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Mixed tensor

{{Short description|Tensor having both covariant and contravariant indices}}

{{redirect|Tensor type|the array data type|Tensor type (computing)}}

{{No footnotes|date=October 2021}}

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 \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.

Changing the tensor type

{{main|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 {{math|gμν}}, and a given covariant index can be raised using the inverse metric tensor {{math|gμν}}. Thus, {{math|gμν}} could be called the index lowering operator and {{math|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).

=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 {{math|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.

See also

References

  • {{cite book |author=D.C. Kay| title=Tensor Calculus| publisher= Schaum’s Outlines, McGraw Hill (USA)| year=1988 | isbn=0-07-033484-6}}
  • {{cite book |first1=J.A. |last1=Wheeler |first2=C. |last2=Misner |first3=K.S. |last3=Thorne |chapter=§3.5 Working with Tensors |title=Gravitation |pages=85–86 |publisher=W.H. Freeman & Co |year=1973 |isbn=0-7167-0344-0}}
  • {{cite book |author=R. Penrose| title=The Road to Reality| publisher= Vintage books| year=2007 | isbn=978-0-679-77631-4}}

External links

  • [http://mathworld.wolfram.com/IndexGymnastics.html Index Gymnastics], Wolfram Alpha

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Category:Tensors