Principle of Equivalence: Understanding the Foundations of General Relativity

[Morgoth]

I have just began my journey in General Theory of relativity through the book of Weinberg-Gravitation and cosmology principles and applications of the general theory of relativity

I don't know if this topic is just a repeat of an older one, if it is please let me know the link to it...

I started reading about the Principle of Equivalence, and it somehow confused me. It says that the Strong P.o.E tells us that:
At every spacetime point in an arbitary gravitational field it is possible to choose a "locally inertial coordinate sysyem" such that within sufficiently small region of the point in question, laws of nature take the same form they have in unaccelerated Cartesian Coordinate systems in the absence of gravity (meaning the laws we get from special relativity).

At this point I thought like this:
If for example I get a spacetime point x in an arbitrary curved space of metric g_αβ, then the metric on that point is equal to the metric of a Minkowski space (Gauss's curvature is equal to 0- flat space). This would mean that:
g_αβ(x)=n_αβ(x)
Going to another point on the same space, point X'=x+Δx, I can still use the P.o.E. to write on that point:
g_αβ(X')=n_αβ(X')

Doesn't that imply that ,by keeping doing the same work over and over again, a space of an arbitrary curvature can in fact be described by a flat space? That is wrong (for example the Sphere cannot).
Where is the "gap" in my approach and so my understanding?

 

[pumila]

The crucial word is "local". To be pedantic it can apply only at one point in space, although it is approximately true love a small region.

 

[atyy]

In addition to pumila's point, although at any particular point the metric can be Minkowski even in a curved spacetime, second derivatives of the metric will not disappear in a curved spacetime by any trick.

 

[WannabeNewton]

What you are talking about are Riemann normal coordinates. Taking p in M, there exists a normal neighborhood U of p which is a neighborhood that is diffeomorphic to a neighborhood V of the origin in Tp(M) under the exponential map at p. We define a coordinate chart on U by $\psi = \varphi ^{-1}\circ exp_{p}^{-1}:U\rightarrow \mathbb{R}^{n}$ where $\varphi$ is any isomorphism between euclidean n - space and Tp(M). At p the riemannian metric reduces to the kronecker delta (and accordingly for pseudo - riemannian metrics to the minkowski metric) and the connection coefficients at this point all vanish. Note that this is at p and the coordinates themselves are only defined on a certain neighborhood of p. This was very informal but if you want to see how they are constructed rigorously then you can check out any text on riemannian geometry (Lee or Do Carmo).

 

[Morgoth]

This might sound a little strange (the questioner trying to give an answer to his question).
I just thought that on a sphere we can always use for infinitesimal lines the Pythagorean Theorem, writing the line ds^2 in a quadratic form... but globally we cannot express distances as lines on it...
This stands true for a LINE, but I don't know how it would stand true for a coordinate system(moving axis-es) as well.

thanks for the rest of answers, I will for sure check what you said WannabeNewton...