Antisymmetric tensor

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. The index subset must generally either be all covariant or all contravariant.

For example, $$T_{ijk\dots} = -T_{jik\dots} = T_{jki\dots} = -T_{kji\dots} = T_{kij\dots} = -T_{ikj\dots}$$ 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 $$k$$ may be referred to as a differential $k$-form, and a completely antisymmetric contravariant tensor field may be referred to as a $k$-vector field.

Antisymmetric and symmetric tensors
A tensor A that is antisymmetric on indices $$i$$ and $$j$$ has the property that the contraction with a tensor B that is symmetric on indices $$i$$ and $$j$$ is identically 0.

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



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

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, $$M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}),$$ and for an order 3 covariant tensor T, $$T_{[abc]} = \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).$$

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

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

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

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

Examples
Totally antisymmetric tensors include:


 * Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric).
 * The electromagnetic tensor, $$F_{\mu\nu}$$ in electromagnetism.
 * The Riemannian volume form on a pseudo-Riemannian manifold.