Page:The Meaning of Relativity - Albert Einstein (1922).djvu/28

this lies its importance. The proof follows from the equation defining tensors.

Special Tensors.

I. The quantities $\delta _{\rho \sigma }$ (4) are tensor components (fundamental tensor).

Proof. If in the right-hand side of the equation of transformation $A_{\mu \nu }'=b_{\mu \alpha }b_{\nu \beta }A_{\alpha \beta }$, we substitute for $A_{\alpha \beta }$ the quantities $\delta _{\alpha \beta }$ (which are equal to 1 or 0 according as $\alpha =\beta$ or $\alpha {\text{ }}\beta$), we get

$A_{\mu \nu }'=b_{\mu \alpha }b_{\nu \alpha }=\delta _{\mu \nu }.$

The justification for the last sign of equality becomes evident if one applies (4) to the inverse substitution (5).

II. There is a tensor $(\delta _{\mu \nu \rho }...)$ skew-symmetrical with respect to all pairs of indices, whose rank is equal to the number of dimensions, $n$, and whose components are equal to $+1$ or $-1$ according as $\mu \nu \rho$ is an even or odd permutation of 123 ...

The proof follows with the aid of the theorem proved above $\left|b_{\rho \sigma }\right|=1$.

These few simple theorems form the apparatus from the theory of invariants for building the equations of pre-relativity physics and the theory of special relativity.

We have seen that in pre-relativity physics, in order to specify relations in space, a body of reference, or a space of reference, is required, and, in addition, a Cartesian system of co-ordinates. We can fuse both these concepts into a single one by thinking of a Cartesian system of co-ordinates as a cubical frame-work formed of rods each of unit length. The co-ordinates of the lattice points of