Simultaneity and generalized connections in general relativity

Stationary extended frames in general relativity are considered. The requirement of stationarity allows one to treat the spacetime as a principal fibre bundle over the one-dimensional group of time translations. Over this bundle a connection form establishes the simultaneity between neighbouring events accordingly with the Einstein synchronization convention. The mathematics involved is that of gauge theories where a gauge choice is interpreted as a global simultaneity convention. Then simultaneity in non-stationary frames is investigated: it turns out to be described by a gauge theory in a fibre bundle without structure group, the curvature being given by the Fr¨olicher–Nijenhuis bracket of the connection. The Bianchi identity of this gauge theory is a differential relation between the vorticity field and the acceleration field. In order for the simultaneity connection to be principal, a necessary and sufficient condition on the 4-velocity of the observers is given.