- Source: 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. The index subset must generally either be all covariant or all contravariant.
For example,
T
i
j
k
…
=
−
T
j
i
k
…
=
T
j
k
i
…
=
−
T
k
j
i
…
=
T
k
i
j
…
=
−
T
i
k
j
…
{\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
k
{\displaystyle k}
may be referred to as a differential
k
{\displaystyle k}
-form, and a completely antisymmetric contravariant tensor field may be referred to as a
k
{\displaystyle k}
-vector field.
Antisymmetric and symmetric tensors
A tensor A that is antisymmetric on indices
i
{\displaystyle i}
and
j
{\displaystyle j}
has the property that the contraction with a tensor B that is symmetric on indices
i
{\displaystyle i}
and
j
{\displaystyle j}
is identically 0.
For a general tensor U with components
U
i
j
k
…
{\displaystyle U_{ijk\dots }}
and a pair of indices
i
{\displaystyle i}
and
j
,
{\displaystyle j,}
U has symmetric and antisymmetric parts defined as:
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
i
j
k
…
=
U
(
i
j
)
k
…
+
U
[
i
j
]
k
…
.
{\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,
M
[
a
b
]
=
1
2
!
(
M
a
b
−
M
b
a
)
,
{\displaystyle M_{[ab]}={\frac {1}{2!}}(M_{ab}-M_{ba}),}
and for an order 3 covariant tensor T,
T
[
a
b
c
]
=
1
3
!
(
T
a
b
c
−
T
a
c
b
+
T
b
c
a
−
T
b
a
c
+
T
c
a
b
−
T
c
b
a
)
.
{\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
M
[
a
b
]
=
1
2
!
δ
a
b
c
d
M
c
d
,
T
[
a
b
c
]
=
1
3
!
δ
a
b
c
d
e
f
T
d
e
f
.
{\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
δ
a
b
…
c
d
…
{\displaystyle \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
{\displaystyle p}
indices may be expressed as
T
[
a
1
…
a
p
]
=
1
p
!
δ
a
1
…
a
p
b
1
…
b
p
T
b
1
…
b
p
.
{\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:
T
i
j
=
1
2
(
T
i
j
+
T
j
i
)
+
1
2
(
T
i
j
−
T
j
i
)
.
{\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,
F
μ
ν
{\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 associated to any 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 – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback
Notes
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
Kata Kunci Pencarian:
- Antisymmetric tensor
- Antisymmetric
- Levi-Civita symbol
- Symmetric tensor
- Electromagnetic tensor
- Ricci calculus
- Tensor (intrinsic definition)
- Tensor product
- Covariant formulation of classical electromagnetism
- Glossary of tensor theory
