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.12 The index subset must generally either be all covariant or all contravariant.

For example,

$T_{ijk\dots} = -T_{jik\dots} = T_{jki\dots} = -T_{kji\dots} = T_{kij\dots} = -T_{ikj\dots}$

holds when the tensor is antisymmetric on it first three indices.

If a tensor changes sign under exchange of any pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.

Antisymmetric and symmetric tensors

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

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

and for an order 3 covariant tensor T,

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

In any number of dimensions, these are equivalent to

$M_{[ab]} = \frac{1}{2!} \, \delta_{ab}^{cd} M_{cd} ,$
$T_{[abc]} = \frac{1}{3!} \, \delta_{abc}^{def} T_{def} .$

More generally, irrespective of the number of dimensions, antisymmetrization over p indices may be expressed as

$S_{[a_1 \dots a_p]} = \frac{1}{p!} \delta_{a_1 \dots a_p}^{b_1 \dots b_p} S_{b_1 \dots b_p} .$

In the above,

$\delta_{ab\dots}^{cd\dots}$

is the generalized Kronecker delta of the appropriate order.

Example

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

2. ^ Juan Ramón Ruíz-Tolosa, Enrique Castillo (2005). From Vectors to Tensors. Springer. ISBN 978-3-540-22887-5. Unknown parameter |other= ignored (|others= suggested) (help), google books