Page:The Meaning of Relativity - Albert Einstein (1922).djvu/26

We shall consider these laws now, in order to be able to apply them later. We shall deal first only with the properties of tensors with respect to the transformation from one Cartesian system to another in the same space of reference, by means of linear orthogonal transformations. As the laws are wholly independent of the number of dimensions, we shall leave this number, $n$, indefinite at first.

Definition. If a figure is defined with respect to every system of Cartesian co-ordinates in a space of reference of $n$ dimensions by the $n^{a}$ numbers $A_{\mu \nu \rho }...$ ($a$ = number of indices), then these numbers are the components of a tensor of rank $a$ if the transformation law is

$A_{\mu '\nu '\rho '}'...=b_{\mu '\mu }b_{\nu '\nu }b_{\rho '\rho }...A_{\mu \nu \rho }...$

(7)

Remark. From this definition it follows that

$A_{\mu \nu \rho }=B_{\mu }C_{\nu }D_{\rho }...$

(8)

is an invariant, provided that $(B),(C),(D)...$ are vectors. Conversely, the tensor character of $(A)$ may be inferred, if it is known that the expression (8) leads to an invariant for an arbitrary choice of the vectors $(B),(C),$ etc.

Addition and Subtraction. By addition and subtraction of the corresponding components of tensors of the same rank, a tensor of equal rank results:

$A_{\mu \nu \rho }...\pm B_{\mu \nu \rho }...=C_{\mu \nu \rho }...$

(9)

The proof follows from the definition of a tensor given above.

Multiplication. From a tensor of rank $a$ and a tensor