Abstract
We extract the relativistic classical radial action from scattering amplitudes, to all orders in perturbation theory, in the probe limit. Our sources include point charges and monopoles, as well as the Schwarzschild and pure-NUT gravitational backgrounds. A characteristic relativistic effect, that scattering trajectories may wind around these sources any number of times, can be recovered when all-order amplitudes are available. We show that the amplitude for scattering a probe off a pure NUT is given by the solution of a transcendental equation involving continued fractions, and explain how to solve this equation to any desired loop order.
Article PDF
Similar content being viewed by others
References
L. Blanchet, T. Damour, G. Esposito-Farese and B.R. Iyer, Gravitational radiation from inspiralling compact binaries completed at the third post-Newtonian order, Phys. Rev. Lett. 93 (2004) 091101 [gr-qc/0406012] [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, Rept. Prog. Phys. 83 (2020) 075901 [arXiv:1807.01699] [INSPIRE].
D. Bini, T. Damour and A. Geralico, Sixth post-Newtonian nonlocal-in-time dynamics of binary systems, Phys. Rev. D 102 (2020) 084047 [arXiv:2007.11239] [INSPIRE].
D. Bini, T. Damour and A. Geralico, Sixth post-Newtonian local-in-time dynamics of binary systems, Phys. Rev. D 102 (2020) 024061 [arXiv:2004.05407] [INSPIRE].
D. Bini, T. Damour, A. Geralico, S. Laporta and P. Mastrolia, Gravitational scattering at the seventh order in G: nonlocal contribution at the sixth post-Newtonian accuracy, Phys. Rev. D 103 (2021) 044038 [arXiv:2012.12918] [INSPIRE].
D. Bini, T. Damour, A. Geralico, S. Laporta and P. Mastrolia, Gravitational dynamics at O(G6): perturbative gravitational scattering meets experimental mathematics, arXiv:2008.09389 [INSPIRE].
D. Bini, T. Damour and A. Geralico, Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders, Phys. Rev. D 102 (2020) 024062 [arXiv:2003.11891] [INSPIRE].
D. Bini, T. Damour and A. Geralico, Radiative contributions to gravitational scattering, Phys. Rev. D 104 (2021) 084031 [arXiv:2107.08896] [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].
K.G. Arun, A. Buonanno, G. Faye and E. Ochsner, Higher-order spin effects in the amplitude and phase of gravitational waveforms emitted by inspiraling compact binaries: Ready-to-use gravitational waveforms, Phys. Rev. D 79 (2009) 104023 [Erratum ibid. 84 (2011) 049901] [arXiv:0810.5336] [INSPIRE].
T. Damour and A. Nagar, An Improved analytical description of inspiralling and coalescing black-hole binaries, Phys. Rev. D 79 (2009) 081503 [arXiv:0902.0136] [INSPIRE].
T. Damour and A. Nagar, Effective One Body description of tidal effects in inspiralling compact binaries, Phys. Rev. D 81 (2010) 084016 [arXiv:0911.5041] [INSPIRE].
Y. Pan, A. Buonanno, R. Fujita, E. Racine and H. Tagoshi, Post-Newtonian factorized multipolar waveforms for spinning, non-precessing black-hole binaries, Phys. Rev. D 83 (2011) 064003 [Erratum ibid. 87 (2013) 109901] [arXiv:1006.0431] [INSPIRE].
A. Taracchini et al., Effective-one-body model for black-hole binaries with generic mass ratios and spins, Phys. Rev. D 89 (2014) 061502 [arXiv:1311.2544] [INSPIRE].
D. Bini and T. Damour, Gravitational self-force corrections to two-body tidal interactions and the effective one-body formalism, Phys. Rev. D 90 (2014) 124037 [arXiv:1409.6933] [INSPIRE].
A. Nagar et al., Factorization and resummation: A new paradigm to improve gravitational wave amplitudes. III: the spinning test-body terms, Phys. Rev. D 100 (2019) 104056 [arXiv:1907.12233] [INSPIRE].
