Mixed tensor

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In tensor analysis, a mixed tensor is a tensor which is neither covariant nor 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 $\begin{pmatrix} M \\ N \end{pmatrix}$ , 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.

[edit] 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 $\begin{pmatrix} M \\ N \end{pmatrix}$ , yields a tensor of type $\begin{pmatrix} M - 1 \\ N + 1 \end{pmatrix}$ , whereas its contravariant inverse, contracted with a tensor of type $\begin{pmatrix} M \\ N \end{pmatrix}$ , yields a tensor of type $\begin{pmatrix} M + 1 \\ N - 1 \end{pmatrix}$ .

[edit] Examples

As an example, a mixed tensor of type $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ can be obtained by raising an index of a covariant tensor of type $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$ ,