Locally Flat Coordinates and Christoffel Symbols

Question

We know in general relativity (or rather in differential geometry) that you can have some locally flat coordinates (I think they are called Riemann normal coordinates) at a point $P$ in our manifold (space time). At this point $P$, the metric is Euclidean up to second order deviation, i.e. $$ g_{\tau \mu} \approx \eta_{\tau \mu} + B_{\tau \mu \ ,\lambda \sigma} \ x^\lambda x^\sigma + ... $$ where $B_{\tau \mu \ ,\lambda \sigma}$ are just the Taylor coefficient terms (second order in $g$).

Now I was led to believe that the Christoffel symbols should vanish at this point $P$ in locally flat coordinates, but under the definition of them, I get

\begin{split} \Gamma_{\rho \nu}^\lambda & \equiv \frac{1}{2} g^{\lambda \tau} (\partial_\rho g_{\nu \tau} + \partial_{\nu} g_{\rho \tau} - \partial_{\tau} g_{\rho \nu}) \\ & = \eta^{\lambda \tau}(B_{\tau \nu \ , \kappa \rho} + B_{\tau \rho \ , \kappa \nu} - B_{\rho \nu \ , \kappa \tau})\ x^\kappa + ... \end{split}

This is a non vanishing Christoffel symbol. If I am misunderstanding, then when exactly do the symbols vanish?

Answer 1

Your expression for the Cristoffel's symbols seems to be correct. In any case, it is definitely true that they should only vanish for $x=0$. The reason is as follows:

By choosing a coordinate system, you label the different points in a piece of your manifold by a set of numbers $x^\mu$. By construction, the point $P$ has the coordinates $x=0$, and non-zero $x$ correspond to points around $P$. The statement that in Riemann normal coordinates around $P$, the Cristoffels's symbols vanish at $P$ means that the symbols vanish for $x=0$.

If the Cristoffel's symbols were to vanish for $x$ in some neighbourhood of $0$, this would mean the curvature tensor vanishes in that neighbourhood. This is only true if the manifold is actually flat at $P$.

Answer 2

In some coordinate systems they do vanish, I think, which to me makes sense as you have the freedom to choose whichever system they vanish in.

Vanishing of Christoffel Symbols

Question : Are the values of the christoffel symbols the same for all coordinate systems on a surface/manifold? I would love to see an example for the cone in two different parametrizations.

Answer : The answer is no. The reason is that the Christoffel symbols are not scalar fields, nor tensor fields, they might vanish completely in a coordinate system and yet be non-vanishing in another. As a simple example consider the plane in cartesian coordinates: All Christoffel symbols vanish. Now consider polar coordinates, there will be some which are non-vanishing