Antisymmetric tensor#Notation

{{Short description|Tensor equal to the negative of any of its transpositions}}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.{{cite book|author1=K.F. Riley |author2=M.P. Hobson |author3=S.J. Bence | title=Mathematical methods for physics and engineering|url=https://archive.org/details/mathematicalmeth00rile |url-access=registration | publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}{{cite book|author1=Juan Ramón Ruíz-Tolosa |author2=Enrique Castillo | title=From Vectors to Tensors | publisher=Springer| year=2005| isbn=978-3-540-22887-5 |url=https://books.google.com/books?id=vgGQUrQMzwYC&pg=PA225 |page=225}} section §7. 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

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

where $\delta_{ab\dots}^{cd\dots}$ is the generalized Kronecker delta, and the Einstein summation convention is in use.

More generally, irrespective of the number of dimensions, antisymmetrization over $p$ 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

References

{{cite book|last=Penrose|first=Roger|author-link=Roger Penrose|title=The Road to Reality|publisher=Vintage books|year=2007|isbn=978-0-679-77631-4}}
{{cite book|author1=J.A. Wheeler|author2=C. Misner|author3=K.S. Thorne|title=Gravitation| publisher=W.H. Freeman & Co|year=1973|pages=85–86, §3.5|isbn=0-7167-0344-0}}

External links

[http://mathworld.wolfram.com/AntisymmetricTensor.html Antisymmetric Tensor – mathworld.wolfram.com]

Category:Tensors