Abstract
The massless (or ultrarelativistic) limit of a Schwarzschild black hole with fixed energy was determined long ago in the form of the Aichelburg-Sexl shockwave, but the status of the same limit for a Kerr black hole is less clear. In this paper, we explore the ultrarelativistic limit of Kerr in the class of Kerr-Schild impulsive pp-waves by exploiting a relation between the metric profile and the eikonal phase associated with scattering between a scalar and the source of the metric. This gives a map between candidate metrics and tree-level, 4-point scattering amplitudes. At large distances from the source, we find that all candidates for the massless limit of Kerr in this class do not have spin effects. This includes the metric corresponding to the massless limit of the amplitude for gravitational scattering between a scalar and a massive particle of infinite spin. One metric, discovered by Balasin and Nachbagauer, does have spin and finite size effects at short distances, leading to a remarkably compact scattering amplitude with many interesting properties. We also discuss the classical single copy of the ultrarelativistic limit of Kerr in electromagnetism.
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M.J. Duff, Quantum Tree Graphs and the Schwarzschild Solution, Phys. Rev. D 7 (1973) 2317 [INSPIRE].
D. Neill and I.Z. Rothstein, Classical Space-Times from the S Matrix, Nucl. Phys. B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
D.A. Sardelis, The tree graphs of quantum gravity and the Reissner-Nordstrom solution, IC/73/186 (1973) [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum corrections to the Schwarzschild and Kerr metrics, Phys. Rev. D 68 (2003) 084005 [hep-th/0211071] [INSPIRE].
S. Mougiakakos and P. Vanhove, Schwarzschild-Tangherlini metric from scattering amplitudes in various dimensions, Phys. Rev. D 103 (2021) 026001 [arXiv:2010.08882] [INSPIRE].
G.U. Jakobsen, Schwarzschild-Tangherlini Metric from Scattering Amplitudes, Phys. Rev. D 102 (2020) 104065 [arXiv:2006.01734] [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 A.K. Ridgway, Radiation and the classical double copy for color charges, Phys. Rev. D 95 (2017) 125010 [arXiv:1611.03493] [INSPIRE].
A. Koemans Collado, P. Di Vecchia, R. Russo and S. Thomas, The subleading eikonal in supergravity theories, JHEP 10 (2018) 038 [arXiv:1807.04588] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
A. Cristofoli, R. Gonzo, D.A. Kosower and D. O’Connell, Waveforms from amplitudes, Phys. Rev. D 106 (2022) 056007 [arXiv:2107.10193] [INSPIRE].
A. Cristofoli, Gravitational shock waves and scattering amplitudes, JHEP 11 (2020) 160 [arXiv:2006.08283] [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].
R. Monteiro, S. Nagy, D. O’Connell, D. Peinador Veiga and M. Sergola, NS-NS spacetimes from amplitudes, JHEP 06 (2022) 021 [arXiv:2112.08336] [INSPIRE].
J. Vines, unpublished notes (2021).
A. Guevara, Reconstructing Classical Spacetimes from the S-Matrix in Twistor Space, arXiv:2112.05111 [INSPIRE].
J.F. Plebanski and M. Demianski, Rotating, charged, and uniformly accelerating mass in general relativity, Annals Phys. 98 (1976) 98 [INSPIRE].
P.C. Aichelburg and R.U. Sexl, On the Gravitational field of a massless particle, Gen. Rel. Grav. 2 (1971) 303 [INSPIRE].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
G.W. Gibbons and M.S. Volkov, Zero mass limit of Kerr spacetime is a wormhole, Phys. Rev. D 96 (2017) 024053 [arXiv:1705.07787] [INSPIRE].
H. Balasin and H. Nachbagauer, Boosting the Kerr geometry into an arbitrary direction, Class. Quant. Grav. 13 (1996) 731 [gr-qc/9508044] [INSPIRE].
J. Podolsky and J.B. Griffiths, Boosted static multipole particles as sources of impulsive gravitational waves, Phys. Rev. D 58 (1998) 124024 [gr-qc/9809003] [INSPIRE].
C. Barrabes and P.A. Hogan, Light-like boost of the Kerr gravitational field, Phys. Rev. D 67 (2003) 084028 [gr-qc/0303055] [INSPIRE].
A. Guevara, A. Ochirov and J. Vines, Scattering of Spinning Black Holes from Exponentiated Soft Factors, JHEP 09 (2019) 056 [arXiv:1812.06895] [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].
V. Ferrari and P. Pendenza, Boosting the Kerr metric, Gen. Rel. Grav. 22 (1990) 1105 [INSPIRE].
H. Balasin and H. Nachbagauer, The Ultrarelativistic Kerr geometry and its energy momentum tensor, Class. Quant. Grav. 12 (1995) 707 [gr-qc/9405053] [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].
