We may consider a "local" region in curved spacetime (local in respect of the spatial and the temporal coordinates). A "local inertial frame" may be constructed by some transformation that produces flat spacetime locally. This transformation produces the diagonal [1,-1,-1,-1] in an approximate manner.
A physical point:
You are performing some experiment in a small laboratory room (stationary w.r.t. the Earth) for a small period of time. Are you in a local inertial frame? If there was a freely falling lift in front of you, it would have been a better approximation to the inertial frame concept. So your laboratory does not qualify to be an inertial frame when compared with the falling lift. (The difference becomes even more conspicuous if you imagine "gravity" to be a 100 or a thousand times stronger.)
Now, let's move into the problem:
We take a small region of curved spacetime and use some suitable transformation to produce a "local inertial" frame. For this transformation we are disregarding all the anisotropies and inhomogeneities of the surrounding space in the original manifold.Incidentally the objective behind creating local inertial frames is to apply SR. The associated Lorentz transformations are supposed to hold good in the ideal conditions of isotropy and homogeneity of space
Our transformation transforms a small region of curved spacetime to a small region of flat spacetime. To this small/finite flat spacetime, we add the rest of it, ignoring all the anisotropies and inhomogeneities of the surroundings in the original curved spacetime [original manifold]. In effect we are assuming a "Global Transformation" without working it out! (And we expect correct results when we go back to the original space by the reverse transformations) We are in effect assuming a "Global Transformation" from curved spacetime to flat spacetime. Is it not as sacrilegious as "Flattening a Sphere"?
How much unreliability is the aforesaid error going to cast on calculations performed in the local inertial frames?To what extent will this error affect the Principle of Equivalence?