Abstract
The double copy is a much-studied relationship between scattering amplitudes in gauge and gravity theories, that has subsequently been extended to classical field solutions. In nearly all previous examples, the graviton field is defined around Minkowski space. Recently, it has been suggested that one may set up a double copy for gravitons defined around a non-trivial background. We investigate this idea from the point of view of the classical double copy. First, we use Kerr-Schild spacetimes to construct graviton solutions in curved space, as double copies of gauge fields on non-zero gauge backgrounds. Next, we find that we can reinterpret such cases in terms of a graviton on a non-Minkowski background, whose single copy is a gauge field in the same background spacetime. The latter type of double copy persists even when the background is not of Kerr-Schild form, and we provide examples involving conformally flat metrics. Our results will be useful in extending the remit of the double copy, including to possible cosmological applications.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
S. Stieberger, Open and Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [INSPIRE].
B. Feng, R. Huang and Y. Jia, Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].
S.H. Henry Tye and Y. Zhang, Dual Identities inside the Gluon and the Graviton Scattering Amplitudes, JHEP 06 (2010) 071 [Erratum ibid. 04 (2011) 114] [arXiv:1003.1732] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, R. Monteiro and D. O’Connell, Algebras for Amplitudes, JHEP 06 (2012) 061 [arXiv:1203.0944] [INSPIRE].
H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [INSPIRE].
M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].
Z. Bern, J.S. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 super Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].
J.J. Carrasco and H. Johansson, Five-Point Amplitudes in N = 4 super-Yang-Mills Theory and N = 8 Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled N ≤ 4 supergravity, JHEP 03 (2013) 056 [arXiv:1212.1146] [INSPIRE].
C.R. Mafra and O. Schlotterer, The Structure of n-Point One-Loop Open Superstring Amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].
R.H. Boels, R.S. Isermann, R. Monteiro and D. O’Connell, Colour-Kinematics Duality for One-Loop Rational Amplitudes, JHEP 04 (2013) 107 [arXiv:1301.4165] [INSPIRE].
N.E.J. Bjerrum-Bohr, T. Dennen, R. Monteiro and D. O’Connell, Integrand Oxidation and One-Loop Colour-Dual Numerators in N = 4 Gauge Theory, JHEP 07 (2013) 092 [arXiv:1303.2913] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, Y.-t. Huang and J. Nohle, Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops, Phys. Rev. D 92 (2015) 045041 [arXiv:1303.6605] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, The Ultraviolet Structure of Half-Maximal Supergravity with Matter Multiplets at Two and Three Loops, Phys. Rev. D 88 (2013) 065007 [arXiv:1305.4876] [INSPIRE].
J. Nohle, Color-Kinematics Duality in One-Loop Four-Gluon Amplitudes with Matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].
S.G. Naculich, H. Nastase and H.J. Schnitzer, All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes, JHEP 04 (2013) 114 [arXiv:1301.2234] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, Dual-color decompositions at one-loop level in Yang-Mills theory, JHEP 06 (2014) 157 [arXiv:1402.6805] [INSPIRE].
C.R. Mafra and O. Schlotterer, Towards one-loop SYM amplitudes from the pure spinor BRST cohomology, Fortsch. Phys. 63 (2015) 105 [arXiv:1410.0668] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in \( \mathcal{N} \) = 5 supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].
C.R. Mafra and O. Schlotterer, Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, JHEP 10 (2015) 124 [arXiv:1505.02746] [INSPIRE].
S. He, R. Monteiro and O. Schlotterer, String-inspired BCJ numerators for one-loop MHV amplitudes, JHEP 01 (2016) 171 [arXiv:1507.06288] [INSPIRE].
Z. Bern, S. Davies and J. Nohle, Double-Copy Constructions and Unitarity Cuts, Phys. Rev. D 93 (2016) 105015 [arXiv:1510.03448] [INSPIRE].
G. Mogull and D. O’Connell, Overcoming Obstacles to Colour-Kinematics Duality at Two Loops, JHEP 12 (2015) 135 [arXiv:1511.06652] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies, JHEP 06 (2017) 064 [arXiv:1511.01740] [INSPIRE].
Z. Bern, J.J.M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, The Five-Loop Four-Point Integrand of N = 8 Supergravity as a Generalized Double Copy, arXiv:1708.06807 [INSPIRE].
S. Oxburgh and C.D. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].
C.D. White, Factorization Properties of Soft Graviton Amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [INSPIRE].
S. Melville, S.G. Naculich, H.J. Schnitzer and C.D. White, Wilson line approach to gravity in the high energy limit, Phys. Rev. D 89 (2014) 025009 [arXiv:1306.6019] [INSPIRE].
A. Luna, S. Melville, S.G. Naculich and C.D. White, Next-to-soft corrections to high energy scattering in QCD and gravity, JHEP 01 (2017) 052 [arXiv:1611.02172] [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].
