Mixed tensor

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 ${\textstyle {\binom {M}{N}}}${\textstyle {\binom {M}{N}}}, 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.

Consider the following octet of related tensors: $${\displaystyle T_{\alpha \beta \gamma },\ T_{\alpha \beta }{}^{\gamma },\ T_{\alpha }{}^{\beta }{}_{\gamma },\ T_{\alpha }{}^{\beta \gamma },\ T^{\alpha }{}_{\beta \gamma },\ T^{\alpha }{}_{\beta }{}^{\gamma },\ T^{\alpha \beta }{}_{\gamma },\ T^{\alpha \beta \gamma }.}$${\displaystyle 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).

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

Likewise, $${\displaystyle T_{\alpha }{}^{\lambda }{}_{\gamma }=T_{\alpha \beta \gamma },円g^{\beta \lambda },}$${\displaystyle T_{\alpha }{}^{\lambda }{}_{\gamma }=T_{\alpha \beta \gamma },円g^{\beta \lambda },} $${\displaystyle T_{\alpha }{}^{\lambda \epsilon }=T_{\alpha \beta \gamma },円g^{\beta \lambda },円g^{\gamma \epsilon },}$${\displaystyle T_{\alpha }{}^{\lambda \epsilon }=T_{\alpha \beta \gamma },円g^{\beta \lambda },円g^{\gamma \epsilon },} $${\displaystyle T^{\alpha \beta }{}_{\gamma }=g_{\gamma \lambda },円T^{\alpha \beta \lambda },}$${\displaystyle T^{\alpha \beta }{}_{\gamma }=g_{\gamma \lambda },円T^{\alpha \beta \lambda },} $${\displaystyle T^{\alpha }{}_{\lambda \epsilon }=g_{\lambda \beta },円g_{\epsilon \gamma },円T^{\alpha \beta \gamma }.}$${\displaystyle T^{\alpha }{}_{\lambda \epsilon }=g_{\lambda \beta },円g_{\epsilon \gamma },円T^{\alpha \beta \gamma }.}

Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta, $${\displaystyle g^{\mu \lambda },円g_{\lambda \nu }=g^{\mu }{}_{\nu }=\delta ^{\mu }{}_{\nu },}$${\displaystyle g^{\mu \lambda },円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.

• Covariance and contravariance of vectors
• Einstein notation
• Ricci calculus
• Tensor (intrinsic definition)
• Two-point tensor

• D.C. Kay (1988). Tensor Calculus. Schaum’s Outlines, McGraw Hill (USA). ISBN 0-07-033484-6.
• Wheeler, J.A.; Misner, C.; Thorne, K.S. (1973). "§3.5 Working with Tensors". Gravitation . W.H. Freeman & Co. pp. 85–86. ISBN 0-7167-0344-0.
• R. Penrose (2007). The Road to Reality . Vintage books. ISBN 978-0-679-77631-4.

• Index Gymnastics, Wolfram Alpha

• Tensors

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