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In attempt to introduce gr concepts

When there are gravitational accelerations present, as for example in the gravitational field of the earth, the space cannot be the flat Minkowski space. Indeed, in the Minkowski space we can have $$\Gamma^{\lambda}_{\mu\nu}=0$$ everywhere. However we don't know this We can only make them vanish at one point, or approximately in a small region, by the use of an appropriate coordinate system. For the sentence in bold, what experience that tells us so about gravitational accelerations (that we can not make them vanish everywhere except in a small region)? How do we know that?

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  • $\begingroup$ Quote is from which reference? $\endgroup$ – Qmechanic Aug 4 '15 at 12:32
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Schutz's A First Course in General Relativity explains it pretty well, I think. The pertinent page is here, but if international copyright law won't let you read it, the basic gist of it is the following: If the gravitational acceleration was the same everywhere in space, then we could shift into a freely falling reference frame, and freely falling observers at other points would not accelerate at all relative to us. In other words, we wouldn't observe a gravitational force at all in this frame; we would have a global inertial frame. Note that this can be thought of as a change of coordinates: $t' = t, x' = x - \frac{1}{2} g t^2$.

However, the gravitational acceleration does not in general have a constant direction or magnitude throughout space & time. For example, two freely falling observers near the surface of the Earth will be accelerating in slightly different directions; in particular, this means that they will be accelerating relative to each other, and each one will think that the other has a force acting on him/her. If they're pretty close to each other, then this relative acceleration will be negligible; so we can still define an approximate local inertial frame for practical purposes. (This would be a local coordinate redefinition.) But as we try to extend out the size of our inertial frame (i.e., we try to describe the motions of freely falling objects farther & farther away from us), eventually these relative accelerations will become too large to ignore. Thus, a global inertial frame does not in general exist.

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It is not clear what OP is asking, but consider the following chain of reasoning:

  1. Scalar curvature $R$ is an invariant independent of choice of coordinates.

  2. In GR, regions of matter, e.g., a star, is modelled with non-zero curvature.

  3. The Minkowski space has zero curvature.

  4. Hence, if a spacetime $(M,g)$ has non-zero scalar curvature $R$ at a point $p\in M$, there doesn't exist an open coordinate neighborhood $U\subseteq M$ of $p$ such that the metric $g$ becomes on Minkowski form $g_{\mu\nu}=\eta_{\mu\nu}$ in the entire open neighborhood $U$, no matter how small. In short: We can not always bring the metric $g$ on Minkowski-form in a (sufficiently small) open neighborhood $U$ of a point $p$.

  5. Conversely, for an arbitrary point $p\in M$ on a Lorentzian manifold $(M,g)$, there exist Riemann normal coordinates in a sufficiently small coordinate neighborhood $U\subseteq M$ of the point $p$ such that the metric $g_{\mu\nu}$ becomes on Minkowski-form with vanishing (Levi-Civita) Christoffel symbols $\Gamma^{\lambda}_{\mu\nu}$ locally in the point $p$ (but not necessarily in the punctured neighborhood $U\backslash\{p\}$ and the manifold $M$ is not necessarily flat in $U$). In particular, the (Levi-Civita) Riemann curvature tensor $R^{\sigma}{}_{\mu\nu\lambda}$ does not necessarily vanish at $p$. In short: We can always bring the metric $g$ and Christoffel symbols $\Gamma$ on Minkowski-form in a point $p$.

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