I confirmed the contraction identity by crafting down Each and every circumstance for all indices. But that is not a evidence, simply a verification, and I do think verifying in a similar style for 4th dimension will probably be cumbersome.
I'm utilizing the Einstein notation, so several indices are summed) or they provide two three-indices symbols with distinct indication
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where by gab is the illustration with the metric in that coordinate technique. We are able to similarly take into consideration a contravariant Levi-Civita tensor by increasing the indices Together with the metric as typical,
This tensor should not be baffled with the tensor density area stated above. The presentation On this part intently follows Carroll 2004.
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