Abstract
We provide evidence that the classical scattering of two spinning black holes is controlled by the soft expansion of exchanged gravitons. We show how an exponentiation of Cachazo-Strominger soft factors, acting on massive higher-spin amplitudes, can be used to find spin contributions to the aligned-spin scattering angle, conjecturally extending previously known results to higher orders in spin at one-loop order. The extraction of the classical limit is accomplished via the on-shell leading-singularity method and using massive spinor-helicity variables. The three-point amplitude for arbitrary-spin massive particles minimally coupled to gravity is expressed in an exponential form, and in the infinite-spin limit it matches the effective stress-energy tensor of the linearized Kerr solution. A four-point gravitational Compton amplitude is obtained from an extrapolated soft theorem, equivalent to gluing two exponential three-point amplitudes, and becomes itself an exponential operator. The construction uses these amplitudes to: 1) recover the known tree-level scattering angle at all orders in spin, 2) recover the known one-loop linear-in-spin interaction, 3) match a previous conjectural expression for the one-loop scattering angle at quadratic order in spin, 4) propose new one-loop results through quartic order in spin. These connections link the computation of higher-multipole interactions to the study of deeper orders in the soft expansion.
Highlights
K (b) observed in [5, 6] that the subleading soft theorem follows from gauge invariance, and because of this, it adopts a universal form up to subleading order
We provide evidence that the classical scattering of two spinning black holes is controlled by the soft expansion of exchanged gravitons
We present a complementary picture to the one of [22] by employing the soft theorem in the conservative sector, focusing on rotating black holes and at the same time extending the soft factor in (1.1) to higher orders in the soft expansion. This is achieved in the following way: It was shown by one of the authors in [29] that the classical ( -independent) piece of the spin-s amplitude can be extracted from a covariant Holomorphic Classical Limit (HCL), which sets the external kinematics such that the momentum transfer k between the massive sources is null
Summary
We start our discussion of the multipole expansion by dissecting the case of graviton emission by two massive vector fields. We face our first challenge: as explained in [27, 28, 30], the spin-1 amplitude contains up to quadrupole interactions, i.e. quadratic in spin, whereas only the linear piece is apparent in eq (2.1) To rewrite this contribution in terms of multipoles, we can use a redefined spin tensor. It is introduced in appendix B via a two-particle expectation value/matrix element, which we call the generalized expectation value (GEV). In this paper we find this condition to be crucial for the matching to the rotating-blackhole computation of [32], as the classical spin tensor Sμν (1.4) satisfies the above SSC by definition The purpose of this SSC is to constrain the mass-dipole components S0i of the spin tensor of an object to vanish in its rest frame. In order to expose the exponential structure described in the introduction and construct such spin operators at any order, we are going to recast the multipole expansion in terms of spinor-helicity variables
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