Page:The Meaning of Relativity - Albert Einstein (1922).djvu/85

Accordingly,

$B_{\mu }'{A^{\mu }}'={\frac {\delta x_{\alpha }}{\delta x_{\mu }'}}{\frac {\delta x_{\mu }'}{\delta x_{\beta }}}B_{\alpha }A^{\beta }=B_{\alpha }A^{\alpha }.$

In particular, the derivatives ${\frac {\delta \phi }{\delta x_{\alpha }}}$ of a scalar $\phi$, are components of a co-variant vector, which, with the co-ordinate differentials, form the scalar ${\frac {\delta \phi }{\delta x_{\alpha }}}dx_{\alpha }$; we see from this example how natural is the definition of the co-variant vectors.

There are here, also, tensors of any rank, which may have co-variant or contra-variant character with respect to each index; as with vectors, the character is designated by the position of the index. For example, $A_{\mu }^{\nu }$ denotes a tensor of the second rank, which is co-variant with respect to the index $\mu$, and contra-variant with respect to the index $\nu$. The tensor character indicates that the equation of transformation is

${A_{\mu }^{\nu }}'={\frac {\delta x_{\alpha }}{\delta x_{\mu }'}}{\frac {\delta x_{\nu }'}{\delta x_{\beta }}}A_{\alpha }^{\beta }.$

(58)

Tensors may be formed by the addition and subtraction of tensors of equal rank and like character, as in the theory of invariants of orthogonal linear substitutions, for example,

$A_{\mu }^{\nu }+B_{\mu }^{\nu }=C_{\mu }^{\nu }.$

(59)

The proof of the tensor character of $C_{\mu }^{\nu }$ depends upon (58).

Tensors may be formed by multiplication, keeping the character of the indices, just as in the theory of invariants of linear orthogonal transformations, for example,

$A_{\mu }^{\nu }B_{\sigma \tau }=C_{\mu \sigma \tau }^{\nu }.$

(60)