A. Antonelli, M. van de Meent, A. Buonanno, J. Steinhoff and J. Vines, Quasicircular inspirals and plunges from nonspinning effective-one-body Hamiltonians with gravitational self-force information, Phys. Rev. D 101 (2020) 024024 [arXiv:1907.11597] [INSPIRE].
A. Nagar, G. Pratten, G. RiemenSchneider and R. Gamba, Multipolar effective one body model for nonspinning black hole binaries, Phys. Rev. D 101 (2020) 024041 [arXiv:1904.09550] [INSPIRE].
S. Albanesi, A. Nagar and S. Bernuzzi, Effective one-body model for extreme-mass-ratio spinning binaries on eccentric equatorial orbits: Testing radiation reaction and waveform, Phys. Rev. D 104 (2021) 024067 [arXiv:2104.10559] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and P. Vanhove, On-shell Techniques and Universal Results in Quantum Gravity, JHEP 02 (2014) 111 [arXiv:1309.0804] [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].
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].
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 the Post-Minkowskian Expansion, Phys. Rev. Lett. 121 (2018) 251101 [arXiv:1808.02489] [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].
P.H. Damgaard, K. Haddad and A. Helset, Heavy Black Hole Effective Theory, JHEP 11 (2019) 070 [arXiv:1908.10308] [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].
N.E.J. Bjerrum-Bohr, A. Cristofoli, P.H. Damgaard and H. Gomez, Scalar-Graviton Amplitudes, JHEP 11 (2019) 148 [arXiv:1908.09755] [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].
A. Brandhuber and G. Travaglini, On higher-derivative effects on the gravitational potential and particle bending, JHEP 01 (2020) 010 [arXiv:1905.05657] [INSPIRE].
A. Guevara, A. Ochirov and J. Vines, Black-hole scattering with general spin directions from minimal-coupling amplitudes, Phys. Rev. D 100 (2019) 104024 [arXiv:1906.10071] [INSPIRE].
B. Maybee, D. O’Connell and J. Vines, Observables and amplitudes for spinning particles and black holes, JHEP 12 (2019) 156 [arXiv:1906.09260] [INSPIRE].
N. Arkani-Hamed, Y.-t. Huang and D. O’Connell, Kerr black holes as elementary particles, JHEP 01 (2020) 046 [arXiv:1906.10100] [INSPIRE].
P. Di Vecchia, A. Luna, S.G. Naculich, R. Russo, G. Veneziano and C.D. White, A tale of two exponentiations in \( \mathcal{N} \) = 8 supergravity, Phys. Lett. B 798 (2019) 134927 [arXiv:1908.05603] [INSPIRE].
N.E.J. Bjerrum-Bohr, A. Cristofoli and P.H. Damgaard, Post-Minkowskian Scattering Angle in Einstein Gravity, JHEP 08 (2020) 038 [arXiv:1910.09366] [INSPIRE].
P. Di Vecchia, S.G. Naculich, R. Russo, G. Veneziano and C.D. White, A tale of two exponentiations in \( \mathcal{N} \) = 8 supergravity at subleading level, JHEP 03 (2020) 173 [arXiv:1911.11716] [INSPIRE].
M.-Z. Chung, Y.-T. Huang and J.-W. Kim, Classical potential for general spinning bodies, JHEP 09 (2020) 074 [arXiv:1908.08463] [INSPIRE].
T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102 (2020) 124008 [arXiv:2010.01641] [INSPIRE].
A. Cristofoli, P.H. Damgaard, P. Di Vecchia and C. Heissenberg, Second-order Post-Minkowskian scattering in arbitrary dimensions, JHEP 07 (2020) 122 [arXiv:2003.10274] [INSPIRE].
G. Kälin, Z. Liu and R.A. Porto, Conservative Dynamics of Binary Systems to Third Post-Minkowskian Order from the Effective Field Theory Approach, Phys. Rev. Lett. 125 (2020) 261103 [arXiv:2007.04977] [INSPIRE].
C. Cheung and M.P. Solon, Classical gravitational scattering at \( \mathcal{O} \)(G3) from Feynman diagrams, JHEP 06 (2020) 144 [arXiv:2003.08351] [INSPIRE].
