Page:The Meaning of Relativity - Albert Einstein (1922).djvu/23

where the integrand in the last integral is the functional determinant of the $x_{\nu }'$ with respect to the $x_{\nu }$, and this by (3) is equal to the determinant $|b_{\mu \nu }|$ of the coefficients of substitution, $b_{\nu \alpha }$. If we form the determinant of the $\delta _{\mu \alpha }$ from equation (4), we obtain, by means of the theorem of multiplication of determinants,

$1=\left|\delta _{\alpha \beta }\right|=\left|\sum \limits _{\nu }b_{\nu \alpha }b_{\nu \beta }\right|=\left|b_{\mu \nu }\right|^{2};\left|b_{\mu \nu }\right|=\pm 1$

(6)

If we limit ourselves to those transformations which have the determinant +1,^[1] and only these arise from continuous variations of the systems of co-ordinates, then $V$ is an invariant.

Invariants, however, are not the only forms by means of which we can give expression to the independence of the particular choice of the Cartesian co-ordinates. Vectors and tensors are other forms of expression. Let us express the fact that the point with the current co-ordinates $x_{\nu }$ lies upon a straight line. We have

$x_{\nu }-A_{\nu }=\lambda B_{\nu }{\text{ (}}\nu {\text{ from 1 to 3).}}$

Without limiting the generality we can put

$\sum B_{\nu }^{2}=1.$

If we multiply the equations by $b_{\beta \nu }$ (compare (3a) and (5)) and sum for all the $\nu$'s, we get

$x_{\beta }'-A_{\beta }'=\lambda B_{\beta }'$

1. There are thus two kinds of Cartesian systems which are designated as "right-handed" and "left-handed" systems. The difference between these is familiar to every physicist and engineer. It is interesting to note that these two kinds of systems cannot be defined geometrically, but only the contrast between them.