Page:The Meaning of Relativity - Albert Einstein (1922).djvu/25

In this, and in analogous cases, we shall omit the sign of summation, and understand that the summation is to be carried out for those indices that appear twice. We thus write the equation of the surface

$a_{\mu \nu }\xi _{\mu }\xi _{\nu }=1.$

The quantities $a_{\mu \nu }$ determine the surface completely, for a given position of the centre, with respect to the chosen system of Cartesian co-ordinates. From the known law of transformation for the $\xi _{\nu }$, (3a) for linear orthogonal transformations, we easily find the law of transformation for the $a_{\mu \nu }$^[1]:

$a_{\sigma \tau }'=b_{\sigma \mu }b_{\tau \nu }a_{\mu \nu }.$

This transformation is homogeneous and of the first degree in the $a_{\mu \nu }$. On account of this transformation, the $a_{\mu \nu }$ are called components of a tensor of the second rank (the latter on account of the double index). If all the components, $a_{\mu \nu }$, of a tensor with respect to any system of Cartesian co-ordinates vanish, they vanish with respect to every other Cartesian system. The form and the position of the surface of the second degree is described by this tensor ($a$).

Analytic tensors of higher rank (number of indices) may be defined. It is possible and advantageous to regard vectors as tensors of rank 1, and invariants (scalars) as tensors of rank 0. In this respect, the problem of the theory of invariants may be so formulated: according to what laws may new tensors be formed from given tensors?

1. The equation $a_{\sigma \tau }'\xi _{\sigma }'\xi _{\tau }'=1$ may, by (5), be replaced by $a_{\sigma \tau }'b_{\mu \sigma }b_{\nu \tau }\xi _{\sigma }\xi _{\tau }=1$, from which the result stated immediately follows.