- #1

BiGyElLoWhAt

Gold Member

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At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate

##D_s D_r T_m - D_r D_s T_m##

And on the last term I'm having some difficulties, the second christoffel symbol.

So we have

##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]## after taking the first derivative. The first part of the second derivative is easy, but the second, I think I may have figured it out, but I'm not 100%, and would like someone to look at this to see if I'm doing this correctly, and if not, to correct me.

Second derivative:

##\partial_s [\partial_r T_m - \Gamma_{rm}^t T_t ] - \Gamma_{sm}^q [\partial_r T_q - \Gamma_{rq}^t T_t]##

My question is the running index (I think that's what it's called) on the second term, and how to replace the indices on, particularly, the very last christoffel symbol. I believe I need a different running index on the second derivative than I do for the first, so hence the q on ##\Gamma_{sm}^q## . However, my concern is in my ability to change the christoffel symbol ##\Gamma_{rm}^t## in the first derivative to ##\Gamma_{rq}^t## in the second. Part of me wants to do this:

##\Gamma_{sm}^q [\partial_r T_m - \Gamma_{rm}^t T_t]_q## **Edit** I realized after looking that I messed up with the m's here. Too many m's in the lower indices.

But I'm not sure if that's applicable. Also not sure what that would mean. I don't like changing anything within the [...] brackets, but I'm not sure how to introduce a new running index, as I'm relatively positive my result should be of the form ##S_{srm}## with S some tensor. Perhaps I should use T, but it's a different tensor of different rank, so I used S.

Hellp, someone learn me some knowledge.