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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-form s and ''N'' Vector s 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 ''gμν'', and a given covariant index can be raised using the inverse metric tensor ''gμν''. Thus, ''gμν'' could be called the ''index lowering operator'' and ''gμν'' 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} .


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} ,
: T_{\alpha \beta} {}^ au = T_{\alpha \beta \gamma} \, g^{\gamma au} ,
where T_{\alpha \beta} {}^ au is the same tensor as T_{\alpha \beta} {}^\gamma , because
: T_{\alpha \beta} {}^ au \, \delta_ au {}^\gamma = T_{\alpha \beta} {}^\gamma ,
with Kronecker ''δ'' acting here like an identity matrix.

Likewise,
: T_\alpha {}^ au {}_\gamma = T_{\alpha \beta \gamma} \, g^{\beta au},
: T_\alpha {}^{ au \epsilon} = T_{\alpha \beta \gamma} \, g^{\beta au} \, g^{\gamma \epsilon},
: T^{\alpha \beta} {}_\gamma = g_{\gamma au} \, T^{\alpha \beta au},
: T^\alpha {}_{ au \epsilon} = g_{ au \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
u} = g^\mu {}_
u = \delta^\mu {}_
u ,
so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.


SEE ALSO