relativity ∗

We present some basic facts concerning simultaneity in both special and general relativity. We discuss Weyl’s proof of the consistence of Einstein’s synchronization convention and consider the general relativistic problem of assigning a time function to a congruence of timelike curves. 1 Simultaneity in special relativity In special relativity the possibility of synchronizing distant clocks so as to obtain a global coordinate time was proved by Weyl [15] in a fundamental and unfortunately overlooked proof. Let a light beam travel from point A to point B, leaving A at time tA according to the clock at A. The Einstein convention states (in Weyl’s version) that the clock at B (say clock B for short) is Einstein synchronized with clock A if the time of arrival according to clock B reads tB = tA + AB/c where AB is the Euclidean distance between the two points. This synchronization convention should satisfy at least the following three properties in order to lead to a spacetime foliation into (equal time) simultaneity slices (i) Time homogeneity. If clock B is set accordingly to the procedure above then repeating the same experiment the equation tB = tA + AB/c holds for whatever value of tA without the need of setting again clock B (i.e. it is meaningful to say that clock B is synchronized with clock A). (ii) Symmetry. If clock A is synchronized with clock B then clock B is synchronized with clock A (i.e. it is meaningful to say that the two clocks are synchronized). ∗To appear in the proceedings of the school Relativistic Coordinates, Reference and Positioning Systems, Salamanca (Spain), January 21-25, 2005