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,

$${\displaystyle T_{ijk\dots }=-T_{jik\dots }=T_{jki\dots }=-T_{kji\dots }=T_{kij\dots }=-T_{ikj\dots }}$$

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

$${\displaystyle {\begin{aligned}M_{[ab]}&={\frac {1}{2!}}\,\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!}}\,\delta _{abc}^{def}T_{def}.\end{aligned}}}$$

 

where ${\displaystyle \delta _{ab\dots }^{cd\dots }}$  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:

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

See also

• Antisymmetric matrix – Form of a matrixPages displaying short descriptions of redirect targets
• Exterior algebra – Algebra of exterior/ wedge products
• 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

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

• 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