A. Sabio Vera, E. Serna Campillo and M.A. Vazquez-Mozo, Color-Kinematics Duality and the Regge Limit of Inelastic Amplitudes, JHEP 04 (2013) 086 [arXiv:1212.5103] [INSPIRE].
H. Johansson, A. Sabio Vera, E. Serna Campillo and M. Á. Vázquez-Mozo, Color-Kinematics Duality in Multi-Regge Kinematics and Dimensional Reduction, JHEP 10 (2013) 215 [arXiv:1307.3106] [INSPIRE].
H. Johansson, A. Sabio Vera, E. Serna Campillo and M.A. Vazquez-Mozo, Color-kinematics duality and dimensional reduction for graviton emission in Regge limit, arXiv:1310.1680 [INSPIRE].
R. Monteiro and D. O’Connell, The Kinematic Algebras from the Scattering Equations, JHEP 03 (2014) 110 [arXiv:1311.1151] [INSPIRE].
M. Tolotti and S. Weinzierl, Construction of an effective Yang-Mills Lagrangian with manifest BCJ duality, JHEP 07 (2013) 111 [arXiv:1306.2975] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, Note on Construction of Dual-trace Factor in Yang-Mills Theory, JHEP 10 (2013) 069 [arXiv:1305.2996] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, The Construction of Dual-trace Factor in Yang-Mills Theory, JHEP 07 (2013) 057 [arXiv:1304.2978] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, An algebraic approach to BCJ numerators, JHEP 03 (2013) 050 [arXiv:1212.6168] [INSPIRE].
S.G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].
S.G. Naculich, Scattering equations and virtuous kinematic numerators and dual-trace functions, JHEP 07 (2014) 143 [arXiv:1404.7141] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Scattering amplitudes in \( \mathcal{N} \) = 2 Maxwell-Einstein and Yang-Mills/Einstein supergravity, JHEP 01 (2015) 081 [arXiv:1408.0764] [INSPIRE].
J.J.M. Carrasco, R. Kallosh, R. Roiban and A.A. Tseytlin, On the U(1) duality anomaly and the S-matrix of N = 4 supergravity, JHEP 07 (2013) 029 [arXiv:1303.6219] [INSPIRE].
S. Litsey and J. Stankowicz, Kinematic numerators and a double-copy formula for N = 4 super-Yang-Mills residues, Phys. Rev. D 90 (2014) 025013 [arXiv:1309.7681] [INSPIRE].
S. Nagy, Chiral Squaring, JHEP 07 (2016) 142 [arXiv:1412.4750] [INSPIRE].
A. Anastasiou, L. Borsten, M.J. Hughes and S. Nagy, Global symmetries of Yang-Mills squared in various dimensions, JHEP 01 (2016) 148 [arXiv:1502.05359] [INSPIRE].
H. Johansson and A. Ochirov, Color-Kinematics Duality for QCD Amplitudes, JHEP 01 (2016) 170 [arXiv:1507.00332] [INSPIRE].
S. Lee, C.R. Mafra and O. Schlotterer, Non-linear gauge transformations in D = 10 SYM theory and the BCJ duality, JHEP 03 (2016) 090 [arXiv:1510.08843] [INSPIRE].
S.M. Barnett, Maxwellian theory of gravitational waves and their mechanical properties, New J. Phys. 16 (2014) 023027 [INSPIRE].
M. Chiodaroli, Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities, arXiv:1607.04129 [INSPIRE].
G.L. Cardoso, S. Nagy and S. Nampuri, A double copy for \( \mathcal{N} \) = 2 supergravity: a linearised tale told on-shell, JHEP 10 (2016) 127 [arXiv:1609.05022] [INSPIRE].
S. Mizera, Inverse of the String Theory KLT Kernel, JHEP 06 (2017) 084 [arXiv:1610.04230] [INSPIRE].
Z. Bern, J.J. Carrasco, W.-M. Chen, H. Johansson and R. Roiban, Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes, Phys. Rev. Lett. 118 (2017) 181602 [arXiv:1701.02519] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy, JHEP 07 (2017) 002 [arXiv:1703.00421] [INSPIRE].
H. Johansson, G. Kälin and G. Mogull, Two-loop supersymmetric QCD and half-maximal supergravity amplitudes, JHEP 09 (2017) 019 [arXiv:1706.09381] [INSPIRE].
H. Johansson and J. Nohle, Conformal Gravity from Gauge Theory, arXiv:1707.02965 [INSPIRE].
C. Cheung and G.N. Remmen, Twofold Symmetries of the Pure Gravity Action, JHEP 01 (2017) 104 [arXiv:1612.03927] [INSPIRE].
