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.
R.O. Hansen, Multipole moments of stationary space-times, J. Math. Phys.15 (1974) 46 [INSPIRE].
A. Ross and B.R. Holstein, Spin effects in the effective quantum field theory of general relativity, J. Phys.A 40 (2007) 6973 [INSPIRE].
B.R. Holstein and A. Ross, Spin effects in long range gravitational scattering, arXiv:0802.0716 [INSPIRE].
V. Vaidya, Gravitational spin Hamiltonians from the S matrix, Phys. Rev.D 91 (2015) 024017 [arXiv:1410.5348] [INSPIRE].
A. Guevara, Holomorphic classical limit for spin effects in gravitational and electromagnetic scattering, JHEP04 (2019) 033 [arXiv:1706.02314] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, Scattering amplitudes for all masses and spins, arXiv:1709.04891 [INSPIRE].
E. Conde and A. Marzolla, Lorentz constraints on massive three-point amplitudes, JHEP09 (2016) 041 [arXiv:1601.08113] [INSPIRE].
E. Conde, E. Joung and K. Mkrtchyan, Spinor-helicity three-point amplitudes from local cubic interactions, JHEP08 (2016) 040 [arXiv:1605.07402] [INSPIRE].
A. Guevara, A. Ochirov and J. Vines, Scattering of spinning black holes from exponentiated soft factors, JHEP09 (2019) 056 [arXiv:1812.06895] [INSPIRE].
Y.F. Bautista and A. Guevara, From scattering amplitudes to classical physics: universality, double copy and soft theorems, arXiv:1903.12419 [INSPIRE].
J. Vines, Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin and effective-one-body mappings, Class. Quant. Grav.35 (2018) 084002 [arXiv:1709.06016] [INSPIRE].
M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, The simplest massive S-matrix: from minimal coupling to black holes, JHEP04 (2019) 156 [arXiv:1812.08752] [INSPIRE].
M. Levi and J. Steinhoff, Spinning gravitating objects in the effective field theory in the post-Newtonian scheme, JHEP09 (2015) 219 [arXiv:1501.04956] [INSPIRE].
LIGO SCIENTIFIC and VIRGO collaborations, Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett.116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
A. Buonanno and B.S. Sathyaprakash, Sources of gravitational waves: theory and observations, arXiv:1410.7832 [INSPIRE].
A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev.D 59 (1999) 084006 [gr-qc/9811091] [INSPIRE].
A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Phys. Rev.D 62 (2000) 064015 [gr-qc/0001013][INSPIRE].
T. Damour, Coalescence of two spinning black holes: an effective one-body approach, Phys. Rev.D 64 (2001) 124013 [gr-qc/0103018] [INSPIRE].
T. Damour, P. Jaranowski and G. Schaefer, Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling, Phys. Rev.D 78 (2008) 024009 [arXiv:0803.0915] [INSPIRE].
E. Barausse, E. Racine and A. Buonanno, Hamiltonian of a spinning test-particle in curved spacetime, Phys. Rev.D 80 (2009) 104025 [ Erratum ibid.D 85 (2012) 069904] [arXiv:0907.4745] [INSPIRE].
E. Barausse and A. Buonanno, An improved effective-one-body Hamiltonian for spinning black-hole binaries, Phys. Rev.D 81 (2010) 084024 [arXiv:0912.3517] [INSPIRE].
E. Barausse and A. Buonanno, Extending the effective-one-body Hamiltonian of black-hole binaries to include next-to-next-to-leading spin-orbit couplings, Phys. Rev.D 84 (2011) 104027 [arXiv:1107.2904] [INSPIRE].
T. Damour and A. Nagar, New effective-one-body description of coalescing nonprecessing spinning black-hole binaries, Phys. Rev.D 90 (2014) 044018 [arXiv:1406.6913] [INSPIRE].
D. Bini and T. Damour, Gravitational scattering of two black holes at the fourth post-Newtonian approximation, Phys. Rev.D 96 (2017) 064021 [arXiv:1706.06877] [INSPIRE].
D. Bini and T. Damour, Gravitational spin-orbit coupling in binary systems, post-Minkowskian approximation and e ffective one-body theory, Phys. Rev.D 96 (2017) 104038 [arXiv:1709.00590] [INSPIRE].
D. Bini and T. Damour, Gravitational spin-orbit coupling in binary systems at the second post-Minkowskian approximation, Phys. Rev.D 98 (2018) 044036 [arXiv:1805.10809] [INSPIRE].
J. Vines, J. Steinhoff and A. Buonanno, Spinning-black-hole scattering and the test-black-hole limit at second post-Minkowskian order, Phys. Rev.D 99 (2019) 064054 [arXiv:1812.00956] [INSPIRE].
