Page:The Meaning of Relativity - Albert Einstein (1922).djvu/24

where we have written

$B_{\beta }'=\sum \limits _{\nu }b_{\beta \nu }B_{\nu }{\text{; }}A_{\beta }'=\sum \limits _{\nu }b_{\beta \nu }A_{\nu }.$

These are the equations of straight lines with respect to a second Cartesian system of co-ordinates $K'$. They have the same form as the equations with respect to the original system of co-ordinates. It is therefore evident that straight lines have a significance which is independent of the system of co-ordinates. Formally, this depends upon the fact that the quantities $(x_{\nu }-A_{\nu })-\lambda B_{\nu }$ are transformed as the components of an interval, $\Delta x_{\nu }$. The ensemble of three quantities, defined for every system of Cartesian co-ordinates, and which transform as the components of an interval, is called a vector. If the three components of a vector vanish for one system of Cartesian co-ordinates, they vanish for all systems, because the equations of transformation are homogeneous. We can thus get the meaning of the concept of a vector without referring to a geometrical representation. This behaviour of the equations of a straight line can be expressed by saying that the equation of a straight line is co-variant with respect to linear orthogonal transformations.

We shall now show briefly that there are geometrical entities which lead to the concept of tensors. Let $P_{0}$ be the centre of a surface of the second degree, $P$ any point on the surface, and $\xi _{\nu }$ the projections of the interval $P_{0}P$ upon the co-ordinate axes. Then the equation of the surface is

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