Antisymmetric tensor

{{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.<ref>{{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}}</ref><ref>{{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.</ref> The index subset must generally either be all ''covariant'' or all ''contravariant''.

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

==Antisymmetric and symmetric tensors==

A tensor '''A''' that is antisymmetric on indices <math>i</math> and <math>j</math> has the property that the contraction with a tensor '''B''' that is symmetric on indices <math>i</math> and <math>j</math> is identically 0.

For a general tensor '''U''' with components <math>U_{ijk\dots}</math> and a pair of indices <math>i</math> and <math>j,</math> '''U''' has symmetric and antisymmetric parts defined as:

:{| |- | <math>U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots})</math> || || (symmetric part) |- | <math>U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})</math> || ||(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 <math display=block>U_{ijk\dots} = U_{(ij)k\dots} + U_{[ij]k\dots}.</math>

==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''', <math display=block>M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}),</math> and for an order 3 covariant tensor '''T''', <math display=block>T_{[abc]} = \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).</math>

In any 2 and 3 dimensions, these can be written as <math display=block>\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}</math> where <math>\delta_{ab\dots}^{cd\dots}</math> is the generalized Kronecker delta, and the Einstein summation convention is in use.

More generally, irrespective of the number of dimensions, antisymmetrization over <math>p</math> indices may be expressed as <math display=block>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}.</math>

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as: <math display=block>T_{ij} = \frac{1}{2}(T_{ij} + T_{ji}) + \frac{1}{2}(T_{ij} - T_{ji}).</math>

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, <math>F_{\mu\nu}</math> in electromagnetism. * The Riemannian volume form on a pseudo-Riemannian manifold.

== See also ==

* {{annotated link|Antisymmetric matrix}} * {{annotated link|Exterior algebra}} * {{annotated link|Levi-Civita symbol}} * {{annotated link|Ricci calculus}} * {{annotated link|Symmetric tensor}} * {{annotated link|Symmetrization}}

==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