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=δμα≠ημα. 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: ημν≠ημν. 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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