Frame described by locally flat metric is in general not an inerial frame - true or false? In general relativity there is always a locally flat system, for which the transformed metric is the Minkowski metric up to second order at one point. But this does not mean, that the system, described by the new coordinates is automatically an inertial frame. Is that right?
If this is true: why can I apply Lorentz transformation of special relativity in that metric? The metric is locally flat so SR applies - but the frame is nevertheless no inertial frame. Isn't that a contradiction, because Lorentz-transformations are valid only in inertial systems?
I'm confused...
 A: I tis very important to notice that you can do this at a given point, but not at any given curve. In fact, it is true that for geodesics you can define a locally flat coordinate system such that the Christoffel symbols vanish along the curve. This is compatible with the idea of the equivalence principle for $inertial$ observers.
However, for more general trajectories, it is not possible to define a such a coordinate system: the Christoffel symbols get corrections due to the acceleration of the trajectory, and these are compatible with the notion of fictious forces that are seen due to non-inertiality.
The way to analyze the problem above is by means of the Fermi normal coordinates (FNC). These are a coordinate system that is associated with a timelike trajectory and describes "what an observer undergoing this trajectory sees". In case you want learn more about this, I recommend https://arxiv.org/abs/1102.0529. In any case, it is possible to show that the metric in FNC can be written locally around the curve (up to first order in acceleration and curvature) as
$$\begin{aligned}
&g_{\tau \tau}=-\left(1+2 a_{\mathrm{i}}(\tau) x^{i}+R_{0 \mathrm{i} 0 \mathrm{j}}(\tau) x^{i} x^{j}\right), \\
&g_{\tau i}=-\frac{2}{3} R_{0 \mathrm{jik}}(\tau) x^{j} x^{k} \\
&g_{i j}=\delta_{\mathrm{ij}}-\frac{1}{3} R_{\mathrm{ikjl}}(\tau) x^{k} x^{l}
\end{aligned},$$
where $\tau$ is the proper time of the trajectory (and the timelike coordinate of the FNC) and $x^i$ denote the proper distance from the trajectory, which are the spacelike trajectories of the FNC. $a(\tau)$ is the trajectories proper acceleration and $R(\tau)$ is the curvature tensor along its motion.
The Christoffel symbols to first order in acceleration and curvature then read
$$\begin{aligned}
\Gamma_{i j}^{\tau} &=\frac{1}{3}\left(R_{0 \mathrm{ijm}}+R_{0 \mathrm{jim}}\right) x^{m} \\
\Gamma_{\tau i}^{\tau} &=a_{\mathrm{i}}+R_{0 \mathrm{i} 0 \mathrm{~m}} x^{m} \\
\Gamma_{\tau \tau}^{\tau} &=0 \\
\Gamma_{j k}^{i} &=\frac{1}{3}\left(R_{\mathrm{j} \mathrm{km}}^{\mathrm{i}}+R_{\mathrm{k} \mathrm{jm}}^{\mathrm{i}}\right) x^{m} \\
\Gamma_{\tau j}^{i} &=R_{0 \mathrm{mj}}{\mathrm{j}} x^{m} \\
\Gamma_{\tau \tau}^{i} &=a^{\mathrm{i}}+R_{0}^{\mathrm{i}}{ }_{0 \mathrm{~m}} x^{m}
\end{aligned},$$
which can be seen to vanish if $a^i = 0$ (inertial motion) and $x^i = 0$ (on top of the curve).
