Einstein’s elevator: sealed in a freely falling lab, you are supposed to be unable to distinguish it from a lab floating in deep space. You have two marbles and as much patience as you like. Can you nevertheless prove there is a planet outside?
Solution
Yes — the equivalence principle is local, and two marbles are enough to probe “non-local.”
Release the marbles a horizontal distance apart: each falls toward the center of the Earth, so their paths converge, and they drift together at tidal acceleration . Release them one above the other instead: the lower one sits in a stronger field and pulls ahead, so they separate at . A lab in empty space shows neither. Squeeze in the horizontal plane, stretch along the vertical — that residue is what no choice of freely falling frame can erase.
The formal statement: coordinates can flatten the metric and kill its first derivatives at a point (that is the freely falling frame), but the second derivatives — the Riemann tensor — stay. Gravity’s irreducible signature is not the pull, which you can transform away, but the squeeze, which you cannot. Relative acceleration of nearby free-fallers is geodesic deviation, and it is the curvature tensor read off with rulers and marbles.
Deeper in the notebook: 01. Tidal Forces and Geodesic Deviation · 02. Weak, Einstein, and Strong Equivalence Principles · 01. Jacobi Fields