Einstein Relatively Easy

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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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"Five or six weeks elapsed between the conception of the idea for the special theory of relativity and the completion of the relevant publication" Einstein to Carl Seeling on March 11, 1952

"Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
David Hilbert

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