For any point m in a spacetime Riemannian manifold M, there exists a local coordinate system at m such that:

we call such a coordinate system a **local inertial frame** or a **normal frame.**

**Remark1**: the possibility of the existence of this local referential is fully demonstrated in our article Local Flatness or Local Inertial Frames and SpaceTime curvature.

**Remark2: **as all the first order derivatives of the metric are null, given the Christoffel symbol expression:

then in a local inertial referential the vanishing of the partial derivatives of the metric tensor at any point of M **is equivalent to the vanishing of Christoffel symbols** at that point and in this referential **the geodesics are straight lines**.

**Remark3**: if the metric first derivatives can always been nullified,** it is not the case for all the second derivatives** which can only be nullified in a **flat spacetime** (refer to the same above article for more details)

**Remark4**: below is a link to an excellent youtube tutorial which gives an overview of how a local inertial frame (black colour) can be obtained by a general coordinate transformation at any point P of a spacetime manifold (blue colour)

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