Mixed tensor

Changing the tensor type

Main article: Raising and lowering indices

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

Likewise, T_\alpha {}^\lambda {}\gamma = T{\alpha \beta \gamma} , g^{\beta \lambda}, T_\alpha {}^{\lambda \epsilon} = T_{\alpha \beta \gamma} , g^{\beta \lambda} , g^{\gamma \epsilon}, T^{\alpha \beta} {}\gamma = g{\gamma \lambda} , T^{\alpha \beta \lambda}, T^\alpha {}{\lambda \epsilon} = g{\lambda \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.

References