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

16 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

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