Simultaneity and generalized connections in general relativity

TL;DR

This paper explores the mathematical structure of simultaneity in general relativity using gauge theory concepts, analyzing stationary and non-stationary frames and their associated connections.

Contribution

It introduces a gauge-theoretic framework for understanding simultaneity in both stationary and non-stationary spacetimes, linking it to fiber bundle connections and curvature.

Findings

Simultaneity in stationary frames can be modeled as a principal fiber bundle with a connection form.

Non-stationary frames require a gauge theory without a structure group, with curvature related to the Frölicher-Nijenhuis bracket.

A necessary and sufficient condition on observer 4-velocity ensures the principal nature of the simultaneity connection.

Abstract

Stationary extended frames in general relativity are considered. The requirement of stationarity allows to treat the spacetime as a principal fiber bundle over the one-dimensional group of time translations. Over this bundle a connection form establishes the simultaneity between neighboring 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 to be described by a gauge theory in a fiber 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.