Scattering Amplitudes and Binary Black Holes
Post-Newtonian (PN) theory has been the established perturbative method to tackle the general relativistic two-body problem in the slow-veloctiy and weak-gravity regime. These calculations provide crucial information for gravitational waveform models such as the effective-on-body (EOB) formalism or phenomenological approaches, modeling the gravitational wave emission from binary compact objects. The post-Newtonian approximation can be obtained from a re-expansion in small velocities of the post-Minkowskian (PM) approximation. A recent resurgence of interest in fully PM computations and results has been associated with the prospect of applying advanced techniques for calculating quantum scattering amplitudes to the classical two-body problem.
In this reference, I showed, under the supervision of Jan Steinhoff and Justin Vines, that at leading order in the PN expansion at each order in spin, the infinite tower of spin interactions between the black holes can be resummed, leading to compact expressions valid to all orders in the black holes's spins. These gravitational waves emitted from a binary on quasi-circular (closed) orbits, can, in fact, be obtained from quantum amplitudes describing the scattering of massive spin-s particles. This I showed, in collaboration with Yilber Fabian Bautista in this reference. We worked out how to relate the gravitational waves emitted from a spinning binary black hole to the classical limit of the 5-point quantum scattering amplitude shown in the figure on the right. We were able to show explicitly that waveforms, needed to detect gravitational waves from inspiraling binary black holes, can be derived consistently, to the orders considered, from the classical limit of quantum scattering amplitudes.
In a work with Justin Vines in this reference, we considered the PM approximation to the scattering of a spinning black hole off of a Kerr black hole in the extreme mass ratio limit employing the decomposition of the associated 4-point amplitude, as shown in the figure above. We confronted this approach with results from general-relativistic “self-force” calculations of the linear perturbations of a Kerr spacetime sourced by a small orbiting body in the equatorial plane. We were able to translate between scattering and circular-orbit results by assuming the existence of a local-in-time canonical Hamiltonian governing the conservative dynamics of generic (bound and unbound) aligned-spin orbits, while employing the associated first law of spinning binary mechanics. To the extent possible with available self-force results, we confirmed, through linear order in the mass ratio, some previous conjectures which would begin to fill in the spin-dependent parts of the conservative dynamics for arbitrary-mass-ratio aligned-spin binary black holes at the fourth-and-a-half and fifth post-Newtonian orders. With this, we were able to fix coefficients in the effective field theory Lagrangian of spinning black holes at second order in the gravitational perturbations. Further details can be found in here.