Mixed Tensor - Changing The Tensor Type

Changing The Tensor Type

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 (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).

Read more about this topic:  Mixed Tensor

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