Ehlers as EM duality in the double copy

Abstract

Given a solution to 4D Einstein gravity with an isometry direction, it is known that the equations of motion are identical to those of a 3D $\sigma$-model with target space geometry $SU(1,1)/U(1)$. Thus, any transformation by $SU(1,1)\cong SL(2,\mathbb{R})$ is a symmetry for the action and allows one to generate new solutions in 4D. Here we clarify and extend recent work on electromagnetic (EM) duality in the context of the classical double copy. In particular, for pure gravity, we identify an explicit map between the Maxwell field of the single copy and the scalars in the target space, allowing us to identify the $U(1)\subset SL(2,\mathbb{R})$ symmetry dual to EM duality in the single copy. Moreover, we extend the analysis to Einstein-Maxwell theory, where we highlight the role of Ehlers-Harrison transformations and, for spherically symmetric charged black hole solutions, we interpret the equations of motion as a truncation of the putative single copy for Einstein-Yang-Mills theory.

v1 A short note describing the key observation of 1911.06797 in hopefully a simpler way; v2 Some meat added to the bones; v3 comments on the classical double copy interpretation of Einstein-Maxwell theory added, draft restructured to better highlight results; v4 matches published version

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• A double copy for asymptotic symmetries in the self-dual sector
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• New heavenly double copies
• Asymptotic Weyl Double Copy
• From Moyal deformations to chiral higher-spin theories and to celestial algebras
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• Scattering Amplitudes and the Double Copy in Topologically Massive Theories
• The Off-Shell Recursion for Gravity and the Classical Double Copy for currents
• Twistor Space Origins of the Newman-Penrose Map
• Black Holes in Klein Space
• The double copy of the multipole expansion
• Non-perturbative aspects of the self-dual double copy
• Isometries and the double copy
• Higher-derivative Heterotic Double Field Theory and Classical Double Copy
• Aligned Fields Double Copy to Kerr-NUT-(A)dS
• Kaluza-Klein from Colour-Kinematics Duality for Massive Fields
• Generalized Black Holes in 3D Kerr-Schild Double Copy
• Gauge $\times$ Gauge $=$ Gravity on Homogeneous Spaces using Tensor Convolutions
• The Classical Double Copy for Half-Maximal Supergravities and T-duality
• Tree-Level Color-Kinematics Duality Implies Loop-Level Color-Kinematics Duality up to Counterterms
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• Gravitational Dyonic Amplitude at One-Loop and its Inconsistency with the Classical Impulse
• Quantization Conditions and the Double Copy
• The double copy: from optics to quantum gravity
• Compactifying the Kerr-Schild Double Copy
• Manifest electric-magnetic duality in linearized conformal gravity
• The Weyl double copy from twistor space
• Hawking radiation from the double copy
• Massive Covariant Colour-Kinematics in 3D
• The Kerr-Schild Double Copy in Lifshitz Spacetime