G. Kälin and R.A. Porto, Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics, JHEP 11 (2020) 106 [arXiv:2006.01184] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Universality of ultra-relativistic gravitational scattering, Phys. Lett. B 811 (2020) 135924 [arXiv:2008.12743] [INSPIRE].
L. de la Cruz, B. Maybee, D. O’Connell and A. Ross, Classical Yang-Mills observables from amplitudes, JHEP 12 (2020) 076 [arXiv:2009.03842] [INSPIRE].
M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, From amplitudes to gravitational radiation with cubic interactions and tidal effects, Phys. Rev. D 103 (2021) 045015 [arXiv:2012.06548] [INSPIRE].
A. Guevara, B. Maybee, A. Ochirov, D. O’connell and J. Vines, A worldsheet for Kerr, JHEP 03 (2021) 201 [arXiv:2012.11570] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Radiation Reaction from Soft Theorems, Phys. Lett. B 818 (2021) 136379 [arXiv:2101.05772] [INSPIRE].
Y.F. Bautista, A. Guevara, C. Kavanagh and J. Vines, From Scattering in Black Hole Backgrounds to Higher-Spin Amplitudes: Part I, arXiv:2107.10179 [INSPIRE].
P.H. Damgaard and P. Vanhove, Remodeling the effective one-body formalism in post-Minkowskian gravity, Phys. Rev. D 104 (2021) 104029 [arXiv:2108.11248] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Planté and P. Vanhove, The amplitude for classical gravitational scattering at third Post-Minkowskian order, JHEP 08 (2021) 172 [arXiv:2105.05218] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Planté and P. Vanhove, Classical gravity from loop amplitudes, Phys. Rev. D 104 (2021) 026009 [arXiv:2104.04510] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The eikonal approach to gravitational scattering and radiation at \( \mathcal{O} \)(G3), JHEP 07 (2021) 169 [arXiv:2104.03256] [INSPIRE].
Z. Liu, R.A. Porto and Z. Yang, Spin Effects in the Effective Field Theory Approach to Post-Minkowskian Conservative Dynamics, JHEP 06 (2021) 012 [arXiv:2102.10059] [INSPIRE].
A. Brandhuber, G. Chen, G. Travaglini and C. Wen, A new gauge-invariant double copy for heavy-mass effective theory, JHEP 07 (2021) 047 [arXiv:2104.11206] [INSPIRE].
C. Dlapa, G. Kälin, Z. Liu and R.A. Porto, Dynamics of Binary Systems to Fourth Post-Minkowskian Order from the Effective Field Theory Approach, arXiv:2106.08276 [INSPIRE].
A. Cristofoli, R. Gonzo, D.A. Kosower and D. O’Connell, Waveforms from Amplitudes, arXiv:2107.10193 [INSPIRE].
A. Brandhuber, G. Chen, G. Travaglini and C. Wen, Classical gravitational scattering from a gauge-invariant double copy, JHEP 10 (2021) 118 [arXiv:2108.04216] [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].
W.D. Goldberger and I.Z. Rothstein, Dissipative effects in the worldline approach to black hole dynamics, Phys. Rev. D 73 (2006) 104030 [hep-th/0511133] [INSPIRE].
T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97 (2018) 044038 [arXiv:1710.10599] [INSPIRE].
T. Damour, Classical and quantum scattering in post-Minkowskian gravity, Phys. Rev. D 102 (2020) 024060 [arXiv:1912.02139] [INSPIRE].
A. Guevara, Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering, JHEP 04 (2019) 033 [arXiv:1706.02314] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
J.F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72 (1994) 2996 [gr-qc/9310024] [INSPIRE].
J.F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
D. Neill and I.Z. Rothstein, Classical Space-Times from the S Matrix, Nucl. Phys. B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
F. Cachazo and A. Guevara, Leading Singularities and Classical Gravitational Scattering, JHEP 02 (2020) 181 [arXiv:1705.10262] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [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].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, Black Hole Binary Dynamics from the Double Copy and Effective Theory, JHEP 10 (2019) 206 [arXiv:1908.01493] [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].
