What is the definition of a rank 3 totally antisymmetric tensor?

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SUMMARY

A rank 3 totally antisymmetric tensor is defined similarly to a rank 4 tensor, where the value is 1 for an even permutation of indices, -1 for an odd permutation, and 0 otherwise. However, it is crucial to note that a rank 3 tensor is classified as a true tensor, while a rank 4 tensor is a pseudotensor. The distinction arises particularly when considering the effects of raising and lowering indices, which is not addressed in the basic definition.

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  • Knowledge of raising and lowering tensor indices
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Homework Statement


The totally antisymmetric rank 4 tensor is defined as 1 for an even combination of its indices and -1 for an odd combination of its indices and 0 otherwise.

Is a rank 3 totally antisymmetric tensor defined the same way?


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The Attempt at a Solution

 
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Mmmmmm. Yes! Do you think 4 is special?
 
The definition is the same, but remember that a cyclic permuation is even/odd iff the number of elements being permuted is odd/even. This sometimes causes confusion when moving moving from 3 to 4 dimensions.
 
Yes and no, I think the definition here is incomplete. It does not include what happens when you raise and lower an index. The rank 4 anti-symmetric tensor is a psuedotensor, the rank 3 one is a true tensor. So overall no.
 

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