Geometric transport along circular orbits in stationary axisymmetric spacetimes

TL;DR

This paper provides an explicit description of parallel transport along circular orbits in stationary axisymmetric spacetimes, extending previous Kerr spacetime analysis to a broader class and clarifying the effects of gravitomagnetic fields and orbit parameters.

Contribution

It introduces a detailed framework for analyzing parallel transport in these spacetimes using Lie transport and Lorentz transformations, expanding understanding beyond Kerr to general stationary axisymmetric cases.

Findings

Explicit expression for parallel transport in these spacetimes.

Extension of Kerr analysis to full spacetime outside the horizon.

Insight into gravitomagnetic effects on orbit acceleration.

Abstract

Parallel transport along circular orbits in orthogonally transitive stationary axisymmetric spacetimes is described explicitly relative to Lie transport in terms of the electric and magnetic parts of the induced connection. The influence of both the gravitoelectromagnetic fields associated with the zero angular momentum observers and of the Frenet-Serret parameters of these orbits as a function of their angular velocity is seen on the behavior of parallel transport through its representation as a parameter-dependent Lorentz transformation between these two inner-product preserving transports which is generated by the induced connection. This extends the analysis of parallel transport in the equatorial plane of the Kerr spacetime to the entire spacetime outside the black hole horizon, and helps give an intuitive picture of how competing "central attraction forces" and centripetal accelerations contribute with gravitomagnetic effects to explain the behavior of the 4-acceleration of circular orbits in that spacetime.