Monday, 13 November 2017

tensor calculus - Lowering/raising metric indexes


So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing.


Consider Minkowski spacetime. The trace of a matrix A can be written in terms of the Minkowski metric as ημνAμν=ημνAμν=Aμμ.


What about the trace of the metric? Notice that ημμ cannot be written as ημνημν, because this is equal to 4, not 2. It seemed to us that there is some kind of divine rule that says "You shall not lower nor raise indexes of the metric", because ημνηνα=δμαημα. Is the metric immune to index manipulations? Is this a notation flaw or am I being ultra-dumb?



Answer



The mistake you made is this: ημνημν. When you raise index μ from downstairs to upstairs, the matrix elements change. η00=1, η00=1. That is why if you take the trace of ημν, you get 2, but if you take the trace of ημν you get 4.


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