Page:The Meaning of Relativity - Albert Einstein (1922).djvu/27

Rh of rank $$\beta$$ we may obtain a tensor of rank $$\alpha + \beta$$ by multiplying all the components of the first tensor by all the components of the second tensor:

Contraction. A tensor of rank $$\alpha - 2$$ may be obtained from one of rank $$\alpha$$ by putting two definite indices equal to each other and then summing for this single index:

The proof is

In addition to these elementary rules of operation there is also the formation of tensors by differentiation ("erweiterung"):

New tensors, in respect to linear orthogonal transformations, may be formed from tensors according to these rules of operation.

Symmetrical Properties of Tensors. Tensors are called symmetrical or skew-symmetrical in respect to two of their indices, $$\mu$$ and $$\nu$$, if both the components which result from interchanging the indices $$\mu$$ and $$\nu$$ are equal to each other or equal with opposite signs.


 * Condition for symmetry : $$A_{\mu \nu\rho} = A_{\mu \nu\rho}$$.


 * Condition for skew-symmetry : $$A_{\mu \nu\rho} = -A_{\nu\mu\rho}$$.

Theorem. The character of symmetry or skew-symmetry exists independently of the choice of co-ordinates, and in