# I Christoffel symbols transformation law

1. Jul 5, 2016

### guitarphysics

In Carroll's GR book (pg. 96), the transformation law for Christoffel symbols is derived from the requirement that the covariant derivative be tensorial. I think I understand that, and the derivation Carroll carries out, up until this step (I have a very simple question here, I believe- something stupid I'm not seeing):
$$\Gamma^{\nu'}_{\mu'\lambda'} \frac{\partial x^{\lambda'}}{\partial x^{\lambda}}V^{\lambda} + \frac{\partial x^{\mu}}{\partial x^{\mu'}}V^{\lambda}\frac{\partial}{\partial x^{\mu}}\frac{\partial x^{\nu'}}{\partial x^{\lambda}} = \frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\nu'}}{\partial x^{\nu}} \Gamma^{\nu}_{\mu\lambda}V^{\lambda}$$

Since this must be true for any vector $V^{\lambda}$, that can be eliminated. We can then multiply by $\partial x^{\lambda}/\partial x^{\sigma'}$ on both sides, and relabel $\sigma' \to \lambda'$ to get:
$$\Gamma^{\nu'}_{\mu'\lambda'} = \frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\lambda}}{\partial x^{\lambda'}} \frac{\partial x^{\nu'}}{\partial x^{\nu}} \Gamma^{\nu}_{\mu\lambda} + \frac{\partial x^{mu}}{\partial x^{\mu'}}\frac{\partial x^{\lambda}}{\partial x^{\lambda'}}\frac{\partial^2 x^{\nu'}}{\partial x^{\mu} \partial x^{\lambda}}$$

Now, my question is just this: why the hell is there a plus sign in the last equation, instead of a minus sign? If I follow Carroll's steps directly from the first equation, I get a minus sign! :(
Sorry if it's obvious- any help is appreciated!

2. Jul 5, 2016

3. Jul 5, 2016

### George Jones

Staff Emeritus
My copy of Carroll's book has a minus sign (in (3.10) on page 96).

4. Jul 5, 2016

5. Jul 5, 2016

### George Jones

Staff Emeritus
As Mr-R noted, be careful with the indices!

I have to catch my bus now, but I will try to type in the details after I get home tonight.

6. Jul 5, 2016

### George Jones

Staff Emeritus
Here are the quantitative details:

\begin{align} \frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\frac{\partial x^{\lambda }}{\partial x^{\lambda ^{\prime }}}\frac{\partial ^{2}x^{\nu ^{\prime }}}{\partial x^{\mu }\partial x^{\lambda }} &= \frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\left( \frac{\partial x^{\lambda }}{\partial x^{\lambda ^{\prime }}}\frac{\partial }{\partial x^{\lambda }}\right) \frac{\partial x^{\nu ^{\prime }}}{\partial x^\mu} \\ &= \frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\right) \\ &= \frac{\partial }{\partial x^{\lambda ^{\prime }}}\left( \frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\right) -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\right) \\ &= \frac{\partial }{\partial x^{\lambda ^{\prime }}}\left( \frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu ^{\prime }}}\right) -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\right) \\ &= \frac{\partial }{\partial x^{\lambda ^{\prime }}}\left( \delta _{\mu ^{\prime }}^{\nu ^{\prime }}\right) -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\right) \\ &= -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\frac{\partial ^{2}x^{\mu }}{\partial x^{\lambda ^{\prime }}\partial x^{\mu ^{\prime }}} . \end{align}

Either the expression at the beginning or the term at the end can be used. Not the differing signs (and indices!).

7. Jul 16, 2016

### guitarphysics

Sorry for the late response, I had been procrastinating going over the indices :s
I'm still not really seeing how that affects the original post; the last term you end on is not present, I believe, in Carroll's derivation at all. (It probably is and I'm just not seeing it- in which case I apologize!)