Antisymmetric tensor

Short description: Tensor equal to the negative of any of its transpositions

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.[1][2] The index subset must generally either be all covariant or all contravariant. For example, $\displaystyle{ T_{ijk\dots} = -T_{jik\dots} = T_{jki\dots} = -T_{kji\dots} = T_{kij\dots} = -T_{ikj\dots} }$ holds when the tensor is antisymmetric with respect to its first three indices.

If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order $\displaystyle{ k }$ may be referred to as a differential $\displaystyle{ k }$-form, and a completely antisymmetric contravariant tensor field may be referred to as a $\displaystyle{ k }$-vector field.

Antisymmetric and symmetric tensors

A tensor A that is antisymmetric on indices $\displaystyle{ i }$ and $\displaystyle{ j }$ has the property that the contraction with a tensor B that is symmetric on indices $\displaystyle{ i }$ and $\displaystyle{ j }$ is identically 0.

For a general tensor U with components $\displaystyle{ U_{ijk\dots} }$ and a pair of indices $\displaystyle{ i }$ and $\displaystyle{ j, }$ U has symmetric and antisymmetric parts defined as:

 $\displaystyle{ U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots}) }$ (symmetric part) $\displaystyle{ U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots}) }$ (antisymmetric part).

Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in $\displaystyle{ U_{ijk\dots} = U_{(ij)k\dots} + U_{[ij]k\dots}. }$

Notation

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M, $\displaystyle{ M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}), }$ and for an order 3 covariant tensor T, $\displaystyle{ T_{[abc]} = \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}). }$

In any 2 and 3 dimensions, these can be written as \displaystyle{ \begin{align} M_{[ab]} &= \frac{1}{2!} \, \delta_{ab}^{cd} M_{cd} , \\[2pt] T_{[abc]} &= \frac{1}{3!} \, \delta_{abc}^{def} T_{def} . \end{align} } where $\displaystyle{ \delta_{ab\dots}^{cd\dots} }$ is the generalized Kronecker delta, and we use the Einstein notation to summation over like indices.

More generally, irrespective of the number of dimensions, antisymmetrization over $\displaystyle{ p }$ indices may be expressed as $\displaystyle{ T_{[a_1 \dots a_p]} = \frac{1}{p!} \delta_{a_1 \dots a_p}^{b_1 \dots b_p} T_{b_1 \dots b_p}. }$

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as: $\displaystyle{ T_{ij} = \frac{1}{2}(T_{ij} + T_{ji}) + \frac{1}{2}(T_{ij} - T_{ji}). }$

This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.

Examples

Totally antisymmetric tensors include: