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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"The essence of my theory is precisely that no independent properties are attributed to space on its own. It can be put jokingly this way. If I allow all things to vanish from the world, then following Newton, the Galilean inertial space remains; following my interpretation, however, nothing remains.."
Letter from A.Einstein to Karl Schwarzschild - Berlin, 9 January 1916

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice."
Einstein to Max Born, letter 52, 4th december 1926

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