Clock rate comparison in a uniform gravitational field

Abstract

A partially alternative derivation of the expression for the time dilation effect in a uniform static gravitational field is obtained by means of a thought experiment in which rates of clocks at rest at different heights are compared using as reference a clock bound to a free falling reference system (FFRS). Derivations along these lines have already been proposed, but generally introducing some shortcut in order to make the presentation elementary. The treatment is here exact: the clocks whose rates one wishes to compare are let to describe their world lines (Rindler's hyperbolae) with respect to the FFRS, and the result is obtained by comparing their lengths in space-time. Only at the end of the paper the corresponding GR metric is derived, to the purpose of making a comparison to the solutions of Einstein field equation. The exercise may nonetheless prove pedagogically instructive insofar as it shows that the exact result of General Relativity (GR) can be obtained in terms of physical and geometrical reasoning without having recourse to the general formalism. It also compels to deal with a few subtle points inherent in the very foundations of GR.