Mixed tensor

"Tensor type" redirects here. For the array data type, see Tensor type (computing).

In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant).

A mixed tensor of type or valence $(\genfrac{}{}{0ex}{}{M}{N})$, also written "type (M, N)", with both M > 0 and N > 0, is a tensor which has M contravariant indices and N covariant indices. Such a tensor can be defined as a linear function which maps an (M + N)-tuple of M one-forms and N vectors to a scalar.

Changing the tensor type

Main article: Raising and lowering indices

Consider the following octet of related tensors: $$T_{\alpha \beta \gamma },T_{\alpha \beta }{}^{\gamma },T_{\alpha }{}^{\beta }{}_{\gamma },T_{\alpha }{}^{\beta \gamma },T^{\alpha }{}_{\beta \gamma },T^{\alpha }{}_{\beta }{}^{\gamma },T^{\alpha \beta }{}_{\gamma },T^{\alpha \beta \gamma }.$$ The first one is covariant, the last one contravariant, and the remaining ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. A given contravariant index of a tensor can be lowered using the metric tensor g_μν, and a given covariant index can be raised using the inverse metric tensor g^μν. Thus, g_μν could be called the index lowering operator and g^μν the index raising operator.

Generally, the covariant metric tensor, contracted with a tensor of type (M, N), yields a tensor of type (M − 1, N + 1), whereas its contravariant inverse, contracted with a tensor of type (M, N), yields a tensor of type (M + 1, N − 1).

Examples

As an example, a mixed tensor of type (1, 2) can be obtained by raising an index of a covariant tensor of type (0, 3), $$T_{\alpha \beta }{}^{\lambda }=T_{\alpha \beta \gamma }{g}^{\gamma \lambda },$$ where $T_{\alpha \beta }{}^{\lambda }$ is the same tensor as $T_{\alpha \beta }{}^{\gamma }$, because $$T_{\alpha \beta }{}^{\lambda }{\delta }_{\lambda }{}^{\gamma }=T_{\alpha \beta }{}^{\gamma },$$ with Kronecker δ acting here like an identity matrix.

Likewise, $$T_{\alpha }{}^{\lambda }{}_{\gamma }=T_{\alpha \beta \gamma }{g}^{\beta \lambda },$$ $$T_{\alpha }{}^{\lambda ϵ}=T_{\alpha \beta \gamma }{g}^{\beta \lambda }{g}^{\gamma ϵ},$$ $$T^{\alpha \beta }{}_{\gamma }=g_{\gamma \lambda }{T}^{\alpha \beta \lambda },$$ $$T^{\alpha }{}_{\lambda ϵ}=g_{\lambda \beta }{g}_{ϵ\gamma }{T}^{\alpha \beta \gamma }.$$

Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta, $$g^{\mu \lambda }{g}_{\lambda \nu }=g^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu },$$ so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.

See also

• Covariance and contravariance of vectors
• Einstein notation
• Ricci calculus
• Tensor (intrinsic definition)
• Two-point tensor

External links

• Index Gymnastics, Wolfram Alpha