Abstract
We revisit the question of whether classical general relativity obeys, beyond the linearised order, an analogue of the global U(1) electric-magnetic duality of Maxwell theory, with the Riemann tensor playing the role analogous to the field strength. Following contradictory claims in the literature, we present a simple gauge-invariant argument that the duality does not hold. The duality condition is the conservation of the helicity charge. Scattering amplitudes of gravitons in general relativity, and of gluons in Yang-Mills theory, violate this selection rule already at tree level. Indeed, the maximally-helicity-violating (MHV) amplitudes are famous for their simplicity. The duality in the linearised theories is, therefore, broken by the interactions. In contrast, the tree-level scattering amplitudes in duality-invariant theories of non-linear electromagnetism are known to obey helicity conservation. While the duality is not a symmetry of the full theory of general relativity, it does hold within a sector of the solution space, including vacuum type D solutions, where the duality is known to rotate between mass and NUT charge.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Montonen and D.I. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
M.K. Gaillard and B. Zumino, Duality Rotations for Interacting Fields, Nucl. Phys. B 193 (1981) 221 [INSPIRE].
R. Penrose, A spinor approach to general relativity, Annals Phys. 10 (1960) 171 [INSPIRE].
J.A. Nieto, S duality for linearized gravity, Phys. Lett. A 262 (1999) 274 [hep-th/9910049] [INSPIRE].
C.M. Hull, Duality in gravity and higher spin gauge fields, JHEP 09 (2001) 027 [hep-th/0107149] [INSPIRE].
M. Henneaux and C. Teitelboim, Duality in linearized gravity, Phys. Rev. D 71 (2005) 024018 [gr-qc/0408101] [INSPIRE].
S. Deser and D. Seminara, Free spin 2 duality invariance cannot be extended to GR, Phys. Rev. D 71 (2005) 081502 [hep-th/0503030] [INSPIRE].
C.W. Bunster, S. Cnockaert, M. Henneaux and R. Portugues, Monopoles for gravitation and for higher spin fields, Phys. Rev. D 73 (2006) 105014 [hep-th/0601222] [INSPIRE].
R. Argurio and F. Dehouck, Gravitational duality and rotating solutions, Phys. Rev. D 81 (2010) 064010 [arXiv:0909.0542] [INSPIRE].
G. Barnich and C. Troessaert, Manifest spin 2 duality with electric and magnetic sources, JHEP 01 (2009) 030 [arXiv:0812.0552] [INSPIRE].
S. de Haro, Dual Gravitons in AdS4/CFT3 and the Holographic Cotton Tensor, JHEP 01 (2009) 042 [arXiv:0808.2054] [INSPIRE].
M. Astorino, Enhanced Ehlers Transformation and the Majumdar-Papapetrou-NUT Spacetime, JHEP 01 (2020) 123 [arXiv:1906.08228] [INSPIRE].
J. Boos and I. Kolář, Nonlocality and gravitoelectromagnetic duality, Phys. Rev. D 104 (2021) 024018 [arXiv:2103.10555] [INSPIRE].
M. Astorino and G. Boldi, Plebanski-Demianski goes NUTs (to remove the Misner string), JHEP 08 (2023) 085 [arXiv:2305.03744] [INSPIRE].
S. Deser and C. Teitelboim, Duality Transformations of Abelian and Nonabelian Gauge Fields, Phys. Rev. D 13 (1976) 1592 [INSPIRE].
U. Kol, Duality in Einstein’s Gravity, arXiv:2205.05752 [INSPIRE].
U. Kol and S.-T. Yau, Duality in Gauge Theory, Gravity and String Theory, arXiv:2311.07934 [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].
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].
A. Banerjee, E.Ó. Colgáin, J.A. Rosabal and H. Yavartanoo, Ehlers as EM duality in the double copy, Phys. Rev. D 102 (2020) 126017 [arXiv:1912.02597] [INSPIRE].
W.T. Emond et al., Amplitudes from Coulomb to Kerr-Taub-NUT, JHEP 05 (2022) 055 [arXiv:2010.07861] [INSPIRE].
N. Moynihan and J. Murugan, On-shell electric-magnetic duality and the dual graviton, Phys. Rev. D 105 (2022) 066025 [arXiv:2002.11085] [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].
E. Crawley, A. Guevara, N. Miller and A. Strominger, Black holes in Klein space, JHEP 10 (2022) 135 [arXiv:2112.03954] [INSPIRE].
