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.^{[1] [2]} The index subset must generally either be all covariant or all contravariant.

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

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

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

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

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

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, $${\displaystyle M_{[ab]}={\frac {1}{2!}}(M_{ab}-M_{ba}),}$${\displaystyle M_{[ab]}={\frac {1}{2!}}(M_{ab}-M_{ba}),} and for an order 3 covariant tensor T, $${\displaystyle T_{[abc]}={\frac {1}{3!}}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).}$${\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 $${\displaystyle {\begin{aligned}M_{[ab]}&={\frac {1}{2!}},円\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!}},円\delta _{abc}^{def}T_{def}.\end{aligned}}}$${\displaystyle {\begin{aligned}M_{[ab]}&={\frac {1}{2!}},円\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!}},円\delta _{abc}^{def}T_{def}.\end{aligned}}} where ${\displaystyle \delta _{ab\dots }^{cd\dots }}${\displaystyle \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 ${\displaystyle p}${\displaystyle p} indices may be expressed as $${\displaystyle T_{[a_{1}\dots a_{p}]}={\frac {1}{p!}}\delta _{a_{1}\dots a_{p}}^{b_{1}\dots b_{p}}T_{b_{1}\dots b_{p}}.}$${\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: $${\displaystyle T_{ij}={\frac {1}{2}}(T_{ij}+T_{ji})+{\frac {1}{2}}(T_{ij}-T_{ji}).}$${\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.

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, ${\displaystyle F_{\mu \nu }}${\displaystyle F_{\mu \nu }} in electromagnetism.
• The Riemannian volume form on a pseudo-Riemannian manifold.

• Antisymmetric matrix – Form of a matrixPages displaying short descriptions of redirect targets
• Exterior algebra – Algebra associated to any 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 – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback

• 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.

• Antisymmetric Tensor – mathworld.wolfram.com

• Tensors

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