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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.