The Einstein Equivalence Principle

For a mass that is inertial in a reference frame $$R'$$ (i.e. in a freely falling elevator where $$R'$$  is uniformly accelerating, all masses in the system are inertial in that reference frame) that can be at any point $$P$$ in curved spacetime in a real gravitational field, all the laws of physics are the same in $$R'$$ as an intertial reference frame $$R$$ in flat Minkowski spacetime (i.e. elevator floating in empty space). The laws of mechanics would be the same for the observer $$O$$ who is in an inertial reference frame in an elevator in empty space where $$\vec{g}$$ can be regarded as zero. Both observers will feel weightless. If $$O$$ lets go of an apple in his elevator, in his reference frame with his coordinate system $$x^μ$$, the apple will float. If he gave the apple a push (with the same force as $$O'$$ so $$\vec{F}_{O',apple}=\vec{F}_{O,apple}$$ over the same time interval $$Δt'=Δt$$), that apple would move away with the same veolocity $$\vec{v}$$ in his reference frame. Not only are all the laws of mechanics the same in both reference frames, but all the laws of physics are the same in both reference frames. This means that all phenomena are indistinguishable in the two reference frames. $$m_i=m_g$$ is very important; otherwise objects would fall at different rates.

According to the EEP, in the neighborhood around any point $$P$$ in curved spacetime all the laws of physics are the same as in a uniformly accelerating reference frame in Minkowski spacetime. For a reference frame $$R'$$ attached to a mass $$m$$ freely falling in a gravitational field $$\vec{g}$$, over very small distances in curved spacetime, $$R'$$ can be treated as an inertial reference frame in flat Minkowski spacetime. In that reference frame, the force acting on particles is $$Γ=0$$ because everything is inertial??

Suppose the observer $$O'$$  is in a freely falling elevator. Imagine that as he is falling, he is carrying a coordinate system $$(t',x',y',z')$$ with him. Consider a second observer $$O$$ who is in an elevator floating in the middle of empty space far away from any gravitational field. $$O$$ will be in an inertial reference frame. If the observer $$O'$$ and the elevator fall through a very small region of spacetime, the tidal forces due to gravity at every point in and along the elevator can be assumed to be zero (difference in values of $$\vec{g}$$ at every point is zero). Therefore, all particles in the system (including the elevator and observer $$O'$$) fall at the same rate and have the same trajectory through space. If we imagine that $$O'$$ is carrying with him the coordinate axes $$(t',x',y',z')$$, all particles in the system are stationary in that coordinate system. For example, if $$O'$$ let go of an apple (while him, the elevator, and the apple are in free fall) the apple (in his reference frame) would be floating in space. Furthermore, if he gave the apple a push, a force $$\vec{F}_{O',apple}$$ would act on the apple for a short time interval $$Δt'$$. This would cause the apple to undergo an acceleration for the time interval $$Δt'$$. At the instant in time when  $$O’$$'s hand is no longer making contact with the apple (from that instant in time and onwards), there would no longer be a force acting on the apple and it would move at a constant speed $$v$$ in a straight line across the elevator.