A globally well-behaved simultaneity connection for stationary frames in the weak field limit

TL;DR

This paper introduces a new global simultaneity convention for stationary frames in the weak gravitational field limit, which improves upon Einstein's convention under certain conditions and has practical applications.

Contribution

It proposes a specific simultaneity convention based on acceleration and vorticity, with better global behavior in weak fields than traditional methods.

Findings

The new convention outperforms Einstein's in weak gravitational fields.

It reduces to Einstein's convention when certain vector conditions are met.

Practical methods for implementing the convention are provided.

Abstract

Local simultaneity conventions are mathematically represented by connections on the bundle of timelike curves that defines the frame. Those simultaneity conventions having an holonomy proportional to the Riemann tensor have a special interest since they exhibit a good global behavior in the weak field limit. By requiring that the simultaneity convention depends on the the acceleration, vorticity vector and the angle between them we are able to restrict considerably the allowed simultaneity conventions. In particular, we focus our study on the simplest among the allowed conventions. It should be preferred over Einstein's in flat spacetime and in general if the tidal force between neighboring particles is weaker than the centrifugal force. It reduces to the Einstein convention if the vectorial product between the acceleration and the vorticity vector vanishes. We finally show how to use this convention in practice.