I'm helping write some supplementary notes for an undergraduate GR class at my university. I'm writing about the equivalence principle, do you find this way of stating matters acceptable? The equivalence principle as stated by Einstein asserts that in a sufficiently small vicinity of an event in space-time and in a freely falling frame, physics is the same as in a global inertial frame in special relativity. Now i want to know if i can state this in an equivalent way as follows: Saying that physics is the same in a freely falling frame in a small enough vicinity of an event in space-time as in an inertial frame of special relativity amounts to saying that every mesurement one takes in such a frame gives the same results, to first order, as the same measurements performed in a global inertial frame or SR. Space-time thus has the same infinitesimal structure as the space-time of special relativity, up to first order in displacements. This means that a reference frame can always be set up such that at a single event in space-time, the metric has the same components as in an inertial frame of special relativity, with the christoffel symbols vanishing there simultaneously, and that a freely falling frame is such a frame. A consequence of the above mentioned is that freely falling particles are represented by time-like geodesics in space-time. Thanks in advance.