The Motion of Test Bodies around Kerr Black Holes

TL;DR

This thesis investigates the motion of finite-size, spinning test bodies around Kerr black holes using a multipole expansion up to quadrupole order, providing new Hamiltonian formulations and insights into conserved quantities in curved spacetime.

Contribution

It introduces a covariant Hamiltonian framework for spinning bodies in Kerr spacetime, extending previous models by including spin-induced quadrupole moments and deriving associated conserved quantities.

Findings

Derived Hamiltonian reproducing MPD equations with quadrupole moments

Identified conserved quantities for spinning bodies in Kerr spacetime

Connected Hamilton-Jacobi solutions to constants of motion

Abstract

This thesis aims to explore the properties of the motion of finite size, compact test bodies around a Kerr black hole in the small mass-ratio approximation. The small body is modelled as a perturbation of Kerr geometry, neglecting its gravitational back-reaction but including deviations from a purely geodesic motion by allowing it to possess a non-trivial internal structure. Such a body can be accurately described by a worldline endowed with a collection of multipole moments. Hereafter, we shall always consider the multipole expansion truncated at quadrupole order. Moreover, only spin-induced quadrupole moment will be taken into account, thus discarding the presence of any tidal-type deformation. For astrophysically realistic objects, this approximation is consistent with expanding the equations of motion up to second order in the body's spin magnitude. The text is structured as follows. The first part is devoted to an extended review of geodesic motion in Kerr spacetime, including Hamiltonian formulation and classification of timelike geodesics, with a particular emphasis put on near-horizon geodesics of high spin black holes. The second part introduces the equations of motion for extended test bodies in generic curved spacetime, also known as Mathisson-Papapetrou-Dixon (MPD) equations. The third part discusses conserved quantities for the MPD equations in Kerr spacetime, restricting to the aforementioned quadrupole approximation. Finally, the covariant Hamiltonian formulation of test body motion in curved spacetime is presented, and an Hamiltonian reproducing the spin-induced quadrupole MPD equations is derived. It is shown that the constants of motion obtained in the previous part directly arise while solving the Hamilton-Jacobi equation at first order in the spin magnitude. Some expectations regarding the computation at quadratic order close the discussion.