In general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds.
Let $\xi^{\alpha}$ the coordinates in the local inertial frame, so we get $$ds^2=\eta_{\alpha \beta}d \xi^{\alpha} d \xi^{\beta}.$$ If we switch the frame of reference to coordinates $x^{\mu}$ : $\xi^{\alpha}= \xi^{\alpha}(x^0, x^1, x^2, x^3)$
and with $$g_{\mu \nu} (x)= \eta_{\alpha \beta} \frac{\partial \xi^{\alpha}}{\partial x^{\mu}} \frac{\partial \xi^{\beta}}{\partial x^{\nu}}$$ we get:
$$ds^2=g_{\mu \nu}d x^{\mu}(x) d x^{\nu}.$$
I don't understand why it isn't possible to find a transformation to get $$ds^2=\eta_{\alpha \beta}d \xi^{\alpha} d \xi^{\beta}$$ on the whole or almost the whole manifold? Because $g_{\mu \nu}(x)$ is still the same on the whole manifold?
Best Answer
If $ds^2=\eta_{\alpha \beta}d \xi^{\alpha} d \xi^{\beta}$ were true for all points of space, we would have no curvature, hence no gravity!
Take for example a sphere (the Earth), locally we can measure distances by $ds^2=dx^2+dy^2$, but this can't hold for two arbitrary points on the sphere. In fact, this coordinate system changes from point to point (think of a tangent plane on the sphere).
We would have to replace the local coordinates, which you called $\xi^\alpha$ (the cartesian coordinates $x$ and $y$ in this case) and replace them by some other global coordinates, such as the angles $\theta$ and $\phi$. (Note that we would still need to patches to cover the total sphere). Then, the distance between two arbitrary points would be calculated using $$ds^2=r^2 \sin^2 \theta d\phi^2 + r^2 d\theta^2$$
So curvature is what makes us introduce $g_{\mu\nu}$ and the global coordinates $x^\mu$.
A local inertial frame would see no gravity and would be able to do Special Relativity, for a small region there is no significant curvature. To continue the analogy of the Earth, you wouldn't appreciate curvature in many kilometers, but the local region would be much smaller than the whole patch. Note that any world map (a whole patch) will present distorsion because of curvature, but a small road map won't have any distorsion.