R. Monteiro et al., NS-NS spacetimes from amplitudes, JHEP 06 (2022) 021 [arXiv:2112.08336] [INSPIRE].
G. Compère, R. Oliveri and A. Seraj, Metric reconstruction from celestial multipoles, JHEP 11 (2022) 001 [arXiv:2206.12597] [INSPIRE].
M.G. Calkin, An invariance property of the free electromagnetic field, Am. J. Phys. 33 (1965) 958.
J. Novotný, Self-duality, helicity conservation and normal ordering in nonlinear QED, Phys. Rev. D 98 (2018) 085015 [arXiv:1806.02167] [INSPIRE].
A.A. Rosly and K.G. Selivanov, Helicity conservation in Born-Infeld theory, in the proceedings of the Workshop on String Theory and Complex Geometry, Novgorod, Russian Federation, June 01–07 (2002) [hep-th/0204229] [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].
Z. Bern et al., Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].
D.Z. Freedman et al., Absence of U(1) Anomalous Superamplitudes in \(\mathcal{N}\) ≥ 5 Supergravities, JHEP 05 (2017) 067 [arXiv:1703.03879] [INSPIRE].
Z. Bern, J. Parra-Martinez and R. Roiban, Canceling the U(1) Anomaly in the S Matrix of N = 4 Supergravity, Phys. Rev. Lett. 121 (2018) 101604 [arXiv:1712.03928] [INSPIRE].
N.H. Pavao, Effective observables for electromagnetic duality from novel amplitude decomposition, Phys. Rev. D 107 (2023) 065020 [arXiv:2210.12800] [INSPIRE].
J.J.M. Carrasco and N.H. Pavao, Virtues of a symmetric-structure double copy, Phys. Rev. D 107 (2023) 065005 [arXiv:2211.04431] [INSPIRE].
J.J.M. Carrasco and N.H. Pavao, Even-point multi-loop unitarity and its applications: exponentiation, anomalies and evanescence, JHEP 01 (2024) 019 [arXiv:2307.16812] [INSPIRE].
I. Agullo, A. del Rio and J. Navarro-Salas, Electromagnetic duality anomaly in curved spacetimes, Phys. Rev. Lett. 118 (2017) 111301 [arXiv:1607.08879] [INSPIRE].
A.K.H. Bengtsson, I. Bengtsson and L. Brink, Cubic Interaction Terms for Arbitrarily Extended Supermultiplets, Nucl. Phys. B 227 (1983) 41 [INSPIRE].
H. Elvang, M. Hadjiantonis, C.R.T. Jones and S. Paranjape, All-Multiplicity One-Loop Amplitudes in Born-Infeld Electrodynamics from Generalized Unitarity, JHEP 03 (2020) 009 [arXiv:1906.05321] [INSPIRE].
C. Wen and S.-Q. Zhang, D3-Brane Loop Amplitudes from M5-Brane Tree Amplitudes, JHEP 07 (2020) 098 [arXiv:2004.02735] [INSPIRE].
H. Elvang, M. Hadjiantonis, C.R.T. Jones and S. Paranjape, Electromagnetic Duality and D3-Brane Scattering Amplitudes Beyond Leading Order, JHEP 04 (2021) 173 [arXiv:2006.08928] [INSPIRE].
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].
Z. Bern, A. De Freitas and H.L. Wong, On the coupling of gravitons to matter, Phys. Rev. Lett. 84 (2000) 3531 [hep-th/9912033] [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].
P.A. Cano and Á. Murcia, Duality-invariant extensions of Einstein-Maxwell theory, JHEP 08 (2021) 042 [arXiv:2104.07674] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
N. Arkani-Hamed et al., Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [https://doi.org/10.1017/CBO9781316091548] [INSPIRE].
Z. Bern and Y.-T. Huang, Basics of Generalized Unitarity, J. Phys. A 44 (2011) 454003 [arXiv:1103.1869] [INSPIRE].
C. Csaki et al., Scattering amplitudes for monopoles: pairwise little group and pairwise helicity, JHEP 08 (2021) 029 [arXiv:2009.14213] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
G. Chalmers and W. Siegel, Simplifying algebra in Feynman graphs. Part 2. Spinor helicity from the space-cone, Phys. Rev. D 59 (1999) 045013 [hep-ph/9801220] [INSPIRE].
S. Ananth, L. Brink, R. Heise and H.G. Svendsen, The N = 8 Supergravity Hamiltonian as a Quadratic Form, Nucl. Phys. B 753 (2006) 195 [hep-th/0607019] [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].
