12}&\phi _{14}&\phi _{24}&\phi _{34}\\\mathbf {h} _{23}&\mathbf {h} _{31}&\mathbf {h} _{12}&\mathbf {-ie} _{x}&\mathbf {-ie} _{y}&\mathbf {-ie} _{z}\end{matrix}}\right\}$

(30a)

$\left.{\begin{matrix}J_{1}&J_{2}&J_{3}&J_{4}\\{\frac {1}{c}}\mathbf {i} _{x}&{\frac {1}{c}}\mathbf {i} _{y}&{\frac {1}{c}}\mathbf {i} _{z}&\mathbf {i} \rho \end{matrix}}\right\}$

(31)

with the convention that $\phi _{\mu \nu }$ shall be equal to $-\phi _{\nu \mu }$. Then Maxwell's equations may be combined into the forms

${\frac {\delta \phi _{\mu \nu }}{\delta x_{n}u}}=J_{\mu }$

(32)

${\frac {\delta \phi _{\mu \nu }}{\delta x_{\sigma }}}+{\frac {\delta \phi _{\nu \sigma }}{\delta x_{\mu }}}+{\frac {\delta \phi _{\sigma \mu }}{\delta x_{\nu }}}=0$

(33)

as one can easily verify by substituting from (30a) and (31). Equations (32) and (33) have a tensor character, and are therefore co-variant with respect to Lorentz transformations, if the $\phi _{\mu \nu }$ and the $J_{\mu }$ have a tensor character, which we assume. Consequently, the laws for

1. In order to avoid confusion from now on we shall use the three-dimensional space indices, $x,y,z$ instead of $1,2,3$, and we shall reserve the numeral indices $1,2,3,4$ for the four-dimensional space-time continuum.