Clocks' synchronization without round-trip conditions

Abstract

Poincar\'e-Einstein's synchronization convention is transitive, and thus leads to a consistent synchronization, only if some form of round-trip property is satisfied. An improved version is given here which does not suffer from this limitation and which therefore may find application in physics, computer science and communication theory. As for the application to physics, the round-trip condition required by the Poincar\'e-Einstein's synchronization convention corresponds to a vanishing Sagnac effect and thus to the selection of an irrotational frame. The corrected method applies also to rotating frames and shows that there is a consistent synchronization for every given measure on space. The correction to Poincar\'e-Einstein's amounts to an average of the Sagnac holonomy over all the possible triangular paths. The mathematics used is reminiscent of Alexander cohomology theory.