# The Orbital Lense-Thirring Precession in a Strong Field

## TL;DR

This paper analytically demonstrates that the orbital angular momentum of particles around a Kerr black hole precesses in a way that closely resembles a circle, extending the weak-field Lense-Thirring precession concept to strong gravitational fields.

## Contribution

It provides an exact analytical description of orbital angular momentum evolution in Kerr spacetime, confirming the near-circular precession pattern for a wide class of orbits and deriving parameters for unstable spherical orbits.

## Key findings

- Orbital hodographs are nearly circular with deviations under 10%.
- Nutation accurately approximates deviations, especially for larger deviations.
- Lense-Thirring precession remains valid in strong gravitational fields.

## Abstract

We study the exact evolution of the orbital angular momentum of a massive particle in the gravitational field of a Kerr black hole. We show analytically that, for a wide class of orbits, the angular momentum's hodograph is always close to a circle. This applies to both bounded and unbounded orbits that do not end up in the black hole. Deviations from the circular shape do not exceed $\approx10\%$ and $\approx7\%$ for bounded and unbounded orbits, respectively. We also find that nutation provides an accurate approximation for those deviations, which fits the exact curve within $\sim 0.01\%$ for the orbits of maximal deviation. Remarkably, the more the deviation, the better the nutation approximates it. Thus, we demonstrate that the orbital Lense-Thirring precession, originally obtained in the weak-field limit, is also a valid description in the general case of (almost) arbitrary exact orbits. As a by-product, we also derive the parameters of unstable spherical timelike orbits as a function of their radii and arbitrary rotation parameter $a$ and Carter's constant $Q$. We verify our results numerically for all the kinds of orbits studied.