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. The index subset must generally either be all covariant or all contravariant.
For example,
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 field of order may be referred to as a differential -form, and a completely antisymmetric contravariant tensor field may be referred to as a -vector field.
Antisymmetric and symmetric tensors
A tensor A that is antisymmetric on indices and has the property that the contraction with a tensor B that is symmetric on indices and is identically 0.
For a general tensor U with components and a pair of indices and 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,
In any 2 and 3 dimensions, these can be written as
More generally, irrespective of the number of dimensions, antisymmetrization over indices may be expressed as
In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:
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, in electromagnetism.
- The Riemannian volume form on a pseudo-Riemannian manifold.
See also
- Antisymmetric matrix
- Exterior algebra – Algebraic construction used in multilinear algebra and geometry
- Levi-Civita symbol – Antisymmetric permutation object acting on tensors
- Ricci calculus – Tensor index notation for tensor-based calculations
- Symmetric tensor – Tensor invariant under permutations of vectors it acts on
- Symmetrization