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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.

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For your 1st question: I think you are correct in so far as concluding that the equation is locally true. However, you may wish to keep in mind that, being a tensor equation, this equation is locally true in every (defined) point of the coordinate chart. As it is always true in a sufficiently small locale about the particle and as the particle, by definition, always exists within this locale, one expects that the geodesic equation (and, I think, the use of the epsilon coordinates in the derivation) will be valid for the entire trajectory. For your 2nd question: the x coordinates need not be the coordinates of the entire manifold - it depends upon how the manifold is defined. However, the great thing about tensor equations is that they are generally true throughout the manifold!

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