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The Motion of Test Bodies around Kerr Black Holes

par Druart, Adrien
Président du jury Argurio, Riccardo
Promoteur Compère, Geoffrey
Publication Non publié, 2023-06-27

Thèse de doctorat

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Résumé :

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. They are derived from a generic action principle, and their physical significance and mathematical consistency is examined in details. The third part discusses conserved quantities for the MPD equations in Kerr spacetime, restricting to the aforementioned quadrupole approximation. The conservation is required to hold perturbatively in the test body's spin magnitude, and the related conserved quantities are build through the explicit resolution of the conservation constraint equations. 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. Two applications of the Hamiltonian formalism are subsequently discussed: (i) the integrability properties of MPD equations in Kerr and Schwarzschild spacetimes and (ii) the Hamilton-Jacobi formulation of MPD equations in Kerr spacetime. 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.