As far as I understand it the equivalence principle in general relativity is the statement that a frame at rest in a uniform gravitational field is equivalent (indistinguishable) to an accelerated (non-inertial) frame in the absence of gravity (i.e. in Minkowski spacetime).
I am trying to reason to myself why free-fall (i.e. a frame in which gravity is the only force acting) defines a locally inertial frame.
Consider an elevator with an observer inside it in free-fall in a uniform gravitational field. Is the reasoning simply that in free-fall, due to the equivalence principle, both the elevator and the observer inside the elevator will accelerate towards the centre of the massive body (that is the source of the gravitational field) at the same rate, hence there will be no relative acceleration between the elevator and the observer inside the elevator. As this is the case for any other objects inside the lift also, assuming that the observer cannot see outside of the lift, then there is no experiment that they can perform that will determine whether they are in free-fall or following uniform unaccelerated motion in deep space (in the absence of gravity), the two scenarios are indistinguishable. Hence, the observer inside the elevator must conclude that they are in an inertial frame (of course, this only holds locally due to tidal forces).
Would this be a correct way to argue the situation?