Antisymmetric tensor explained

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,$$T\_\; =\; -T\_\; =\; T\_\; =\; -T\_\; =\; T\_\; =\; -T\_$$holds when the tensor is antisymmetric with respect to its first three indices.

k may be referred to as a differential k -form, and a completely antisymmetric contravariant tensor field may be referred to as a k -vector field.

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...} and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

U _{(ij)k...} = (1)/(2) (U _{ijk...} + U _{jik...} )		(symmetric part)
U _{[ij]k...} = (1)/(2) (U _{ijk...} - U _{jik...} )		(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$$U\_\; =\; U\_\; +\; U\_.$$

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,$$M\_\; =\; \backslash frac(M\_\; -\; M\_),$$and for an order 3 covariant tensor T,$$T\_\; =\; \backslash frac(T\_-T\_+T\_-T\_+T\_-T\_).$$

In any 2 and 3 dimensions, these can be written aswhere

\delta ^{cd ...} _{ab ...} 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$$T\_\; =\; \backslash frac\; \backslash delta\_^\; T\_.$$

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:$$T\_\; =\; \backslash frac(T\_\; +\; T\_)\; +\; \backslash frac(T\_\; -\; T\_).$$

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.

References

• Book: Penrose, Roger. Roger Penrose. The Road to Reality. Vintage books. 2007. 978-0-679-77631-4.
• Book: J.A. Wheeler. C. Misner. K.S. Thorne. Gravitation. W.H. Freeman & Co. 1973. 85–86, §3.5. 0-7167-0344-0.

External links

• Antisymmetric Tensor – mathworld.wolfram.com

Notes and References

1. Book: K.F. Riley . M.P. Hobson . S.J. Bence . Mathematical methods for physics and engineering. registration . Cambridge University Press. 2010 . 978-0-521-86153-3.
2. Book: Juan Ramón Ruíz-Tolosa . Enrique Castillo . From Vectors to Tensors . Springer. 2005. 978-3-540-22887-5 . 225. section §7.