Antisymmetric tensor

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In mathematics and theoretical physics, a tensor is antisymmetric or alternating 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, T i j k … = − T j i k … = T j k i … = − T k j i … = T k i j … = − T i k j … {\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 k {\displaystyle k} {\displaystyle k} may be referred to as a differential k {\displaystyle k} {\displaystyle k}-form, and a completely antisymmetric contravariant tensor field may be referred to as a k {\displaystyle k} {\displaystyle k}-vector field.

Antisymmetric and symmetric tensors

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

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

U ( i j ) k … = 1 2 ( U i j k … + U j i k … ) {\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)
U [ i j ] k … = 1 2 ( U i j k … − U j i k … ) {\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 U i j k … = U ( i j ) k … + U [ i j ] k … . {\displaystyle U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }.} {\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, M [ a b ] = 1 2 ! ( M a b − M b a ) , {\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, T [ a b c ] = 1 3 ! ( T a b c − T a c b + T b c a − T b a c + T c a b − T c b a ) . {\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 M [ a b ] = 1 2 ! δ a b c d M c d , T [ a b c ] = 1 3 ! δ a b c d e f T d e f . {\displaystyle {\begin{aligned}M_{[ab]}&={\frac {1}{2!}}\,\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!}}\,\delta _{abc}^{def}T_{def}.\end{aligned}}} {\displaystyle {\begin{aligned}M_{[ab]}&={\frac {1}{2!}}\,\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!}}\,\delta _{abc}^{def}T_{def}.\end{aligned}}} where δ a b … c d … {\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 p {\displaystyle p} {\displaystyle p} indices may be expressed as T [ a 1 … a p ] = 1 p ! δ a 1 … a p b 1 … b p T b 1 … 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}}.} {\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: T i j = 1 2 ( T i j + T j i ) + 1 2 ( T i j − T j i ) . {\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.

Examples

Totally antisymmetric tensors include:

See also

Notes

  1. K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.
  2. Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7.

References