J. Ehlers and W. Kundt, Exact solutions of the gravitational field equations, in Gravitation, An Introduction to Current Research, L. Witten ed., Wiley, New York, NY, U.S.A. (1962), p. 49 [INSPIRE].
H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press, Cambridge, U.K. (2003) [https://doi.org/10.1017/CBO9780511535185] [INSPIRE].
J.B. Griffiths and J. Podolsky, Exact Space-Times in Einstein’s General Relativity, in Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K. (2009) [https://doi.org/10.1017/CBO9780511635397] [INSPIRE].
M. Blau, Plane waves and Penrose limits, Université de Neuchâtel (2011) and online pdf version at http://www.blau.itp.unibe.ch/Lecturenotes.html.
G.M. Shore, Memory, Penrose Limits and the Geometry of Gravitational Shockwaves and Gyratons, JHEP 12 (2018) 133 [arXiv:1811.08827] [INSPIRE].
A. Koemans Collado, P. Di Vecchia and R. Russo, Revisiting the second post-Minkowskian eikonal and the dynamics of binary black holes, Phys. Rev. D 100 (2019) 066028 [arXiv:1904.02667] [INSPIRE].
A. Cristofoli et al., The Uncertainty Principle and Classical Amplitudes, SAGEX-21-31-E (2021) [arXiv:2112.07556] [INSPIRE].
H.D.I. Abarbanel and C. Itzykson, Relativistic eikonal expansion, Phys. Rev. Lett. 23 (1969) 53 [INSPIRE].
M. Levy and J. Sucher, Eikonal approximation in quantum field theory, Phys. Rev. 186 (1969) 1656 [INSPIRE].
S.J. Wallace and J.A. McNeil, Relativistic Eikonal Expansion, Phys. Rev. D 16 (1977) 3565 [INSPIRE].
G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198 (1987) 61 [INSPIRE].
D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys. B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
T. Adamo, A. Cristofoli and P. Tourkine, Eikonal amplitudes from curved backgrounds, SciPost Phys. 13 (2022) 032 [arXiv:2112.09113] [INSPIRE].
T. Adamo, A. Cristofoli and A. Ilderton, Classical physics from amplitudes on curved backgrounds, JHEP 08 (2022) 281 [arXiv:2203.13785] [INSPIRE].
R. Penrose, The geometry of impulsive gravitational waves, in General relativity: Papers in honour of J.L. Synge, L. O’Raifeartaigh ed., Clarendon Oxford (1972), pp. 101–115.
V. Ferrari, P. Pendenza and G. Veneziano, Beamlike Gravitational Waves and Their Geodesics, Gen. Rel. Grav. 20 (1988) 1185 [INSPIRE].
H. Balasin, Geodesics for impulsive gravitational waves and the multiplication of distributions, Class. Quant. Grav. 14 (1997) 455 [gr-qc/9607076] [INSPIRE].
R. Steinbauer, Geodesics and geodesic deviation for impulsive gravitational waves, J. Math. Phys. 39 (1998) 2201 [gr-qc/9710119] [INSPIRE].
J. Podolsky and K. Vesely, Continuous coordinates for all impulsive pp-waves, Phys. Lett. A 241 (1998) 145 [gr-qc/9803016] [INSPIRE].
R. Steinbauer and J.A. Vickers, The Use of generalised functions and distributions in general relativity, Class. Quant. Grav. 23 (2006) R91 [gr-qc/0603078] [INSPIRE].
C. Samann and R. Steinbauer, On the completeness of impulsive gravitational wave spacetimes, Class. Quant. Grav. 29 (2012) 245011 [arXiv:1207.2633] [INSPIRE].
A. Lecke, R. Steinbauer and R. Svarc, The regularity of geodesics in impulsive pp-waves, Gen. Rel. Grav. 46 (2014) 1648 [arXiv:1310.1322] [INSPIRE].
M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, The simplest massive S-matrix: from minimal coupling to Black Holes, JHEP 04 (2019) 156 [arXiv:1812.08752] [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].
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].
V.P. Frolov and A. Koek, Gravitational lensing, memory, and the Penrose limit, Phys. Rev. D 106 (2022) 064026 [arXiv:2206.12731] [INSPIRE].
C. Barrabes and P.A. Hogan, Deflection of highly relativistic particles in a gravitational field, Class. Quant. Grav. 21 (2004) 405 [gr-qc/0311010] [INSPIRE].
R.H. Boyer and R.W. Lindquist, Maximal analytic extension of the Kerr metric, J. Math. Phys. 8 (1967) 265 [INSPIRE].