W.D. Goldberger and I.Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev.D 73 (2006) 104029 [hep-th/0409156] [INSPIRE].
R.A. Porto, Post-Newtonian corrections to the motion of spinning bodies in NRGR, Phys. Rev.D 73 (2006) 104031 [gr-qc/0511061] [INSPIRE].
R.A. Porto and I.Z. Rothstein, The hyperfine Einstein-Infeld-Hoffmann potential, Phys. Rev. Lett.97 (2006) 021101 [gr-qc/0604099] [INSPIRE].
R.A. Porto and I.Z. Rothstein, Spin(1)Spin(2) effects in the motion of inspiralling compact binaries at third order in the post-Newtonian expansion, Phys. Rev.D 78 (2008) 044012 [Erratum ibid.D 81 (2010) 029904] [arXiv :0802. 0720] [INSPIRE].
M. Levi, Binary dynamics from spin1-spin2 coupling at fourth post-Newtonian order, Phys. Rev.D 85 (2012) 064043 [arXiv:1107.4322] [INSPIRE].
M. Levi and J. Steinhoff, Complete conservative dynamics for inspiralling compact binaries with spins at fourth post-Newtonian order, arXiv:1607.04252 [INSPIRE].
R.A. Porto, The effective field theorist's approach to gravitational dynamics, Phys. Rept.633 (2016) 1 [arXiv:1601.04914] [INSPIRE].
M. Levi, Effective field theories of post-Newtonian gravity: a comprehensive review, arXiv:1807.01699 [INSPIRE].
J. Vines and J. Steinhoff, Spin-multipole effects in binary black holes and the test-body limit, Phys. Rev.D 97 (2018) 064010 [arXiv:1606.08832] [INSPIRE].
N. Siemonsen, J. Steinhoff and J. Vines, Gravitational waves from spinning binary black holes at the leading post-Newtonian orders at all orders in spin, Phys. Rev.D 97 (2018) 124046 [arXiv:1712.08603] [INSPIRE].
Y. Iwasaki, Quantum theory of gravitation vs. classical theory - fourth-order potential, Frog. Theor. Phys.46 (1971) 1587 [INSPIRE].
M.J. Duff, Quantum tree graphs and the Schwarzschild solution, Phys. Rev.D 7 (1973) 2317 [INSPIRE].
J.F. Donoghue, Leading quantum correction to the N ewtonian potential, Phys. Rev. Lett.72 (1994) 2996 [gr-qc/9310024] [INSPIRE].
J.F. Donoghue, General relativity as an effective field theory: the leading qua nt um corrections, Phys. Rev.D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev.D 67 (2003) 084033 [Erratum ibid.D 71 (2005) 069903] [hep-th/0211072] [INSPIRE].
LB. Khriplovich and G.G. Kirilin, Quantum long range interactions in general relativity, J. Exp. Theor. Phys.98 (2004) 1063 [Zh. Eksp. Tear. Fiz.125 (2004) 1219] [gr-qc/0402018] [INSPIRE].
B.R. Holstein and J.F. Donoghue, Classical physics and quantum loops, Phys. Rev. Lett.93 (2004) 201602 [hep-th/0405239] [INSPIRE].
D. Neill and I.Z. Rothstein, Classical space-times from the S matrix, Nucl. Phys.B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and P. Vanhove, On-shell t echni ques and universal results in quantum gravity, JHEP02 (2014) 111 [arXiv:1309.0804] [INSPIRE].
N.E.J. Bjerrum-Bohr, B.R. Holstein, L. Planté and P. Vanhove, Graviton-photo n scattering, Phys. Rev.D 91 (2015) 064008 [arXiv:1410.4148] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, L. Planté and P. Vanhove, Bending of light in quantum gravity, Phys. Rev. Lett.114 (2015) 061301 [arXiv:1410.7590] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, L. Planté and P. Vanhove, Light-like scattering in quantum gravity, JHEP11 (2016) 117 [arXiv:1609.07477] [INSPIRE].
N.E.J. Bjerrum-Bohr, B.R. Holstein, J.F. Donoghue, L. Planté and P. Vanhove, Illuminating light bending, PoS(CORFU2016)077 (2017) [arXiv:1704.01624] [INSPIRE].
F. Cachazo and A. Guevara, Leading singularities and classical gravitational scattering, arXiv:1705.10262 [INSPIRE].
T. Damour, Gravitational scattering, post-Minkowskian approximation and effective one-body theory, Phys. Rev.D 94 (2016) 104015 [arXiv:1609.00354] [INSPIRE].
