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

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THE MEANING OF RELATIVITY

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

Special Tensors.

I. The quantities (4) are tensor components (fundamental tensor).

Proof. If in the right-hand side of the equation of transformation , we substitute for the quantities (which are equal to 1 or 0 according as or ), 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 skew-symmetrical with respect to all pairs of indices, whose rank is equal to the number of dimensions, , and whose components are equal to or according as is an even or odd permutation of 123 ...

The proof follows with the aid of the theorem proved above .

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