# 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.^{[1]}^{[2]} The index subset must generally either be all *covariant* or all *contravariant*.

For example,

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 of order *p* 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
and a pair of indices *i* and *j*, **U** has symmetric and antisymmetric parts defined as:

(symmetric part) (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

## 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**,

and for an order 3 covariant tensor **T**,

In any number of dimensions, these are equivalent to

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

In the above,

is the generalized Kronecker delta of the appropriate order.

## 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, in electromagnetism
- The Riemannian volume form on a pseudo-Riemannian manifold

## See also

## Notes

- ↑ 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. - ↑ Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005).
*From Vectors to Tensors*. Springer. p. 225. ISBN 978-3-540-22887-5. section §7.

## References

- J.A. Wheeler; C. Misner; K.S. Thorne (1973).
*Gravitation*. W.H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0. - R. Penrose (2007).
*The Road to Reality*. Vintage books. ISBN 0-679-77631-1.