Page:The Meaning of Relativity - Albert Einstein (1922).djvu/87

multiply by $g^{\mu \beta }$ and sum over the $\mu$, we obtain, by the use of (62),

$dx_{\beta }=g^{\beta \mu }d\xi _{\mu }.$

(64)

Since the ratios of the $d\xi _{\mu }$ are arbitrary, and the $dx_{\beta }$ as well as the $dx_{\mu }$ are components of vectors, it follows that the $g^{\mu \nu }$ are the components of a contra-variant tensor ^[1] (contra-variant fundamental tensor). The tensor character of $\delta _{\alpha }^{\beta }$ (mixed fundamental tensor) accordingly follows, by (62). By means of the fundamental tensor, instead of tensors with co-variant index character, we can introduce tensors with contra-variant index character, and conversely. For example,

${\begin{aligned}A^{\mu }&=g^{\mu \alpha }A_{\alpha }\\A_{\mu }&=g_{\mu \alpha }A^{\alpha }\\T_{\mu }^{\sigma }&=g^{\sigma \nu }T_{\mu \nu }.\end{aligned}}$

Volume Invariants. The volume element

$\int dx_{1}dx_{2}dx_{3}dx_{4}=dx$

is not an invariant. For by Jacobi's theorem,

$dx'=\left|{\frac {dx_{\mu }'}{dx_{\nu }}}\right|dx.$

(65)

1. If we multiply (64) by \frac{\delta x_\alpha'}{\delta x_\beta}</math>, sum over the $\beta$, and replace the $d\xi ^{\mu }$ by a transformation to the accented system, we obtain

   $dx_{\alpha }'={\frac {\delta x_{\sigma }'}{\delta x_{\mu }}}{\frac {\delta x_{\alpha }'}{\delta x_{\beta }}}g^{\mu \beta }d\xi _{\sigma }'.$

   The statement made above follows from this, since, by (64), we must also have $dx_{\alpha }'={g^{\sigma \alpha }}'d\xi _{\sigma }'$, and both equations must hold for every choice of the $d\xi _{\sigma }'$.