Advertisement

The classical double copy in maximally symmetric spacetimes

  • Mariana Carrillo González
  • Riccardo Penco
  • Mark Trodden
Open Access
Regular Article - Theoretical Physics

Abstract

The classical double copy procedure relates classical asymptotically-flat gravitational field solutions to Yang-Mills and scalar field solutions living in Minkowski space. In this paper we extend this correspondence to maximally symmetric curved spacetimes. We consider asymptotically (A)dS spacetimes in Kerr-Schild form and construct the corresponding single and zeroth copies. In order to clarify the interpretation of these copies, we study several examples including (A)dS-Schwarzschild, (A)dS-Kerr, black strings, black branes, and waves, paying particular attention to the source terms. We find that the single and zeroth copies of stationary solutions satisfy different equations than those of wave solutions. We also consider how to obtain Einstein-Maxwell solutions using this procedure. Finally, we derive the classical single and zeroth copy of the BTZ black hole.

Keywords

Scattering Amplitudes Black Holes Gauge Symmetry 

Notes

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.

References

  1. [1]
    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].ADSMathSciNetGoogle Scholar
  2. [2]
    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].ADSGoogle Scholar
  3. [3]
    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].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    C. Cheung, C.-H. Shen and C. Wen, Unifying Relations for Scattering Amplitudes, JHEP 02 (2018) 095 [arXiv:1705.03025] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    C. Cheung, TASI Lectures on Scattering Amplitudes, arXiv:1708.03872 [INSPIRE].
  8. [8]
    C. Cheung and C.-H. Shen, Symmetry for Flavor-Kinematics Duality from an Action, Phys. Rev. Lett. 118 (2017) 121601 [arXiv:1612.00868] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α -corrections from the open string, JHEP 06 (2017) 093 [arXiv:1608.02569] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    G. Chen and Y.-J. Du, Amplitude Relations in Non-linear σ-model, JHEP 01 (2014) 061 [arXiv:1311.1133] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    D. Nandan, J. Plefka, O. Schlotterer and C. Wen, Einstein-Yang-Mills from pure Yang-Mills amplitudes, JHEP 10 (2016) 070 [arXiv:1607.05701] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    S. Stieberger and T.R. Taylor, Disk Scattering of Open and Closed Strings (I), Nucl. Phys. B 903 (2016) 104 [arXiv:1510.01774] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    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].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Complete construction of magical, symmetric and homogeneous N = 2 supergravities as double copies of gauge theories, Phys. Rev. Lett. 117 (2016) 011603 [arXiv:1512.09130] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    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 Colliders (TASI 2014), Boulder, Colorado, June 2-27, 2014, pp. 477-557, WSP, WSP (2015) [DOI:10.1142/97898146787660011] [arXiv:1506.00974] [INSPIRE].
  19. [19]
    S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
  20. [20]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    F. Cachazo, Fundamental BCJ Relation in N = 4 SYM From The Connected Formulation, arXiv:1206.5970 [INSPIRE].
  23. [23]
    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].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    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].
  25. [25]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    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].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    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].ADSGoogle Scholar
  28. [28]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Z. Bern, J.S. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 superYang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].
  30. [30]
    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].ADSCrossRefGoogle Scholar
  31. [31]
    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].
  32. [32]
    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].ADSGoogle Scholar
  33. [33]
    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].
  34. [34]
    R. Saotome and R. Akhoury, Relationship Between Gravity and Gauge Scattering in the High Energy Limit, JHEP 01 (2013) 123 [arXiv:1210.8111] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    D. Neill and I.Z. Rothstein, Classical Space-Times from the S Matrix, Nucl. Phys. B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Luna et al., Perturbative spacetimes from Yang-Mills theory, JHEP 04 (2017) 069 [arXiv:1611.07508] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    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].ADSGoogle Scholar
  38. [38]
    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].ADSGoogle Scholar
  39. [39]
    R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    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].ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    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].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    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].ADSMathSciNetGoogle Scholar
