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_αβ^τ is the same tensor as T_αβ^γ, 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

• Covariance and contravariance of vectors
• Tensor (intrinsic definition)
• Two-point tensor