- #1
Pencilvester
- 214
- 52
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?