Abstract
We develop a general formalism for computing classical observables for relativistic scattering of spinning particles, directly from on-shell amplitudes. We then apply this formalism to minimally coupled Einstein-gravity amplitudes for the scattering of massive spin 1/2 and spin 1 particles with a massive scalar, constructed using the double copy. In doing so we reproduce recent results at first post-Minkowskian order for the scattering of spinning black holes, through quadrupolar order in the spin-multipole expansion.
Highlights
Recent work has suggested that an on-shell expression of the no-hair theorem is that black holes correspond to minimal coupling in classical limits of quantum scattering amplitudes for massive spin n particles and gravitons
Further similar work in [4], up to spin 2, suggested that the black-hole multipoles (1.1) up to order l = 2n are faithfully reproduced from tree-level amplitudes for minimally coupled spin n particles. Such amplitudes for arbitrary spin n were computed in [5], using the representation of minimal coupling for arbitrary spins presented in [6] using the massive spinor-helicity formalism — see [7, 8]. Those amplitudes were shown in [9, 10] to lead in the limit n → ∞ to the two-black-hole aligned-spin scattering angle found in [11] at first post-Minkowskian (1PM) order and to all orders in the spin-multipole expansion, while in [12] they were shown to yield the contributions to the interaction potential at the leading post-Newtonian (PN) orders at each order in spin
The dynamics of spinning black holes is of great interest for gravitational-wave astronomy [14], and spin leads to essential corrections which are required for precision analysis of signals from binary black hole mergers [15]
Summary
To describe the incoming and outgoing states for a weak scattering process in asymptotically flat spacetime, we can use special relativistic physics, working as in Minkowski. There, any isolated body has a constant linear momentum vector pμ and an antisymmetric tensor field Jμν(x) giving its total angular momentum about the point x, with the x-dependence determined by J μν(x′) = J μν(x) + 2p[μ(x′ − x)ν], or equivalently ∇λJ μν = 2p[μδν]λ. Center-of-mass (cm) position and intrinsic and orbital angular momenta are frame-dependent concepts, but a natural inertial frame is provided by the direction of the momentum pμ, giving the proper rest frame. Where z can be any point on the proper cm worldline, and where Sμν = Jμν(z) is the intrinsic spin tensor, satisfying. Given the condition (2.2), the complete information of the spin tensor Sμν is encoded in the momentum pμ and the spin pseudo-vector [97], sμ. The total angular momentum tensor Jμν(x) can be reconstructed from pμ, sμ, and a point z on the proper cm worldline, via (2.4) and (2.1)
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