Z. Bern, A. Luna, R. Roiban, C.-H. Shen and M. Zeng, Spinning black hole binary dynamics, scattering amplitudes, and effective field theory, Phys. Rev. D 104 (2021) 065014 [arXiv:2005.03071] [INSPIRE].
Z. Bern, H. Ita, J. Parra-Martinez and M.S. Ruf, Universality in the classical limit of massless gravitational scattering, Phys. Rev. Lett. 125 (2020) 031601 [arXiv:2002.02459] [INSPIRE].
Z. Bern et al., Scattering Amplitudes and Conservative Binary Dynamics at \( \mathcal{O} \)(G4), Phys. Rev. Lett. 126 (2021) 171601 [arXiv:2101.07254] [INSPIRE].
W.B. Bonnor and M.A. Rotenberg, Gravitational waves from isolated sources, Proc. Roy. Soc. Lond. A 289 (1966) 247.
L. Blanchet and T. Damour, Radiative gravitational fields in general relativity. I — General structure of the field outside the source, Phil. Trans. Roy. Soc. Lond. A 320 (1986) 379.
L. Blanchet and T. Damour, Tail Transported Temporal Correlations in the Dynamics of a Gravitating System, Phys. Rev. D 37 (1988) 1410 [INSPIRE].
L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304 [INSPIRE].
L. Blanchet and G. Schaefer, Gravitational wave tails and binary star systems, Class. Quant. Grav. 10 (1993) 2699 [INSPIRE].
L. Blanchet, Gravitational wave tails of tails, Class. Quant. Grav. 15 (1998) 113 [Erratum ibid. 22 (2005) 3381] [gr-qc/9710038] [INSPIRE].
H. Asada and T. Futamase, Propagation of gravitational waves from slow motion sources in Coulomb type potential, Phys. Rev. D 56 (1997) R6062 [gr-qc/9711009] [INSPIRE].
C.R. Galley, A.K. Leibovich, R.A. Porto and A. Ross, Tail effect in gravitational radiation reaction: Time nonlocality and renormalization group evolution, Phys. Rev. D 93 (2016) 124010 [arXiv:1511.07379] [INSPIRE].
T. Marchand, L. Blanchet and G. Faye, Gravitational-wave tail effects to quartic non-linear order, Class. Quant. Grav. 33 (2016) 244003 [arXiv:1607.07601] [INSPIRE].
L. Landau and E. Lifshitz, Mechanics: Volume 1, Elsevier Science (1982).
G. Kälin and R.A. Porto, From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist), JHEP 02 (2020) 120 [arXiv:1911.09130] [INSPIRE].
G. Kälin and R.A. Porto, From Boundary Data to Bound States, JHEP 01 (2020) 072 [arXiv:1910.03008] [INSPIRE].
G. Kälin, Z. Liu and R.A. Porto, Conservative Tidal Effects in Compact Binary Systems to Next-to-Leading Post-Minkowskian Order, Phys. Rev. D 102 (2020) 124025 [arXiv:2008.06047] [INSPIRE].
W.G. Unruh, Absorption Cross-Section of Small Black Holes, Phys. Rev. D 14 (1976) 3251 [INSPIRE].
N.G. Sanchez, Absorption and Emission Spectra of a Schwarzschild Black Hole, Phys. Rev. D 18 (1978) 1030 [INSPIRE].
N.G. Sanchez, Elastic Scattering of Waves by a Black Hole, Phys. Rev. D 18 (1978) 1798 [INSPIRE].
S.A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185 (1973) 635 [INSPIRE].
S.A. Teukolsky and W.H. Press, Perturbations of a rotating black hole. III — Interaction of the hole with gravitational and electromagnet ic radiation, Astrophys. J. 193 (1974) 443 [INSPIRE].
K.W. Ford and J.A. Wheeler, Application of semiclassical scattering analysis, Annals Phys. 7 (1959) 287.
M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rept. Prog. Phys. 35 (1972) 315.
