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, [math]\displaystyle{ 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]\displaystyle{ k }[/math] may be referred to as a differential [math]\displaystyle{ k }[/math]-form, and a completely antisymmetric contravariant tensor field may be referred to as a [math]\displaystyle{ k }[/math]-vector field.
Antisymmetric and symmetric tensors
A tensor A that is antisymmetric on indices [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] has the property that the contraction with a tensor B that is symmetric on indices [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] is identically 0.
For a general tensor U with components [math]\displaystyle{ U_{ijk\dots} }[/math] and a pair of indices [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j, }[/math] U has symmetric and antisymmetric parts defined as:
[math]\displaystyle{ U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots}) }[/math] (symmetric part) [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}), }[/math] and for an order 3 covariant tensor T, [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \delta_{ab\dots}^{cd\dots} }[/math] 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 [math]\displaystyle{ p }[/math] indices may be expressed as [math]\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}. }[/math]
In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as: [math]\displaystyle{ 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]\displaystyle{ F_{\mu\nu} }[/math] in electromagnetism.
- The Riemannian volume form on a pseudo-Riemannian manifold.
See also
- Exterior algebra – Algebra of a vector space
- 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
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. https://archive.org/details/mathematicalmeth00rile.
- ↑ Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. https://books.google.com/books?id=vgGQUrQMzwYC&pg=PA225. 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
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