Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it alternates sign when the two indices are interchanged:[1]

\( T_{ijk\dots} = -T_{jik\dots} \)

An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric. If a tensor changes sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form.

Antisymmetric and symmetric tensors

A tensor A which is antisymmetric on indices i and j has the property that the contraction with a tensor B, which 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 a rank 2 covariant tensor M,

\( M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}) \,, \)

and for a rank 3 covariant tensor T,

\( T_{[abc]} = \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}) \,. \)

In three dimensions, these are equivalent to

\( M_{[ab]} = \varepsilon_{abc} \, \frac{1}{2!} \, \varepsilon^{dec} M_{de} \,, \)
\( T_{[abc]} = \varepsilon_{abc} \, \frac{1}{3!} \, \varepsilon^{def} T_{def} \,. \)

While in four dimensions, these are equivalent to

\( M_{[ab]} = \frac{1}{2!} \, \varepsilon_{abcd} \, \frac{1}{2!} \, \varepsilon^{efcd} M_{ef} \,,
\( T_{[abc]} = \varepsilon_{abcd} \, \frac{1}{3!} \, \varepsilon^{efgd} T_{efg} \,. \)

More generally, in n dimensions

\( S_{[a_1 \dots a_i]} = \frac{1}{(n-i)!} \varepsilon_{a_1 \dots a_i~b_1 \dots b_{n-i}} \frac{1}{i!} \varepsilon^{c_1 \dots c_i~b_1 \dots b_{n-i}} S_{c_1 \dots c_i} \,. \)

Example

An important antisymmetric tensor in physics is the electromagnetic tensor F in electromagnetism.
See also

Symmetric tensor
Antisymmetric matrix
Exterior algebra
Levi-Civita symbol
Ricci calculus

References

^ 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.

J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. p.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

[1] - mathworld, wolfram

Such a φ will be represented by a skew-symmetric matrix A, φ(v, w) = vTAw, once a basis of V is chosen; and conversely an n×n skew-symmetric matrix A on Kn gives rise to an alternating form sending (v, w) to vTAw.

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