Someone argues: "A freely falling lift removes gravity completely inside it
PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry
The Geometry of Gravity, Ch Accelerated observers, Rindler coordinates and the equivalence principle
Problem
Someone argues: "A freely falling lift removes gravity completely inside it, by the equivalence principle. There is nothing in the statement of the principle that mentions a size, so a freely falling elevator the size of a continent, say ℓ=1000km tall, removes gravity just as completely as a small one. And since the exact metric of a real planet never makes any component vanish above the surface, unlike the Rindler g00, there is no coordinate 'edge' of the kind seen in the rocket, so nothing at all limits how large the free-falling region can be."
((a)) Using Earth's radius R⊕=6371km and the formula for the fractional discrepancy between the exact potential difference and the uniform-field (equivalence-principle) approximation over a height ℓ, namely ℓ/R⊕, evaluate this fraction for ℓ=1000km and comment on whether it is small.
((b)) Explain why the absence of a vanishing metric component in the exact planetary case does not by itself show that the equivalence principle holds without limit, contrasting a vanishing component with a quantity every observer agrees on.
((c)) State, in one sentence each, what the weak and the Einstein forms of the equivalence principle actually license, and which one is being over-extended by the claim.