G. Mogull, J. Plefka and J. Steinhoff, Classical black hole scattering from a worldline quantum field theory, JHEP 02 (2021) 048 [arXiv:2010.02865] [INSPIRE].
G.U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, Classical Gravitational Bremsstrahlung from a Worldline Quantum Field Theory, Phys. Rev. Lett. 126 (2021) 201103 [arXiv:2101.12688] [INSPIRE].
C. Shi and J. Plefka, Classical double copy of worldline quantum field theory, Phys. Rev. D 105 (2022) 026007 [arXiv:2109.10345] [INSPIRE].
J.J. Thomson, XXXIV. On momentum in the electric field, Philos. Mag. 8 (1904) 331.
H.J. Lipkin, W.I. Weisberger and M. Peshkin, Magnetic charge quantization and angular momentum, Annals Phys. 53 (1969) 203 [INSPIRE].
D.G. Boulware, L.S. Brown, R.N. Cahn, S.D. Ellis and C.-k. Lee, Scattering on Magnetic Charge, Phys. Rev. D 14 (1976) 2708 [INSPIRE].
J.S. Schwinger, K.A. Milton, W.-y. Tsai, L.L. DeRaad Jr. and D.C. Clark, Nonrelativistic Dyon-Dyon Scattering, Annals Phys. 101 (1976) 451 [INSPIRE].
Y.M. Shnir, Magnetic Monopoles, Text and Monographs in Physics, Springer, Berlin/Heidelberg (2005) [DOI] [INSPIRE].
C. Csáki, S. Hong, Y. Shirman, O. Telem, J. Terning and M. Waterbury, Scattering amplitudes for monopoles: pairwise little group and pairwise helicity, JHEP 08 (2021) 029 [arXiv:2009.14213] [INSPIRE].
C. Csáki, S. Hong, Y. Shirman, O. Telem and J. Terning, Completing Multiparticle Representations of the Poincaré Group, Phys. Rev. Lett. 127 (2021) 041601 [arXiv:2010.13794] [INSPIRE].
A. Luna, R. Monteiro, D. O’Connell and C.D. White, The classical double copy for Taub-NUT spacetime, Phys. Lett. B 750 (2015) 272 [arXiv:1507.01869] [INSPIRE].
S. Caron-Huot and Z. Zahraee, Integrability of Black Hole Orbits in Maximal Supergravity, JHEP 07 (2019) 179 [arXiv:1810.04694] [INSPIRE].
Y.-T. Huang, U. Kol and D. O’Connell, Double copy of electric-magnetic duality, Phys. Rev. D 102 (2020) 046005 [arXiv:1911.06318] [INSPIRE].
R. Alawadhi, D.S. Berman, B. Spence and D. Peinador Veiga, S-duality and the double copy, JHEP 03 (2020) 059 [arXiv:1911.06797] [INSPIRE].
U. Kol and M. Porrati, Gravitational Wu-Yang Monopoles, Phys. Rev. D 101 (2020) 126009 [arXiv:2003.09054] [INSPIRE].
N. Moynihan and J. Murugan, On-Shell Electric-Magnetic Duality and the Dual Graviton, arXiv:2002.11085 [INSPIRE].
W.T. Emond, Y.-T. Huang, U. Kol, N. Moynihan and D. O’Connell, Amplitudes from Coulomb to Kerr-Taub-NUT, arXiv:2010.07861 [INSPIRE].
J.-W. Kim and M. Shim, Gravitational Dyonic Amplitude at One-Loop and its Inconsistency with the Classical Impulse, JHEP 02 (2021) 217 [arXiv:2010.14347] [INSPIRE].
R. Alawadhi, D.S. Berman, C.D. White and S. Wikeley, The single copy of the gravitational holonomy, JHEP 10 (2021) 229 [arXiv:2107.01114] [INSPIRE].
Y. Mino, M. Sasaki and T. Tanaka, Gravitational radiation reaction to a particle motion, Phys. Rev. D 55 (1997) 3457 [gr-qc/9606018] [INSPIRE].
T.C. Quinn and R.M. Wald, An Axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved space-time, Phys. Rev. D 56 (1997) 3381 [gr-qc/9610053] [INSPIRE].