C. Cheung and G.N. Remmen, Hidden Simplicity of the Gravity Action, JHEP 09 (2017) 002 [arXiv:1705.00626] [INSPIRE].
C. Cheung, C.-H. Shen and C. Wen, Unifying Relations for Scattering Amplitudes, arXiv:1705.03025 [INSPIRE].
R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [INSPIRE].
A. Luna, R. Monteiro, I. Nicholson, D. O’Connell and C.D. White, The double copy: Bremsstrahlung and accelerating black holes, JHEP 06 (2016) 023 [arXiv:1603.05737] [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].
A.K. Ridgway and M.B. Wise, Static Spherically Symmetric Kerr-Schild Metrics and Implications for the Classical Double Copy, Phys. Rev. D 94 (2016) 044023 [arXiv:1512.02243] [INSPIRE].
C.D. White, Exact solutions for the biadjoint scalar field, Phys. Lett. B 763 (2016) 365 [arXiv:1606.04724] [INSPIRE].
P.-J. De Smet and C.D. White, Extended solutions for the biadjoint scalar field, Phys. Lett. B 775 (2017) 163 [arXiv:1708.01103] [INSPIRE].
A. Anastasiou, L. Borsten, M.J. Duff, L.J. Hughes and S. Nagy, Yang-Mills origin of gravitational symmetries, Phys. Rev. Lett. 113 (2014) 231606 [arXiv:1408.4434] [INSPIRE].
L. Borsten and M.J. Duff, Gravity as the square of Yang-Mills?, Phys. Scripta 90 (2015) 108012 [arXiv:1602.08267] [INSPIRE].
A. Anastasiou et al., Twin supergravities from Yang-Mills theory squared, Phys. Rev. D 96 (2017) 026013 [arXiv:1610.07192] [INSPIRE].
A. Anastasiou, L. Borsten, M.J. Duff, A. Marrani, S. Nagy and M. Zoccali, Are all supergravity theories Yang-Mills squared?, arXiv:1707.03234 [INSPIRE].
L. Borsten, On D = 6, \( \mathcal{N} \) = (2, 0) and \( \mathcal{N} \) = (4, 0) theories, arXiv:1708.02573 [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].
W.D. Goldberger, S.G. Prabhu and J.O. Thompson, Classical gluon and graviton radiation from the bi-adjoint scalar double copy, Phys. Rev. D 96 (2017) 065009 [arXiv:1705.09263] [INSPIRE].
A. Luna et al., Perturbative spacetimes from Yang-Mills theory, JHEP 04 (2017) 069 [arXiv:1611.07508] [INSPIRE].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Scattering on plane waves and the double copy, arXiv:1706.08925 [INSPIRE].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Amplitudes on plane waves from ambitwistor strings, JHEP 11 (2017) 160 [arXiv:1708.09249] [INSPIRE].
L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].
T. Adamo, E. Casali and D. Skinner, A Worldsheet Theory for Supergravity, JHEP 02 (2015) 116 [arXiv:1409.5656] [INSPIRE].
J.J.M. Carrasco, Gauge and Gravity Amplitude Relations, arXiv:1506.00974 [INSPIRE].
C. Cheung, TASI Lectures on Scattering Amplitudes, arXiv:1708.03872 [INSPIRE].
C.D. White, The double copy: gravity from gluons, arXiv:1708.07056 [INSPIRE].
A.H. Taub, Empty space-times admitting a three parameter group of motions, Annals Math. 53 (1951) 472.
E. Newman, L. Tamubrino and T. Unti, Empty space generalization of the Schwarzschild metric, J. Math. Phys. 4 (1963) 915 [INSPIRE].
Z.W. Chong, G.W. Gibbons, H. Lü and C.N. Pope, Separability and Killing tensors in Kerr-Taub-NUT-de Sitter metrics in higher dimensions, Phys. Lett. B 609 (2005) 124 [hep-th/0405061] [INSPIRE].
H. Stephani, D. Kramer, M.A. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press, (2003).
W. Chen, H. Lü and C.N. Pope, General Kerr-NUT-AdS metrics in all dimensions, Class. Quant. Grav. 23 (2006) 5323 [hep-th/0604125] [INSPIRE].
W. Chen and H. Lü, Kerr-Schild structure and harmonic 2-forms on (A)dS-Kerr-NUT metrics, Phys. Lett. B 658 (2008) 158 [arXiv:0705.4471] [INSPIRE].
A.H. Taub, Generalised Kerr-Schild space-times., Annals Phys. 134 (1981) 326.
A. Luna, I. Nicholson, D. O’Connell and C.D. White, Inelastic Black Hole Scattering from Charged Scalar Amplitudes, arXiv:1711.03901 [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1710.01953
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Bahjat-Abbas, N., Luna, A. & White, C.D. The Kerr-Schild double copy in curved spacetime. J. High Energ. Phys. 2017, 4 (2017). https://doi.org/10.1007/JHEP12(2017)004
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2017)004