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,

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 of order p may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.

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

M_{[ab]} = \frac{1}{2!} \, \delta_{ab}^{cd} M_{cd} ,

T_{[abc]} = \frac{1}{3!} \, \delta_{abc}^{def} T_{def} .

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

Notes

^ 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.
^ Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7.

References

J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0.
R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.

External links

Antisymmetric Tensor – mathworld.wolfram.com

[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.

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