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the three-acceleration and the three-velocity, and
* and
* is the Lorentz factor for the speed (with ). A dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time (in other terms, ).

In an instantaneously co-moving inertial reference frame , and , i.e. in such a reference frame

Geometrically, four-acceleration is a curvature vector of a worldline.

Therefore,

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