L. Barack and A. Ori, Mode sum regularization approach for the selfforce in black hole space-time, Phys. Rev. D 61 (2000) 061502 [gr-qc/9912010] [INSPIRE].
L. Barack, Y. Mino, H. Nakano, A. Ori and M. Sasaki, Calculating the gravitational selfforce in Schwarzschild space-time, Phys. Rev. Lett. 88 (2002) 091101 [gr-qc/0111001] [INSPIRE].
S.L. Detweiler and B.F. Whiting, Selfforce via a Green’s function decomposition, Phys. Rev. D 67 (2003) 024025 [gr-qc/0202086] [INSPIRE].
L. Barack and N. Sago, Gravitational self force on a particle in circular orbit around a Schwarzschild black hole, Phys. Rev. D 75 (2007) 064021 [gr-qc/0701069] [INSPIRE].
S.E. Gralla and R.M. Wald, A Rigorous Derivation of Gravitational Self-force, Class. Quant. Grav. 25 (2008) 205009 [Erratum ibid. 28 (2011) 159501] [arXiv:0806.3293] [INSPIRE].
L. Barack, Gravitational self force in extreme mass-ratio inspirals, Class. Quant. Grav. 26 (2009) 213001 [arXiv:0908.1664] [INSPIRE].
T. Damour, Gravitational Self Force in a Schwarzschild Background and the Effective One Body Formalism, Phys. Rev. D 81 (2010) 024017 [arXiv:0910.5533] [INSPIRE].
L. Barack and N. Sago, Gravitational self-force on a particle in eccentric orbit around a Schwarzschild black hole, Phys. Rev. D 81 (2010) 084021 [arXiv:1002.2386] [INSPIRE].
L. Blanchet, S.L. Detweiler, A. Le Tiec and B.F. Whiting, High-Order Post-Newtonian Fit of the Gravitational Self-Force for Circular Orbits in the Schwarzschild Geometry, Phys. Rev. D 81 (2010) 084033 [arXiv:1002.0726] [INSPIRE].
L. Barack, T. Damour and N. Sago, Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism, Phys. Rev. D 82 (2010) 084036 [arXiv:1008.0935] [INSPIRE].
L. Barack et al., Black holes, gravitational waves and fundamental physics: a roadmap, Class. Quant. Grav. 36 (2019) 143001 [arXiv:1806.05195] [INSPIRE].
L. Barack and A. Pound, Self-force and radiation reaction in general relativity, Rept. Prog. Phys. 82 (2019) 016904 [arXiv:1805.10385] [INSPIRE].
R.E. Langer, On the Connection Formulas and the Solutions of the Wave Equation, Phys. Rev. 51 (1937) 669 [INSPIRE].
N. Gaddam and N. Groenenboom, Soft graviton exchange and the information paradox, arXiv:2012.02355 [INSPIRE].
N. Gaddam, N. Groenenboom and G. ’t Hooft, Quantum gravity on the black hole horizon, JHEP 01 (2022) 023 [arXiv:2012.02357] [INSPIRE].
B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].
NIST Digital Library of Mathematical Functions, Release 1.1.1 of 2021-03-15, http://dlmf.nist.gov/.
C.G. Darwin, On some orbits of an electron, Philos. Mag. 25 (1913) 201.
T.H. Boyer, Unfamiliar trajectories for a relativistic particle in a Kepler or Coulomb potential, Am. J. Phys. 72 (2004) 992.
R. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press (1973).
A. Ghatak, R. Gallawa and I. Goyal, Modified airy function and wkb solutions to the wave equation, Monograph (NIST MN) 176 (1991).
T. Regge and J.A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063 [INSPIRE].
B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968) 1559 [INSPIRE].
G. Scharf, Schwarzschild geodesics in terms of elliptic functions and the related red shift, J. Mod. Phys. 2 (2011) 274 [arXiv:1101.1207] [INSPIRE].
S. Mano, H. Suzuki and E. Takasugi, Analytic solutions of the Regge-Wheeler equation and the postMinkowskian expansion, Prog. Theor. Phys. 96 (1996) 549 [gr-qc/9605057] [INSPIRE].
