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According to Einstein's Equivalence Principle a uniform gravitational field is equivalent to a uniformly accelerating frame of reference. Although one would expect an object in free fall to accelerate uniformly this is not the case. The only reasonable requirement one can make is that an observer in free fall will determine that a supported object accelerate uniformly. However the acceleration of supported objects, seen by a free fall observer, will depend on the objects location. The metric constructed with such considerations is found to be [1][2][3][4][5]

This metric is derived by demanding that the Riemann tensor vanishes in the region where the metric represents a uniform gravitational field.

[1] Principle of Equivalence, F. Rohrlich, Ann. Phys. 22, 169-191, (1963), page 173/
[2] Radiation from a Uniformly Accelerated Charge, David G. Boulware, Ann. Phys., 124, (1980), page174.
[3] Relativistic solutions to the falling body in a uniform gravitational field, Carl G. Adler, Robert W. Brehme, Am. J. Phys. 59 (3), March 1991.
[4] Gravitation, Charles. W. Misner, Kip S. Thorne, John Archibald Wheeler, (1973), sect 6.6.
[5] The uniformly accelerated reference frame, J. Dwayne Hamilton, Am. J. Phys., 46(1), Jan. 1978.

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