Contraction of a rank 4 tensor

[Warren2007]

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.

 

[ystael]

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

 

[Warren2007]

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].