S. Mano, H. Suzuki and E. Takasugi, Analytic solutions of the Teukolsky equation and their low frequency expansions, Prog. Theor. Phys. 95 (1996) 1079 [gr-qc/9603020] [INSPIRE].
E.W. Leaver, Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics, J. Math. Phys. 27 (1986) 1238.
E. Berti and V. Cardoso, Quasinormal ringing of Kerr black holes. I. The Excitation factors, Phys. Rev. D 74 (2006) 104020 [gr-qc/0605118] [INSPIRE].
S.R. Dolan, Scattering and Absorption of Gravitational Plane Waves by Rotating Black Holes, Class. Quant. Grav. 25 (2008) 235002 [arXiv:0801.3805] [INSPIRE].
D. Bini and T. Damour, Analytical determination of the two-body gravitational interaction potential at the fourth post-Newtonian approximation, Phys. Rev. D 87 (2013) 121501 [arXiv:1305.4884] [INSPIRE].
T. Damour, P. Jaranowski and G. Schäfer, Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity, Phys. Rev. D 93 (2016) 084014 [arXiv:1601.01283] [INSPIRE].
M. Sasaki and H. Tagoshi, Analytic black hole perturbation approach to gravitational radiation, Living Rev. Rel. 6 (2003) 6 [gr-qc/0306120] [INSPIRE].
Black Hole Perturbation Toolkit, http://bhptoolkit.org/.
H.E. Haber, Spin formalism and applications to new physics searches, in 21st Annual SLAC Summer Institute on Particle Physics: Spin Structure in High-energy Processes (SSI 93), (School: 26 July–3 August, Topical Conference: 4–6 August), pp. 231–272 (1994) [hep-ph/9405376] [INSPIRE].
P. Baratella, C. Fernandez, B. von Harling and A. Pomarol, Anomalous Dimensions of Effective Theories from Partial Waves, JHEP 03 (2021) 287 [arXiv:2010.13809] [INSPIRE].
H.M. Pilkuhn, Relativistic Particle Physics, Springer Berlin Heidelberg (1979) [DOI].
N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, Scattering amplitudes for all masses and spins, JHEP 11 (2021) 070 [arXiv:1709.04891] [INSPIRE].
M. Jiang, J. Shu, M.-L. Xiao and Y.-H. Zheng, Partial Wave Amplitude Basis and Selection Rules in Effective Field Theories, Phys. Rev. Lett. 126 (2021) 011601 [arXiv:2001.04481] [INSPIRE].
M. Jacob and G.C. Wick, On the General Theory of Collisions for Particles with Spin, Annals Phys. 7 (1959) 404 [INSPIRE].
T.T. Wu and C.N. Yang, Dirac Monopole Without Strings: Monopole Harmonics, Nucl. Phys. B 107 (1976) 365 [INSPIRE].
P.P. Benderet, Zur Theorie singulärer Magnetpole, Helv. Phys. Acta 19 (1946) 503.
W.B. Bonnor, A new interpretation of the nut metric in general relativity, Math. Proc. Cambridge Phil. Soc. 66 (1969) 145.
A.H. Taub, Empty space-times admitting a three parameter group of motions, Annals Math. 53 (1951) 472 [INSPIRE].
E. Newman, L. Tamburino and T. Unti, Empty space generalization of the Schwarzschild metric, J. Math. Phys. 4 (1963) 915 [INSPIRE].
C.W. Misner, The Flatter regions of Newman, Unti and Tamburino’s generalized Schwarzschild space, J. Math. Phys. 4 (1963) 924 [INSPIRE].
J.S. Dowker, The nut solution as a gravitational dyon, Gen. Rel. Grav. 5 (1974) 603.
R.L. Zimmerman and B.Y. Shahir, Geodesics for the Nut Metric and Gravitational Monopoles, Gen. Rel. Grav. 21 (1989) 821 [INSPIRE].
