Underlining some aspects of the equivalence principle

We want to shed light on some discussed aspects of the principle of equivalence. We consider two different phases in the evolution of the principle. We pay attention to the fact that some textbooks do not dwell on the mathematical meaning of 'locally flat spacetime', which can confuse students and, occasionally, can also lead to misinterpretations. For example, the non-eliminability of tidal forces, not even locally, may suggest that the equivalence principle is wrong. By contrast, we want to show that this is in perfect agreement with the Einstein geometrical interpretation of spacetime. Finally, in order to check the approximations used in the Gedankenexperiment of an elevator, we estimate explicitly the magnitude of the deflection of light and the effects of tidal forces in a 'real lift'.