Page:The Meaning of Relativity - Albert Einstein (1922).djvu/35

it is assumed that for symmetrical dilatations in all directions, i.e. when

${\frac {\delta u_{1}}{\delta x_{1}}}={\frac {\delta u_{2}}{\delta x_{2}}}={\frac {\delta u_{3}}{\delta x_{3}}}{\text{; }}{\frac {\delta u_{1}}{\delta x_{2}}}{\text{, etc.}}=0,$

there are no frictional forces present, from which it follows that $\beta =-{\frac {2}{3}}\alpha$. If only ${\frac {\delta u_{1}}{\delta x_{3}}}$ is different from zero, let $p_{31}=-\eta {\frac {\delta u_{1}}{\delta x_{3}}}$, by which $\alpha$ is determined. We then obtain for the complete stress tensor,

$p_{\mu \nu }=p\delta _{\mu \nu }-\eta \left[\left({\frac {\delta u_{\mu }}{\delta x_{\nu }}}+{\frac {\delta u_{\nu }}{\delta x_{\mu }}}\right)-{\frac {2}{3}}\left({\frac {\delta u_{1}}{\delta x_{1}}}+{\frac {\delta u_{2}}{\delta x_{2}}}+{\frac {\delta u_{3}}{\delta x_{3}}}\right)\delta _{\mu \nu }\right]$

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The heuristic value of the theory of invariants, which arises from the isotropy of space (equivalence of all directions), becomes evident from this example.

We consider, finally, Maxwell's equations in the form which are the foundation of the electron theory of Lorentz.

$\left.{\begin{aligned}{\frac {\delta h_{3}}{\delta x_{2}}}-{\frac {\delta h_{2}}{\delta x_{3}}}&={\frac {1}{c}}{\frac {\delta e_{1}}{\delta t}}+{\frac {1}{c}}i_{1}\\{\frac {\delta h_{1}}{\delta x_{3}}}-{\frac {\delta h_{3}}{\delta x_{1}}}&={\frac {1}{c}}{\frac {\delta e_{2}}{\delta t}}+{\frac {1}{c}}i_{2}\\&\cdots \\{\frac {\delta e_{1}}{\delta x_{1}}}+{\frac {\delta e_{2}}{\delta x_{2}}}&+{\frac {\delta e_{3}}{\delta x_{3}}}=\rho \end{aligned}}\right\}$

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$\left.{\begin{aligned}{\frac {\delta e_{3}}{\delta x_{2}}}-{\frac {\delta e_{2}}{\delta x_{3}}}&=-{\frac {1}{c}}{\frac {\delta h_{1}}{\delta t}}\\{\frac {\delta e_{1}}{\delta x_{3}}}-{\frac {\delta e_{3}}{\delta x_{1}}}&=-{\frac {1}{c}}{\frac {\delta h_{2}}{\delta t}}\\&\cdots \\{\frac {\delta h_{1}}{\delta x_{1}}}+{\frac {\delta h_{2}}{\delta x_{2}}}&+{\frac {\delta h_{3}}{\delta x_{3}}}=0\end{aligned}}\right\}$

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