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. The index subset must generally either be all covariant or all contravariant.
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
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.
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
- Levi-Civita symbol
- Symmetric tensor
- Antisymmetric matrix
- Antisymmetric relation
- Exterior algebra
- Ricci calculus
- 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.