# What is the extent of Local Inertial coordinate system

My professor's notes say, that according to the Equivalence principle there is Freely Falling coordinate system in which the equations of motion for a free particle is $$\frac{d^2 \epsilon^{\mu}}{d\tau^2}=0$$ (1) where $$\epsilon^{\mu}$$ are the coordinates in the Local Inertial coordinate system centred at $$\epsilon^{\mu}$$ = (0, 0,0,0) . Then he goes on and does a general coordinate transformation to $$x^{\mu}$$ system to derive the geodesic equation.

My question is what is expanse of the coordinate system $$\epsilon^{\mu}$$. To me it looks that eq (1) is true only in the immediate neighbourhood of the point where the coordinate system is centred and as you move a finite distance away from that point eq (1) will cease to hold true as at some other point $$\epsilon^{\mu}$$ ( centred at (0, 0,0,0) ) won't be a Minkowskian freely falling coordinate system.

Is that correct or is eq (1) valid for the whole trajectory of the particle and that $$\epsilon^{\mu}$$ coordinates are valid for the whole trajectory as well. And also whether $$x^{\mu}$$ are the coordinates for the whole manifold.

Can anyone help in me in this regard.