Expansion of covariant derivative

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[itex](V(s)_{||})^\mu = V(s)^\mu + s \Gamma^\mu_{\nu \lambda} \frac{dx^\nu}{ds} V(s)^\lambda + [/itex] higher-order terms
(Here we have parallel transported vector from point "s" to a very close point)


Hi, I tried to make some calculations to reach the high-order terms for parallel transporting of vector above and I think they may be shaky, so I would like to ask it on forum about how these high-order terms are expanded?? what is the logic of expanding high order terms??? Could you show the expansion of high- order terms using mathematical stuff or approach??? I tried to make some search on internet but have not obtained any valuable information.
 
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  • #2
stevendaryl
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Well, the exact equation for parallel transport is:

[itex]\frac{dV^\mu}{ds} = - \Gamma^\mu_{\nu \lambda} \frac{dx^\nu}{ds} V^\lambda[/itex]

So that's a set of coupled partial differential equations. You can solve it using a power series:

[itex]V^\mu(s) = V[0]^\mu + V[1]^\mu s + V[2]^\mu s^2 + ...[/itex]

To compute the higher-order terms, though, you have to take into account that [itex]\Gamma^\mu_{\nu \lambda}[/itex] is not necessarily constant, either. So you need to expand both [itex]V[/itex] and [itex]\Gamma[/itex] in power series. It's a mess to do in general.
 
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Hmm, the the exact equation is what you wrote
Well, the exact equation for parallel transport is:

[itex]\frac{dV^\mu}{ds} = - \Gamma^\mu_{\nu \lambda} \frac{dx^\nu}{ds} V^\lambda[/itex]

So that's a set of coupled partial differential equations. You can solve it using a power series:

[itex]V^\mu(s) = V[0]^\mu + V[1]^\mu s + V[2]^\mu s^2 + ...[/itex]

To compute the higher-order terms, though, you have to take into account that [itex]\Gamma^\mu_{\nu \lambda}[/itex] is not necessarily constant, either. So you need to expand both [itex]V[/itex] and [itex]\Gamma[/itex] in power series. It's a mess to do in general.
Thanks for your nice explanation "stevendarly"....
 

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