# Prove that two transformation laws of the Christoffel symbols are the same

1. Oct 19, 2011

### 3029298

1. The problem statement, all variables and given/known data

Prove that the transformation law

$\Gamma^{\sigma '}_{\lambda '\rho '}=\frac{\partial x^\nu}{\partial x^{\lambda '}}\frac{\partial x^\rho}{\partial x^{\rho '}}\frac{\partial x^{\sigma '}}{\partial x^{\mu}}\Gamma^{\mu}_{\nu\rho}+\frac{\partial x^{\sigma '}}{\partial x^{\mu}}\frac{\partial^2 x^\mu}{\partial x^{\lambda '}\partial x^{\rho '}}$

is equivalent to

$\Gamma^{\sigma '}_{\lambda '\rho '}=\frac{\partial x^\lambda}{\partial x^{\lambda '}}\frac{\partial x^\rho}{\partial x^{\rho '}}\frac{\partial x^{\sigma '}}{\partial x^{\sigma}}\Gamma^{\sigma}_{\lambda\rho}-\frac{ \partial x^\mu}{\partial x^{\lambda '}}\frac{\partial x^{\lambda}}{\partial x^{\rho '}}\frac{\partial^2 x^{\sigma '}}{\partial x^{\mu}\partial x^{\lambda}}$

3. The attempt at a solution

The first term is easy, just relabel the dummy indices $\nu \rightarrow \lambda$ and $\mu \rightarrow \sigma$. But for the rest of the problem, I have no clue what to do.

Last edited: Oct 19, 2011
2. Oct 21, 2011

### George Jones

Staff Emeritus
Maybe this is too late. It's a trick. Differentiate the far left and right of

$$\delta^{\lambda '}_{\mu '} = \frac{\partial x^{\lambda '}}{\partial x^{\mu '}} = \frac{\partial x^{\lambda '}}{\partial x^\rho} \frac{\partial x^\rho}{\partial x^{\mu '}}$$

with respect to $x^{\alpha '}$.