of mg is needed to keep one from accelerating downward. Four-vector derivations The spacetime equations of this section allow one to address all deviations between proper and coordinate acceleration in a single calculation. For example, let's calculate the Christoffel symbols: for the far-coordinate Schwarzschild metric , where rs is the Schwarzschild radius 2GM/c2. The resulting array of coefficients becomes: From this you can obtain the shell-frame proper acceleration by setting coordinate acceleration to zero and thus requiring that proper acceleration cancel the geometric acceleration of a stationary object i.e. . This does not solve the problem yet, since Schwarzschild coordinates in curved spacetime are book-keeper coordinates but not those of a local observer. The magnitude of the above proper acceleration 4-vector, namely , is however precisely what we want i.e. the upward frame-invariant proper acceleration needed to counteract the downward geometric acceleration felt by dwellers on the surface of a planet. A special case of the above Christoffel symbol set is the flat-space spherical coordinate set obtained by setting rs or M above to zero: From this we can obtain, for example, the centripetal proper acceleration needed to cancel the centrifugal geometric acceleration of an object moving at constant angular velocity at the equator where . Forming the same 4-vector sum as above for the case of and zero yields nothing more than the classical acceleration for rotational motion given above, i.e. so that . Coriolis effects also reside in these connection coefficients, and similarly arise from coordinate-frame geometry alone. See also Acceleration: change in velocity Proper velocity: momentum per mass in special relativity; composed of the spacelike components of the 4-velocity Proper reference frame (flat spacetime): accelerated reference frame in special relativity (Minkowski space) Fictitious force: one name for mass times geometric acceleration Four-vector: making the connection between space and time explicit Kinematics: for studying ways that position changes with time Uniform acceleration: holding coordinate acceleration fixed Footnotes External links Excerpts from the first edition of Spacetime Physics, and other resources posted by Edwin F. Taylor James Hartle's gravity book page including Mathematica programs to calculate Christoffel symbols. Andrew Hamilton's notes and programs for working with local tetrads at U. Colorado, Boulder. Minkowski spacetime Acceleration