Observables and amplitudes for spinning particles and black holes
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Springer
Received: September 2, 2019 Accepted: November 22, 2019 Published: December 23, 2019
Ben Maybee,a Donal O’Connella and Justin Vinesb a
Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tate Road, Edinburgh, EH9 3JZ, Scotland, U.K. b Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨ uhlenberg 1, Potsdam 14476, Germany
E-mail: [email protected], [email protected], [email protected] 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. Keywords: Scattering Amplitudes, Black Holes ArXiv ePrint: 1906.09260
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)156
JHEP12(2019)156
Observables and amplitudes for spinning particles and black holes
Contents 1 Introduction
1
2 Spin and scattering observables in classical gravity 2.1 Linear and angular momenta in asymptotic Minkowski space 2.2 Scattering of spinning black holes in linearized gravity
3 3 5 5 6 6 8 10
4 Classical limits of amplitudes with spin 4.1 Gauge theory amplitudes 4.2 Gravity amplitudes
12 12 15
5 Black hole scattering observables from amplitudes 5.1 Linear impulse 5.2 Angular impulse
16 17 18
6 Discussion
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A Conventions
21
B Explicit evaluation of the QFT spin vector
22
C Spin and scattering observables in electrodynamics
24
1
Introduction
Kerr black holes are very special spinning objects. Any stationary axisymmetric extended body has an infinite tower of mass-multipole moments Iℓ and current-multipole moments Jℓ , which generally depend intricately on its internal structure and composition. For a Kerr black hole, every multipole is determined by only the mass m and spin s, through the simple relation due to Hansen [1], ℓ is Iℓ + iJℓ = m . (1.1) m This distinctive behaviour is a reflection of the no-hair theorem, stating that black holes in general relativity (GR) are uniquely characterised by their mass and spin (and charge).
–1–
JHEP12(2019)156
3 Spin and scattering observables in quantum field theory 3.1 Single particle states 3.2 The Pauli-Lubanski spin pseudovector 3.3 The change in spin during scattering 3.4 Passing to the classical limit
–2–
JHEP12(2019)156
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. Amplitudes for long-range gravitational scattering of spin 1/2 and spin 1 particles were found in [2, 3] to
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