A. Hodges, A simple formula for gravitational MHV amplitudes, arXiv:1204.1930 [INSPIRE].
L.J. Mason and D. Skinner, Gravity, Twistors and the MHV Formalism, Commun. Math. Phys. 294 (2010) 827 [arXiv:0808.3907] [INSPIRE].
P. Mansfield, The Lagrangian origin of MHV rules, JHEP 03 (2006) 037 [hep-th/0511264] [INSPIRE].
A. Brandhuber, B. Spence and G. Travaglini, Amplitudes in Pure Yang-Mills and MHV Diagrams, JHEP 02 (2007) 088 [hep-th/0612007] [INSPIRE].
M. Bianchi, H. Elvang and D.Z. Freedman, Generating Tree Amplitudes in N = 4 SYM and N = 8 SG, JHEP 09 (2008) 063 [arXiv:0805.0757] [INSPIRE].
M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Gravitational Born Amplitudes and Kinematical Constraints, Phys. Rev. D 12 (1975) 397 [INSPIRE].
M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Supergravity and the S Matrix, Phys. Rev. D 15 (1977) 996 [INSPIRE].
J.S. Dowker, The nut solution as a gravitational dyon, Gen. Rel. Grav. 5 (1974) 603.
J.F. Plebanski and M. Demianski, Rotating, charged, and uniformly accelerating mass in general relativity, Annals Phys. 98 (1976) 98 [INSPIRE].
J. Ehlers, Transformations of static exterior solutions of Einstein’s gravitational field equations into different solutions by means of conformal mapping, Colloq. Int. CNRS 91 (1962) 275 [INSPIRE].
R.P. Geroch, A method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971) 918 [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 and D. O’Connell, Type D Spacetimes and the Weyl Double Copy, Class. Quant. Grav. 36 (2019) 065003 [arXiv:1810.08183] [INSPIRE].
Z. Bern et al., The SAGEX review on scattering amplitudes Chapter 2: An invitation to color-kinematics duality and the double copy, J. Phys. A 55 (2022) 443003 [arXiv:2203.13013] [INSPIRE].
D.A. Kosower, R. Monteiro and D. O’Connell, The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes, J. Phys. A 55 (2022) 443015 [arXiv:2203.13025] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Plante and P. Vanhove, The SAGEX review on scattering amplitudes Chapter 13: Post-Minkowskian expansion from scattering amplitudes, J. Phys. A 55 (2022) 443014 [arXiv:2203.13024] [INSPIRE].
A. Buonanno et al., Snowmass White Paper: Gravitational Waves and Scattering Amplitudes, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2204.05194] [INSPIRE].
T. Adamo et al., Snowmass White Paper: the Double Copy and its Applications, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2204.06547] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The gravitational eikonal: from particle, string and brane collisions to black-hole encounters, arXiv:2306.16488 [INSPIRE].
T. Banks and L.J. Dixon, Constraints on String Vacua with Space-Time Supersymmetry, Nucl. Phys. B 307 (1988) 93 [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, Commun. Math. Phys. 383 (2021) 1669 [arXiv:1810.05338] [INSPIRE].
Z. Bern et al., Evanescent Effects Can Alter Ultraviolet Divergences in Quantum Gravity without Physical Consequences, Phys. Rev. Lett. 115 (2015) 211301 [arXiv:1507.06118] [INSPIRE].
Z. Bern, H.-H. Chi, L. Dixon and A. Edison, Two-Loop Renormalization of Quantum Gravity Simplified, Phys. Rev. D 95 (2017) 046013 [arXiv:1701.02422] [INSPIRE].
Z. Bern, D. Kosower and J. Parra-Martinez, Two-loop n-point anomalous amplitudes in N = 4 supergravity, Proc. Roy. Soc. Lond. A 476 (2020) 20190722 [arXiv:1905.05151] [INSPIRE].
J.J.M. Carrasco, M. Lewandowski and N.H. Pavao, Color-Dual Fates of F3, R3, and N = 4 Supergravity, Phys. Rev. Lett. 131 (2023) 051601 [arXiv:2203.03592] [INSPIRE].
Acknowledgments
The author is grateful to John Joseph Carrasco, George Doran, Uri Kol, Lionel Mason, Nathan Moynihan, Congkao Wen and Sam Wikeley for comments or collaboration on related topics. RM is supported by the Royal Society via a University Research Fellowship.
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: 2312.02351
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
Monteiro, R. No U(1) ‘electric-magnetic’ duality in Einstein gravity. J. High Energ. Phys. 2024, 93 (2024). https://doi.org/10.1007/JHEP04(2024)093
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2024)093