Circular Holonomy and Clock Effects in Stationary Axisymmetric Spacetimes

TL;DR

This paper investigates clock effects and holonomy in stationary axisymmetric spacetimes, focusing on the differences between static and nonstatic cases, with specific analysis in Kerr spacetime.

Contribution

It extends the study of clock effects to holonomy group elements and analyzes the differences in holonomy invariance between static and nonstatic spacetimes.

Findings

Holonomy invariance band exists in static cases but not in nonstatic cases.

In Kerr spacetime, rotation frequencies relate to geodesic periods.

Differences in clock effects depend on spacetime staticity.

Abstract

Stationary axisymmetric spacetimes containing a pair of oppositely-rotating periodically-intersecting circular geodesics allow the study of various so-called `clock effects' by comparing either observer or geodesic proper time periods of orbital circuits defined by the observer or the geodesic crossing points. This can be extended from a comparison of clocks to a comparison of parallel transported vectors, leading to the study of special elements of the spacetime holonomy group. The band of holonomy invariance found for a dense subset of special geodesic orbits outside a certain radius in the static case does not exist in the nonstatic case. In the Kerr spacetime the dimensionless frequencies associated with parallel transport rotations can be expressed as ratios of the proper and average coordinate periods of the circular geodesics.