# Analytic solutions for parallel transport along generic bound geodesics in Kerr spacetime

## TL;DR

This paper derives analytical solutions for parallel transport along bound geodesics in Kerr spacetime, aiding the understanding of spin effects in extreme mass-ratio inspirals and identifying resonance locations.

## Contribution

It provides the first closed-form solutions for parallel transport along generic bound geodesics in Kerr spacetime, expressed in Mino time.

## Key findings

- Analytical expressions for parallel transport along geodesics in Kerr spacetime.
- Identification of spin-spin resonance locations in EMRIs.
- Analytical proper time as a function of Mino time.

## Abstract

We provide analytical closed form solutions for the parallel transport along a bound geodesic in Kerr spacetime. This can be considered the lowest order approximation for the motion a spinning black hole in an extreme mass-ratio inspiral. As an illustration of the usefulness of our new found expressions we scope out the locations of spin-spin resonances in quasi-circular EMRIs. All solutions are given as functions of Mino time, which facilitates the decoupling of the equations of motion. To help physical interpretation, we also provide an analytical expression for the proper time along a geodesic as a function of Mino time.