V. Kagramanova, J. Kunz, E. Hackmann and C. Lammerzahl, Analytic treatment of complete and incomplete geodesics in Taub-NUT space-times, Phys. Rev. D 81 (2010) 124044 [arXiv:1002.4342] [INSPIRE].
G. Clément, D. Gal’tsov and M. Guenouche, Rehabilitating space-times with NUTs, Phys. Lett. B 750 (2015) 591 [arXiv:1508.07622] [INSPIRE].
V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Rel. 20 (2017) 6 [arXiv:1705.05482] [INSPIRE].
D. Bini, C. Cherubini, R.T. Jantzen and B. Mashhoon, Massless field perturbations and gravitomagnetism in the Kerr-Taub-NUT space-time, Phys. Rev. D 67 (2003) 084013 [gr-qc/0301080] [INSPIRE].
J. Meixner, F. Schäfke and G. Wolf, Mathieu Functions and Spheroidal Functions and their Mathematical Foundations: Further Studies, Lecture Notes in Mathematics, Springer Berlin Heidelberg (2006) [DOI].
F. Arscott, Periodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions, International Series of Monographs in Pure and Applied Mathematics, Oxford, printed in Poland (1964).
P.E. Falloon, P.C. Abbott and J.B. Wang, Theory and computation of spheroidal wavefunctions, J. Phys. A 36 (2003) 5477.
A.S. Eddington, A Comparison of Whitehead’s and Einstein’s Formulae, Nature 113 (1924) 192.
D. Finkelstein, Past-Future Asymmetry of the Gravitational Field of a Point Particle, Phys. Rev. 110 (1958) 965 [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, Dynamical Structure and Definition of Energy in General Relativity, Phys. Rev. 116 (1959) 1322 [INSPIRE].
G. Schäfer and P. Jaranowski, Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries, Living Rev. Rel. 21 (2018) 7 [arXiv:1805.07240] [INSPIRE].
L. Blanchet and G. Faye, Lorentzian regularization and the problem of point-like particles in general relativity, J. Math. Phys. 42 (2001) 4391 [gr-qc/0006100] [INSPIRE].
T. Damour, P. Jaranowski and G. Schaefer, Dimensional regularization of the gravitational interaction of point masses, Phys. Lett. B 513 (2001) 147 [gr-qc/0105038] [INSPIRE].
L. Blanchet, Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Rel. 17 (2014) 2 [arXiv:1310.1528] [INSPIRE].
R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [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].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Plane wave backgrounds and colour-kinematics duality, JHEP 02 (2019) 198 [arXiv:1810.05115] [INSPIRE].
R. Monteiro, D. O’Connell, D. Peinador Veiga and M. Sergola, Classical solutions and their double copy in split signature, JHEP 05 (2021) 268 [arXiv:2012.11190] [INSPIRE].
M. Campiglia and S. Nagy, A double copy for asymptotic symmetries in the self-dual sector, JHEP 03 (2021) 262 [arXiv:2102.01680] [INSPIRE].
L. Borsten, H. Kim, B. Jurčo, T. Macrelli, C. Sämann and M. Wolf, Double Copy from Homotopy Algebras, Fortsch. Phys. 69 (2021) 2100075 [arXiv:2102.11390] [INSPIRE].
E. Chacón, S. Nagy and C.D. White, The Weyl double copy from twistor space, JHEP 05 (2021) 2239 [arXiv:2103.16441] [INSPIRE].
R. Gonzo and C. Shi, Geodesics from classical double copy, Phys. Rev. D 104 (2021) 105012 [arXiv:2109.01072] [INSPIRE].
H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga and C.N. Pope, Asymptotic Weyl double copy, JHEP 11 (2021) 126 [arXiv:2109.07866] [INSPIRE].
T. Adamo and U. Kol, Classical double copy at null infinity, arXiv:2109.07832 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2109.12092
Rights and permissions
Open Access . 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.
About this article
Cite this article
Kol, U., O’Connell, D. & Telem, O. The radial action from probe amplitudes to all orders. J. High Energ. Phys. 2022, 141 (2022). https://doi.org/10.1007/JHEP03(2022)141
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2022)141