T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev.D 97 (2018) 044038 [arXiv:1710.10599] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, G. Festuccia, L. Planté and P. Vanhove, General relativity from scattering amplitudes, Phys. Rev. Lett.121 (2018) 171601 [arXiv:1806.04920] [INSPIRE].
C. Cheung, I.Z. Rothstein and M.P. Solon, From scattering amplitudes to classical potentials in thepost-Minkowskian expansion, Phys. Rev. Lett. 121 (2018) 251101 [arXiv:1808.02489] [INSPIRE].
Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, Scattering amplitudes and the conservative Hamiltonian for binary systems at third post-Minkowskian order, Phys. Rev. Lett.122 (2019) 201603 [arXiv:1901.04424] [INSPIRE].
A. Cristofoli, N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Post-Minkowskian Hamiltonians in general relativity, Phys. Rev.D 100 (2019) 084040 [arXiv:1906.01579] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev.D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett.105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
Z. Bern, T. Dennen, Y.-T. Huang and M. Kiermaier, Gravity as the square of gauge theory, Phys. Rev.D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].
H. Johansson and A. Ochirov, Pure gravities via color-kinematics duality for fundamental matter, JHEP11 (2015) 046 [arXiv:1407.4772] [INSPIRE].
H. Johansson and A. Ochirov, Color-kinematics duality for QCD amplitudes, JHEP01 (2016) 170 [arXiv:1507.00332] [INSPIRE].
J.J.M. Carrasco, Gauge and gravity amplitude relations, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics. Journeys Through the Precision Frontier: Amplitudes for Collid ers ( TASI 2014), Boulder, CO, U.S.A., 2-27 June 2014, World Scientific, Singapore (2015), pg. 477 [arXiv:1506.00974] [INSPIRE].
Z. Bern et al., Ultraviolet properties of N = 8 supergravity at five loops, Phys. Rev.D 98 (2018) 086021 [arXiv:1804.09311] [INSPIRE].
Z. Bern, D. Kosower and J. Parra-Martinez, Two-loop n-point anomalous amplitudes in N = 4 supergravity, arXiv:1905.05151 [INSPIRE].
R. Monteiro, D. O'Connell and C.D. White, Black holes and the double copy, JHEP12 (2014) 056 [arXiv:1410.0239] [INSPIRE].
A. Luna, R. Monteiro, D. O'Connell and C.D. White, The classical double copy forTaub-NUTspacetime, Phys. Lett.B 750 (2015) 272 [arXiv:1507.01869] [INSPIRE].
A. Luna, R. Monteiro, I. Nicholson, D. O'Connell and C.D. White, The double copy: Bremsstrahlung and accelerating black holes, JHEP06 (2016) 023 [arXiv:1603.05737] [INSPIRE].
W.D. Goldberger and A.K. Ridgway, Radiation and the classical double copy for color charges, Phys. Rev.D 95 (2017) 125010 [arXiv:1611.03493] [INSPIRE].
A. Luna et al., Perturbative spacetimes from Yang-Mills theory, JHEP04 (2017) 069 [arXiv:1611.07508] [INSPIRE].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Scattering on plane waves and the double copy, Class. Quant. Grav.35 (2018) 015004 [arXiv:1706.08925] [INSPIRE].
W.D. Goldberger, S.G. Prabhu and J.O. Thompson, Classical gluon and graviton radiation from the hi-adjoint scalar double copy, Phys. Rev.D 96 (2017) 065009 [arXiv:1705.09263] [INSPIRE].
N. Bahjat-Abbas, A. Luna and C.D. White, The Kerr-Schild double copy in curved spacetime, JHEP12 (2017) 004 [arXiv:1710.01953] [INSPIRE].
M. Carrillo-González, R. Penco and M. Trodden, The classical double copy in maximally symmetric spacetimes, JHEP04 (2018) 028 [arXiv:1711.01296] [INSPIRE].
W.D. Goldberger and A.K. Ridgway, Bound states and the classical double copy, Phys. Rev.D 97 (2018) 085019 [arXiv:1711.09493] [INSPIRE].
C.-H. Shen, Gravitational radiation from color-kinematics duality, J HEP11 (2018) 162 [arXiv:1806.07388] [INSPIRE].
J. Plefka, J. Steinhoff and W. Wormsbecher, Effective action of dilaton gravity as the classical double copy of Yang-Mills theory, Phys. Rev.D 99 (2019) 024021 [arXiv:1807.09859] [INSPIRE].
D.S. Berman, E. Chacón, A. Luna and C.D. White, The self-dual classical double copy and the Eguchi-Hanson instanton, JHEP01 (2019) 107 [arXiv:1809.04063] [INSPIRE].
