# Hamilton-Jacobi equation for spinning particles near black holes

## TL;DR

This paper develops a Hamilton-Jacobi framework for spinning particles near Kerr black holes, accounting for finite-size effects crucial for accurate gravitational wave modeling with LISA.

## Contribution

It formulates and solves the Hamilton-Jacobi equation for spinning particles in Kerr spacetime, incorporating spin-curvature coupling effects.

## Key findings

- Derived solutions for geodesic deviations due to spin effects.
- Computed orbital turning points and fundamental frequencies.
- Provides essential ingredients for LISA waveform models.

## Abstract

A compact stellar-mass object inspiralling onto a massive black hole deviates from geodesic motion due to radiation-reaction forces as well as finite-size effects. Such post-geodesic deviations need to be included with sufficient precision into wave-form models for the upcoming space-based gravitational-wave detector LISA. I present the formulation and solution of the Hamilton-Jacobi equation of geodesics near Kerr black holes perturbed by the so-called spin-curvature coupling, the leading order finite-size effect. In return, this solution allows to compute a number of observables such as the turning points of the orbits as well as the fundamental frequencies of motion. This result provides one of the necessary ingredients for waveform models for LISA and an important contribution useful for the relativistic two-body problem in general.