  43. [43]
    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].ADSCrossRefzbMATHGoogle Scholar
  44. [44]
    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].
  45. [45]
    L. Borsten and M.J. Duff, Gravity as the square of Yang-Mills?, Phys. Scripta 90 (2015) 108012 [arXiv:1602.08267] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    G. Cardoso, S. Nagy and S. Nampuri, Multi-centered \( \mathcal{N} \) = 2 BPS black holes: a double copy description, JHEP 04 (2017) 037 [arXiv:1611.04409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    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].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    Y.-Z. Chu, More On Cosmological Gravitational Waves And Their Memories, Class. Quant. Grav. 34 (2017) 194001 [arXiv:1611.00018] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    N. Bahjat-Abbas, A. Luna and C.D. White, The Kerr-Schild double copy in curved spacetime, JHEP 12 (2017) 004 [arXiv:1710.01953] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  51. [51]
    V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
  52. [52]
    H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press (2004).Google Scholar
  53. [53]
    A. Taub, Generalised kerr-schild space-times, Annals Phys. 134 (1981) 326.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    K.E. Mastronikola, Remarks on generalised kerr-schild metrics, Class. Quant. Grav. 4 (1987) L179.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    T. Malek and V. Pravda, Kerr-Schild spacetimes with (A)dS background, Class. Quant. Grav. 28 (2011) 125011 [arXiv:1009.1727] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    H. Nariai, On some static solutions of einstein’s gravitational field equations in a spherically symmetric case, Sci. Rep. Tohoku Univ. 34 (1951) 160.Google Scholar
  57. [57]
    H. Nariai, On a new cosmological solution of einstein’s field equations of gravitation, Sci. Rep. Tohoku Univ. 35 (1950) 62.MathSciNetzbMATHGoogle Scholar
  58. [58]
    G.W. Gibbons, H. Lü, D.N. Page and C.N. Pope, The General Kerr-de Sitter metrics in all dimensions, J. Geom. Phys. 53 (2005) 49 [hep-th/0404008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    W. Israel, Source of the kerr metric, Phys. Rev. D 2 (1970) 641 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  60. [60]
  61. [61]
    L.J. Romans, Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory, Nucl. Phys. B 383 (1992) 395 [hep-th/9203018] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    J.B. Griffiths and J. Podolsky, Exact Space-Times in Einstein’s General Relativity, Cambridge University Press, Cambridge (2009).CrossRefzbMATHGoogle Scholar
  63. [63]
    T. Hirayama and G. Kang, Stable black strings in anti-de Sitter space, Phys. Rev. D 64 (2001) 064010 [hep-th/0104213] [INSPIRE].ADSMathSciNetGoogle Scholar
  64. [64]
    A. Chamblin, S.W. Hawking and H.S. Reall, Brane world black holes, Phys. Rev. D 61 (2000) 065007 [hep-th/9909205] [INSPIRE].ADSMathSciNetGoogle Scholar
  65. [65]
    R. Gregory, Black string instabilities in Anti-de Sitter space, Class. Quant. Grav. 17 (2000) L125 [hep-th/0004101] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    J.B. Griffiths, P. Docherty and J. Podolsky, Generalized Kundt waves and their physical interpretation, Class. Quant. Grav. 21 (2004) 207 [gr-qc/0310083] [INSPIRE].
  67. [67]
    M. Gurses, T.C. Sisman and B. Tekin, New Exact Solutions of Quadratic Curvature Gravity, Phys. Rev. D 86 (2012) 024009 [arXiv:1204.2215] [INSPIRE].ADSGoogle Scholar
  68. [68]
    M. Hotta and M. Tanaka, Shock wave geometry with nonvanishing cosmological constant, Class. Quant. Grav. 10 (1993) 307 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    G.T. Horowitz and N. Itzhaki, Black holes, shock waves and causality in the AdS/CFT correspondence, JHEP 02 (1999) 010 [hep-th/9901012] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
  72. [72]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    S. Oxburgh and C.D. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    M. Ortaggio, V. Pravda and A. Pravdova, Higher dimensional Kerr-Schild spacetimes, Class. Quant. Grav. 26 (2009) 025008 [arXiv:0808.2165] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    P.C. Vaidya and P.V. Bhatt, A generalized Kerr-Schild metric, Pramana 3 (1974) 28.ADSCrossRefGoogle Scholar
  76. [76]
    B. Ett and D. Kastor, An Extended Kerr-Schild Ansatz, Class. Quant. Grav. 27 (2010) 185024 [arXiv:1002.4378] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    Z.W. Chong, M. Cvetič, H. Lü and C.N. Pope, General non-extremal rotating black holes in minimal five-dimensional gauged supergravity, Phys. Rev. Lett. 95 (2005) 161301 [hep-th/0506029] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  78. [78]
    A.N. Aliev and D.K. Ciftci, A Note on Rotating Charged Black Holes in Einstein-Maxwell-Chern-Simons Theory, Phys. Rev. D 79 (2009) 044004 [arXiv:0811.3948] [INSPIRE].ADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Mariana Carrillo González
    • 1
  • Riccardo Penco
    • 1
  • Mark Trodden
    • 1
  1. 1.Center for Particle Cosmology, Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaU.S.A.

Personalised recommendations