Page:The Meaning of Relativity - Albert Einstein (1922).djvu/27

of rank $\beta$ we may obtain a tensor of rank $\alpha +\beta$ by multiplying all the components of the first tensor by all the components of the second tensor:

$T_{\mu \nu \rho }..._{\alpha \beta }...=A_{\mu \nu \rho }...B_{\alpha \beta \gamma }...$

(10)

Contraction. A tensor of rank $\alpha -2$ may be obtained from one of rank $\alpha$ by putting two definite indices equal to each other and then summing for this single index:

$T_{\rho }...=A_{\mu \mu \rho }...(=\sum \limits _{\mu }A_{\mu \mu \rho }...)$

(11)

The proof is

$A_{\mu \mu \rho }'=b_{\mu \alpha }b_{\mu \beta }b_{\mu \gamma }...A_{\alpha \beta \gamma }...=\delta _{\alpha \beta }b_{\rho \gamma }...A_{\alpha \beta \gamma }...=b_{\rho \gamma }...A_{\alpha \alpha \gamma }...$

In addition to these elementary rules of operation there is also the formation of tensors by differentiation ("erweiterung"):

$T_{\mu \nu \rho ...\alpha }={\frac {\delta A_{\mu \nu \rho }...}{\delta x_{\alpha }}}$

(12)

New tensors, in respect to linear orthogonal transformations, may be formed from tensors according to these rules of operation.

Symmetrical Properties of Tensors. Tensors are called symmetrical or skew-symmetrical in respect to two of their indices, $\mu$ and $\nu$, if both the components which result from interchanging the indices $\mu$ and $\nu$ are equal to each other or equal with opposite signs.

Condition for symmetry : $A_{\mu \nu \rho }=A_{\mu \nu \rho }$.

Condition for skew-symmetry : $A_{\mu \nu \rho }=-A_{\nu \mu \rho }$.

Theorem. The character of symmetry or skew-symmetry exists independently of the choice of co-ordinates, and in