S. Pasterski and A. Puhm, Shifting spin on the celestial sphere, Phys. Rev. D 104 (2021) 086020 [arXiv:2012.15694] [INSPIRE].
R. Gonzo, T. McLoughlin and A. Puhm, Celestial holography on Kerr-Schild backgrounds, JHEP 10 (2022) 073 [arXiv:2207.13719] [INSPIRE].
W. Israel, Source of the kerr metric, Phys. Rev. D 2 (1970) 641 [INSPIRE].
W. Israel, Line sources in general relativity, Phys. Rev. D 15 (1977) 935 [INSPIRE].
R. Sachs and P.G. Bergmann, Structure of Particles in Linearized Gravitational Theory, Phys. Rev. 112 (1958) 674 [INSPIRE].
A.I. Janis and E.T. Newman, Structure of Gravitational Sources, J. Math. Phys. 6 (1965) 902 [INSPIRE].
R.P. Geroch, Multipole moments. Part II. Curved space, J. Math. Phys. 11 (1970) 2580 [INSPIRE].
Z. Perjes, Solutions of the coupled Einstein Maxwell equations representing the fields of spinning sources, Phys. Rev. Lett. 27 (1971) 1668 [INSPIRE].
R.O. Hansen, Multipole moments of stationary space-times, J. Math. Phys. 15 (1974) 46 [INSPIRE].
A. Krasinski, Ellipsoidal Spacetimes, Sources for the Kerr Metric, Annals Phys. 112 (1978) 22 [INSPIRE].
H. Balasin, Distributional energy momentum tensor of the extended Kerr geometry, Class. Quant. Grav. 14 (1997) 3353 [gr-qc/9702060] [INSPIRE].
A. Burinskii, E. Elizalde, S.R. Hildebrandt and G. Magli, Regular sources of the Kerr-Schild class for rotating and nonrotating black hole solutions, Phys. Rev. D 65 (2002) 064039 [gr-qc/0109085] [INSPIRE].
H. Balasin and H. Nachbagauer, Distributional energy momentum tensor of the Kerr-Newman space-time family, Class. Quant. Grav. 11 (1994) 1453 [gr-qc/9312028] [INSPIRE].
H. Yoshino, Lightlike limit of the boosted Kerr black holes in higher-dimensional spacetimes, Phys. Rev. D 71 (2005) 044032 [gr-qc/0412071] [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions, Int. J. Mod. Phys. A 3 (1988) 1615 [INSPIRE].
M. Ciafaloni, D. Colferai, F. Coradeschi and G. Veneziano, Unified limiting form of graviton radiation at extreme energies, Phys. Rev. D 93 (2016) 044052 [arXiv:1512.00281] [INSPIRE].
J.B. Griffiths and J. Podolsky, Null multipole particles as sources of pp-waves, Phys. Lett. A 236 (1997) 8 [INSPIRE].
R. Aoude, K. Haddad and A. Helset, Tidal effects for spinning particles, JHEP 03 (2021) 097 [arXiv:2012.05256] [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].
R. Aoude and A. Ochirov, Classical observables from coherent-spin amplitudes, JHEP 10 (2021) 008 [arXiv:2108.01649] [INSPIRE].
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive moments for scattering amplitudes, Phys. Rev. D 104 (2021) 036006 [arXiv:2011.00037] [INSPIRE].
A.J. Tolley, Z.-Y. Wang and S.-Y. Zhou, New positivity bounds from full crossing symmetry, JHEP 05 (2021) 255 [arXiv:2011.02400] [INSPIRE].
S. Caron-Huot and V. Van Duong, Extremal Effective Field Theories, JHEP 05 (2021) 280 [arXiv:2011.02957] [INSPIRE].
M. Herrero-Valea, R. Santos-Garcia and A. Tokareva, Massless positivity in graviton exchange, Phys. Rev. D 104 (2021) 085022 [arXiv:2011.11652] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, The EFT-Hedron, JHEP 05 (2021) 259 [arXiv:2012.15849] [INSPIRE].
S.D. Chowdhury, K. Ghosh, P. Haldar, P. Raman and A. Sinha, Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds, SciPost Phys. 13 (2022) 051 [arXiv:2112.11755] [INSPIRE].
Z. Bern, D. Kosmopoulos and A. Zhiboedov, Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude, J. Phys. A 54 (2021) 344002 [arXiv:2103.12728] [INSPIRE].
L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley, Reverse Bootstrapping: IR Lessons for UV Physics, Phys. Rev. Lett. 128 (2022) 051602 [arXiv:2111.09226] [INSPIRE].
L.-Y. Chiang, Y.-T. Huang, L. Rodina and H.-C. Weng, De-projecting the EFThedron, arXiv:2204.07140 [INSPIRE].
