Scattering of gravitational waves off spinning compact objects with an effective worldline theory

TL;DR

This paper calculates how gravitational waves scatter off spinning compact objects like black holes using an effective worldline theory, connecting classical and quantum scattering amplitudes and exploring spin effects up to octupole order.

Contribution

It introduces a method to compute classical scattering amplitudes for spinning objects with higher multipole moments using an effective worldline approach, extending previous results to include cubic-in-spin effects.

Findings

Agreement with known black hole results at cubic-in-spin order

Derived new scattering amplitude expressions for general spin-induced multipoles

Established a link between classical scattering and quantum gravitational amplitudes

Abstract

We study the process, within classical general relativity, in which an incident gravitational plane wave, of weak amplitude and long wavelength, scatters off a massive spinning compact object, such as a black hole or neutron star. The amplitude of the asymptotic scattered wave, considered here at linear order in Newton's constant $G$ while at higher orders in the object's multipole expansion, is a valuable characterization of the response of the object to external gravitational fields. This amplitude coincides with a classical ($\hbar\to0$) limit of a quantum 4-point (object and graviton in, object and graviton out) gravitational Compton amplitude, at the tree (linear-in-$G$) level. Such tree-level Compton amplitudes are key building blocks in generalized-unitary-based approaches to the post-Minkowskian dynamics of binaries of spinning compact objects. In this paper, we compute the classical amplitude using an effective worldline theory to describe the compact object, determined by an action functional for translational and rotational degrees of freedom, including couplings of spin-induced higher multipole moments to spacetime curvature. We work here up to the levels of quadratic-in-spin quadrupole and cubic-in-spin octupole couplings, respectively involving Wilson coefficients $C_2$ and $C_3$. For the special case $C_2=C_3=1$ corresponding to a black hole, we find agreement through cubic-in-spin order between our classical amplitude and previous conjectures arising from considerations of quantum scattering amplitudes. We also present new results for general $C_2$ and $C_3$, anticipating instructive comparisons with results from effective quantum theories.