Connection coefficient transformation law

  • #1
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Main Question or Discussion Point

Hello PF, in Carroll’s “Spacetime and Geometry”, he works out the transformation law for connection coefficients in his introduction to covariant derivatives, and I’m wondering if there is a typo in the final equation. He starts with$$\nabla_{\mu} V^{\nu} = \partial_{\mu} V^{\nu} + \Gamma^{\nu}_{\mu \lambda} V^{\lambda}$$and what the transformation law must be if we want the covariant derivative of a vector to be tensorial:$$\nabla_{\mu’} V^{\nu’} = \frac {\partial x^\mu} {\partial x^{\mu’}} \frac {\partial x^{\nu’}} {\partial x^\nu} \nabla_\mu V^\nu$$Starting with these, he eventually gets this:$$\Gamma^{\nu’}_{\mu’ \lambda’} \frac {\partial x^{\lambda'}} {\partial x^\lambda} V^\lambda + \frac {\partial x^\mu} {\partial x^{\mu'}} V^\lambda \frac {\partial^2 x^{\nu'}} {\partial x^\mu \partial x^\lambda} = \frac {\partial x^\mu} {\partial x^{\mu'}} \frac {\partial x^{\nu'}} {\partial x^\nu} \Gamma^\nu_{\mu \lambda} V^\lambda$$I followed him here no problem. The next thing he does is eliminate ##V^\lambda## from both sides, multiplies everything by ##\frac {\partial x^\lambda} {\partial x^{\sigma'}}##, then changes all the ##\sigma'##'s to ##\lambda'##'s (for aesthetics I guess). When I do this I end up with $$\Gamma^{\nu'}_{\mu' \lambda'} = \frac {\partial x^\mu} {\partial x^{\mu'}} \frac {\partial x^\lambda} {\partial x^{\lambda'}} \frac {\partial x^{\nu'}} {\partial x^\nu} \Gamma^\nu_{\mu \lambda} - \frac {\partial x^\mu} {\partial x^{\mu'}} \frac {\partial x^\lambda} {\partial x^{\lambda'}} \frac {\partial^2 x^{\nu'}} {\partial x^\mu \partial x^\lambda}$$but in the book, he has a ##+## between the two terms on the RHS. Is this simply a typo, or am I missing something?
 

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  • #2
Orodruin
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I don't know what he writes in the book, but in his online lecture notes he has a minus sign (equation (3.6)).

In general, I prefer the other form of this transformation rule as I find it easier to remember
$$
\newcommand{\tc}[2]{\frac{\partial x^{#1}}{\partial x^{#2}}}
\Gamma^{\nu'}_{\mu'\lambda'} = \tc{\mu}{\mu'}\tc{\lambda}{\lambda'} \tc{\nu'}{\nu} \Gamma^\nu_{\mu\lambda} + \tc{\nu'}{\nu}\frac{\partial^2 x^\nu}{\partial x^{\mu'}\partial x^{\lambda'}}.
$$
The equivalence of the two is rather straightforward to show.
 
  • #4
strangerep
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[...] but in the book, he has a ##+## between the two terms on the RHS. Is this simply a typo, or am I missing something?
In my copy of Carroll's book (Pearson2014 edition), it is a minus sign -- assuming you're referring to eq(3.10).

Maybe it was a typo in the original edition?

Btw, there is an ongoing errata list here.
 
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