Antisymmetric tensor

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:

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 2 and 3 dimensions, these can be written as \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} where \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 indices may be expressed as 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_{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_{\mu\nu} in electromagnetism.
• The Riemannian volume form on a pseudo-Riemannian manifold.

Notes

References

References

1. (2010). "Mathematical methods for physics and engineering". Cambridge University Press.
2. (2005). "From Vectors to Tensors". Springer.