Page:The Meaning of Relativity - Albert Einstein (1922).djvu/86

The proof follows directly from the rule of transformation.

Tensors may be formed by contraction with respect to two indices of different character, for example,

$A_{\mu \sigma \tau }^{\mu }=B_{\sigma \tau }$

(61)

The tensor character of $A_{\mu \sigma \tau }^{\mu }$ determines the tensor character of $B_{\sigma \tau }$. Proof—

${A_{\mu \sigma \tau }^{\mu }}'={\frac {\delta x_{\alpha }}{\delta x_{\mu }'}}{\frac {\delta x_{\mu }'}{\delta x_{\beta }}}{\frac {\delta x_{s}}{\delta _{\sigma }'}}{\frac {\delta x_{t}}{\delta x_{\tau }'}}={\frac {\delta x_{s}}{\delta x_{\sigma }'}}{\frac {\delta x_{t}}{\delta x_{\tau }'}}A_{\alpha st}^{\alpha }.$

The properties of symmetry and skew-symmetry of a tensor with respect to two indices of like character have the same significance as in the theory of invariants.

With this, everything essential has been said with regard to the algebraic properties of tensors.

The Fundamental Tensor. It follows from the invariance of $ds^{2}$ for an arbitrary choice of the $dx_{\nu }$, in connexion with the condition of symmetry consistent with (55), that the $g_{\mu \nu }$, are components of a symmetrical co-variant tensor (Fundamental Tensor). Let us form the determinant, $g$, of the $g_{\mu \nu }$, and also the minors, divided by $g$, corresponding to the single $g_{\mu \nu }$. These minors, divided by $g$, will be denoted by $g^{\mu \nu }$ and their co-variant character is not yet known. Then we have

$g_{\mu \alpha }g^{\mu \beta }=\delta _{\alpha }^{\beta }={\begin{matrix}1{\text{ if }}\alpha =\beta \\0{\text{ if }}\alpha \neq \beta \end{matrix}}.$

(62)

If we form the infinitely small quantities (co-variant vectors)

$d\xi _{\mu }=g_{\mu }\alpha dx_{\alpha }$

(63)