Mixed tensor
Template:Short description Template:Redirect Template:No footnotes 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 or valence <math display="inline">\binom{M}{N}</math>, 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 a tensor can be defined as a linear function which maps an (M + N)-tuple of M one-forms and N vectors to a scalar.
Changing the tensor typeEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Consider the following octet of related tensors: <math display="block"> 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} .</math> 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 Template:Math, and a given covariant index can be raised using the inverse metric tensor Template:Math. Thus, Template:Math could be called the index lowering operator and Template:Math 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).
ExamplesEdit
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), <math display="block"> T_{\alpha \beta} {}^\lambda = T_{\alpha \beta \gamma} \, g^{\gamma \lambda} ,</math> where <math> T_{\alpha \beta} {}^\lambda </math> is the same tensor as <math> T_{\alpha \beta} {}^\gamma </math>, because <math display="block"> T_{\alpha \beta} {}^\lambda \, \delta_\lambda {}^\gamma = T_{\alpha \beta} {}^\gamma, </math> with Kronecker Template:Math acting here like an identity matrix.
Likewise, <math display="block"> T_\alpha {}^\lambda {}_\gamma = T_{\alpha \beta \gamma} \, g^{\beta \lambda}, </math> <math display="block"> T_\alpha {}^{\lambda \epsilon} = T_{\alpha \beta \gamma} \, g^{\beta \lambda} \, g^{\gamma \epsilon},</math> <math display="block"> T^{\alpha \beta} {}_\gamma = g_{\gamma \lambda} \, T^{\alpha \beta \lambda},</math> <math display="block"> T^\alpha {}_{\lambda \epsilon} = g_{\lambda \beta} \, g_{\epsilon \gamma} \, T^{\alpha \beta \gamma}. </math>
Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta, <math display="block"> g^{\mu \lambda} \, g_{\lambda \nu} = g^\mu {}_\nu = \delta^\mu {}_\nu ,</math> so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.
See alsoEdit
- Covariance and contravariance of vectors
- Einstein notation
- Ricci calculus
- Tensor (intrinsic definition)
- Two-point tensor
ReferencesEdit
External linksEdit
- Index Gymnastics, Wolfram Alpha