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. 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:

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

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 $$U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }.$$

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 $$\begin{array}{rl}{M}_{[ab]}& =\frac{1}{2!}{\delta }_{ab}^{cd}{M}_{cd},\\ T_{[abc]}& =\frac{1}{3!}{\delta }_{abc}^{def}{T}_{def}.\end{array}$$ where ${\delta }_{ab\dots }^{cd\dots }$ 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 $p$ 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.

See also

• Exterior algebra – Algebra of a vector space
• Levi-Civita symbol – Antisymmetric permutation object acting on tensors
• Ricci calculus – Tensor index notation for tensor-based calculations
• Symmetric tensor – Tensor invariant under permutations of vectors it acts on
• Symmetrization

References

• Penrose, Roger (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4. 
• J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0. 

External links

• Antisymmetric Tensor – mathworld.wolfram.com