Mixed tensor

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 ${\begin{pmatrix}M\\N\end{pmatrix}}$, 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 tensor can be defined as a linear function which maps an M+N-tuple of M one-forms and N vectors to a scalar.

Index raising and lowering

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),

$T_{\alpha \beta }{}^{\tau }=T_{\alpha \beta \gamma }\,g^{\gamma \tau }$,

where $T_{\alpha \beta }{}^{\tau }$ is the same tensor as $T_{\alpha \beta }{}^{\gamma }$, because

$T_{\alpha \beta }{}^{\tau }\,\delta _{\tau }{}^{\gamma }=T_{\alpha \beta }{}^{\gamma }$,

with Kronecker δ acting here like an identity matrix.

Likewise,

$T_{\alpha }{}^{\tau }{}_{\gamma }=T_{\alpha \beta \gamma }\,g^{\beta \tau },$

$T_{\alpha }{}^{\tau \epsilon }=T_{\alpha \beta \gamma }\,g^{\beta \tau }\,g^{\gamma \epsilon },$

$T^{\alpha \beta }{}_{\gamma }=g_{\gamma \tau }\,T^{\alpha \beta \tau },$

$T^{\alpha }{}_{\tau \epsilon }=g_{\tau \beta }\,g_{\epsilon \gamma }\,T^{\alpha \beta \gamma }.$

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

• Tensor (intrinsic definition)