M. Carrillo González, B. Melcher, K. Ratliff, S. Watson and C.D. White, The classical double copy in three spacetime dimensions, JHEP07 (2019) 167 [arXiv:1904.11001] [INSPIRE].
A. Luna, R. Monteiro, I. Nicholson and D. O'Connell, Type D spacetime s and the Weyl double copy, Class. Quant. Grav.36 (2019) 065003 [arXiv:1810.08183] [INSPIRE].
W.D. Goldberger, J. Li and S.G. Prabhu, Spi nning particles, axion radiation and the classical double copy, Phys. Rev.D 97 (2018) 105018 [arXiv:1712.09250] [INSPIRE].
J. Li and S.G. Prabhu, Gravitational radiation from the classical spinning double copy, Phys. Rev.D 97 (2018) 105019 [arXiv:1803.02405] [INSPIRE].
A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent and J. Vines, Energetics of two-body Hamiltonians in post-Minkowskian gravity, Phys. Rev.D 99 (2019) 104004 [arXiv:1901.07102] [INSPIRE].
D.A. Kosower, B. Maybee and D. O'Connell, Amplitudes, observables and classical scattering, JHEP02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
A. Luna, I. Nicholson, D. O'Connell and C.D. White, Inelastic black hole scattering from charged scalar amplitudes, JHEP03 (2018) 044 [arXiv:1711.03901] [INSPIRE].
A.D. Fokker, Relativiteitstheorie (in Dutch), P. Noordhoff, The Netherlands (1929).
W.M. Tulczyjew, Motion of multipole particles in general relativity theory, Acta Phys. Polan.18 (1959) 393.
M. Mathisson, Neue mechanik materieller systemes (in German), Acta Phys. Pola n.6 (1937) 163 [INSPIRE].
M. Mathisson, Republication of: New mechanics of material systems, Gen. Rel. Grav.42 (2010) 1011.
A. Papapetrou, Spinning test particles in general relativity. 1, Proc. Roy. Soc. Land.A 209 (1951) 248 [INSPIRE].
W.G. Dixon, A covariant multipole formalism for extended test bodies in general relativity, in Proceedings of the International School of Physics Enrico Fermi LXVII, J. Ehlers ed., North Holland, The Netherlands (1979), pg. 156.
W.G. Dixon, The new mechanics of Myron Mathisson and its subsequent development, Fund. Theor. Phys.179 (2015) 1 [INSPIRE].
M.H.L. Pryce, Commuting co-ordinates in the new field theory, Proc. Roy. Soc. Land.A 150 (1935) 166 [INSPIRE].
M.H.L. Pryce, The mass center in the restricted theory of relativity and its connection with the quantum theory of elementary particles, Proc. Roy. Soc. Land.A 195 (1948) 62 [INSPIRE].
T.D. Newton and E.P. Wigner, Localized states for elementary systems, Rev. Mod. Phys.21 (1949) 400 [INSPIRE].
J. Vines, D. Kunst, J. Steinhoff and T. Hinderer, Canonical Hamiltonian for an extended test body in curved spacetime: to quadratic order in spin, Phys. Rev.D 93 (2016) 103008 [arXiv:1601.07529] [INSPIRE].
S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity, Wiley, U.S.A. (1972).
S. Cotogno, C. Lorcé and P. Lowdon, Poincaré constraints on the gravitational form factors for massive states with arbitrary spin, Phys. Rev.D 100 (2019) 045003 [arXiv:1905.11969] [INSPIRE].
C. Lorcé and P. Lowdon, Universality of the Poincare gravitational form factor constraints, arXiv:1908.02567 [INSPIRE].
S. Ferrara, M. Porrati and V.L. Telegdi, g = 2 as the natural value of the tree-level gyromagnetic ratio of elementary particles, Phys. Rev.D 46 (1992) 3529 [INSPIRE].
C. Lorcé, New explicit expressions for Dirac bilinears, Phys. Rev.D 97 (2018) 016005 [arXiv:1705.08370] [INSPIRE].
M.-Z. Chung, Y.-T. Huang and J.-W. Kim, From quantized spins to rotating black holes, arXiv:1908.08463 [INSPIRE].
N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of qua ntized fields, Wiley, U.S.A. (1980).
S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press, Cambridge, U.K. (2005).
J.D. Jackson, Classical el ectrodynamics, Wiley, U.S.A. (1999).
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
ArXiv ePrint: 1906.09260
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Maybee, B., O’Connell, D. & Vines, J. Observables and amplitudes for spinning particles and black holes. J. High Energ. Phys. 2019, 156 (2019). https://doi.org/10.1007/JHEP12(2019)156
- Scattering Amplitudes
- Black Holes