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Contraction of a rank 4 tensor

  • Thread starter Warren2007
  • Start date
  • #1
I'm trying to contract a rank 4 tensor with covariant rank 2 and contravariant rank 2 with four different indices

[T[ab][cd]]

to get a scalar value T and I have no idea how to do it as I'm sure a or b does not equal c or d.

Any help would be much appreciated.
 

Answers and Replies

  • #2
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You contract a tensor by putting the same index in one up slot and one down slot and summing. So one way to contract [tex]T^{ab}{}_{cd}[/tex] down to a scalar would be as [tex]T^{ab}{}_{ab} = \sum_{i, j} T^{ij}{}_{ij}[/tex]. Note that this is not necessarily equal to [tex]T^{ab}{}_{ba} = \sum_{i, j} T^{ij}{}_{ji}[/tex]! (Here [tex]i, j[/tex] are indices; if you are dealing with tensors over an [tex]n[/tex]-dimensional space, then [tex]1 \leq i, j \leq n[/tex] is the range of the sums.)
 
  • #3
But if a does not equal c and b does not equal d then how do you convert T[ab/cd] to T[ab/ab].
 

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