S.B. Giddings and M. Srednicki, High-energy gravitational scattering and black hole resonances, Phys. Rev. D 77 (2008) 085025 [arXiv:0711.5012] [INSPIRE].
S.B. Giddings and R.A. Porto, The Gravitational S-matrix, Phys. Rev. D 81 (2010) 025002 [arXiv:0908.0004] [INSPIRE].
S.B. Giddings, The gravitational S-matrix: Erice lectures, in Subnuclear Series 48, World Scientific, Singapore (2013), pp. 93–147 [https://doi.org/10.1142/9789814522489_0005] [arXiv:1105.2036] [INSPIRE].
N. Arkani-Hamed, M. Pate, A.-M. Raclariu and A. Strominger, Celestial amplitudes from UV to IR, JHEP 08 (2021) 062 [arXiv:2012.04208] [INSPIRE].
R. Saotome and R. Akhoury, Relationship Between Gravity and Gauge Scattering in the High Energy Limit, JHEP 01 (2013) 123 [arXiv:1210.8111] [INSPIRE].
S.G. Naculich, All-loop-orders relation between Regge limits of = 4 SYM and = 8 supergravity four-point amplitudes, JHEP 02 (2021) 044 [arXiv:2012.00030] [INSPIRE].
S.G. Naculich and T.W. Wecker, Proof of a three-loop relation between the Regge limits of four-point amplitudes in 𝒩 = 4 SYM and 𝒩 = 8 supergravity, JHEP 07 (2022) 043 [arXiv:2204.02417] [INSPIRE].
R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [INSPIRE].
E.T. Newman and A.I. Janis, Note on the Kerr spinning particle metric, J. Math. Phys. 6 (1965) 915 [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. Moynihan, Kerr-Newman from Minimal Coupling, JHEP 01 (2020) 014 [arXiv:1909.05217] [INSPIRE].
K. Haddad, Exponentiation of the leading eikonal phase with spin, Phys. Rev. D 105 (2022) 026004 [arXiv:2109.04427] [INSPIRE].
W.-M. Chen, M.-Z. Chung, Y.-T. Huang and J.-W. Kim, The 2PM Hamiltonian for binary Kerr to quartic in spin, JHEP 08 (2022) 148 [arXiv:2111.13639] [INSPIRE].
R. Aoude, K. Haddad and A. Helset, Searching for Kerr in the 2PM amplitude, JHEP 07 (2022) 072 [arXiv:2203.06197] [INSPIRE].
R. Aoude, K. Haddad and A. Helset, Classical Gravitational Spinning-Spinless Scattering at (G2S∞), Phys. Rev. Lett. 129 (2022) 141102 [arXiv:2205.02809] [INSPIRE].
G. Menezes and M. Sergola, NLO deflections for spinning particles and Kerr black holes, JHEP 10 (2022) 105 [arXiv:2205.11701] [INSPIRE].
P.H. Damgaard, J. Hoogeveen, A. Luna and J. Vines, Scattering angles in Kerr metrics, Phys. Rev. D 106 (2022) 124030 [arXiv:2208.11028] [INSPIRE].
W.B. Bonnor, Spinning null fluid in general relativity, Int. J. Theor. Phys. 3 (1970) 257 [INSPIRE].
V.P. Frolov, W. Israel and A. Zelnikov, Gravitational field of relativistic gyratons, Phys. Rev. D 72 (2005) 084031 [hep-th/0506001] [INSPIRE].
V.P. Frolov and D.V. Fursaev, Gravitational field of a spinning radiation beam-pulse in higher dimensions, Phys. Rev. D 71 (2005) 104034 [hep-th/0504027] [INSPIRE].
J. Podolsky, R. Steinbauer and R. Svarc, Gyratonic pp-waves and their impulsive limit, Phys. Rev. D 90 (2014) 044050 [arXiv:1406.3227] [INSPIRE].
A. Gruzinov and G. Veneziano, Gravitational Radiation from Massless Particle Collisions, Class. Quant. Grav. 33 (2016) 125012 [arXiv:1409.4555] [INSPIRE].
M. Soldate, Partial Wave Unitarity and Closed String Amplitudes, Phys. Lett. B 186 (1987) 321 [INSPIRE].
M. Correia, A. Sever and A. Zhiboedov, An analytical toolkit for the S-matrix bootstrap, JHEP 03 (2021) 013 [arXiv:2006.08221] [INSPIRE].
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Adamo, T., Cristofoli, A. & Tourkine, P. The ultrarelativistic limit of Kerr. J. High Energ. Phys. 2023, 107 (2023). https://doi.org/10.1007/JHEP02(2023)107
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DOI: https://doi.org/10.1007/JHEP02(2023)107