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Antisymmetric Tensor

An antisymmetric tensor (also called an alternating tensor) over a field of field characteristic zero is a tensor which changes sign when two indices of the same variance are switched. For example, a tensor A^(x_1,...,x_n) such that

A^(x_1,...,x_i,...,x_j,...,x_n)=-A^(x_1,...,x_j,...,x_i,...,x_n)
(1)

is antisymmetric.

The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies

A^(mn)=-A^(nm).
(2)

Furthermore, any rank-2 tensor can be written as a sum of symmetric and antisymmetric parts as

A^(mn)=1/2(A^(mn)+A^(nm))+1/2(A^(mn)-A^(nm)).
(3)

The antisymmetric part of a tensor A^(ab) is sometimes denoted using the special notation

A^([ab])=1/2(A^(ab)-A^(ba)).
(4)

For a general rank-n tensor,

A^([a_1...a_n])=1/(n!)sum_(sigma in S_n)sgn(sigma)A^(a_(sigma(1))...a_(sigma(n))),
(5)

where S_n is the symmetric group and sgn(sigma) is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example

T^((ab)c)_([de])=1/4(T^(abc)_(de)+T^(bac)_(de)-T^(abc)_(ed)-T^(bac)_(ed)).
(6)

(Wald 1984, p. 26).

See also

Alternating Multilinear Form, Exterior Algebra, Symmetric Tensor, Wedge Product

Portions of this entry contributed by Todd Rowland

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References

Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984.

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Antisymmetric Tensor

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Antisymmetric Tensor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AntisymmetricTensor.html

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