Antisymmetric Tensor

An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched. For example, a tensor such that

(1)

is antisymmetric.

The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies

(2)

Furthermore, any rank-2 tensor can be written as a sum of symmetric and antisymmetric parts as

(3)

The antisymmetric part of a tensor is sometimes denoted using the special notation

(4)

For a general rank- tensor,

(5)

where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example

(6)

(Wald 1984, p. 26).