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.
Highlights
The classical double copy is an intriguing connection between gravity and gauge theory [1], which has been motivated from a relationship between perturbative scattering amplitudes in gauge theory and gravity [2,3,4].1 In its simplest form, the central observation is that solutions to Einstein gravity, or “the double copy,” can be mapped to solutions of Maxwell’s equations,2 or “the single copy,” through a Kerr-Schild (KS) decomposition of the spacetime
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
In the earlier part of this work we married the KS ansatz of pure gravity with a natural 4D to 3D dimensional reduction, which allowed us to identify the Maxwell field strengths of the double copy formalism directly in terms of the scalars parametrizing a hyperbolic coset geometry in 3D. This can be done for generic spacetime geometries and it is the rotation of the scalars under a Uð1Þ ⊂ SLð2; RÞ that is mapped to EM duality in the Maxwell fluxes of the double copy formalism
Summary
The classical double copy is an intriguing connection between gravity and gauge theory [1], which has been motivated from a relationship between perturbative scattering amplitudes in gauge theory and gravity [2,3,4].1 In its simplest form, the central observation is that solutions to Einstein gravity, or “the double copy,” can be mapped to solutions of Maxwell’s equations, or “the single copy,” through a Kerr-Schild (KS) decomposition of the spacetime. We are combining the classical double copy in pure gravity with Kaluza-Klein reduction and the beauty of this approach is that a manifest Uð1Þ symmetry in 4D leads to an SLð2; RÞ symmetry in the lower-dimensional theory Through this process, we show how the electric and magnetic Maxwell field strengths of the single copy are related to the derivatives of the scalars of the 3D σ model in the double copy, thereby providing a succinct way to understand observations made in [29]. We illustrate how the Ehlers transformation once again plays the counterpart of EM duality in this extended setting
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