Double copy of electric-magnetic duality

We argue that the complex transformation relating the Schwarzschild to the Taub-NUT metric, introduced by Talbot, is in fact an electric-magnetic duality transformation. We show that at null infinity, the complex transformation is equivalent to a complexified Bondi-Metzner-Sachs supertranslation, which rotates the supertranslation and the dual (magnetic) supertranslation charges. This can also be seen from the cubic coupling between the classical source and its background, which for Taub-NUT is given by a complex phase rotation acting on gravitational minimal couplings. The same phase rotation generates dyons from electrons at the level of minimally coupled amplitudes, manifesting the double copy relation between the two solutions.

Figure 1

The maps from the Schwarzschild solution to the Taub-NUT and to Kerr metrics. The underlying physical mechanism behind these two maps is different. In the first case the map is a duality operation while in the later case it results from exponentiation of spin operators [3]. It would be interesting to extend this diagram and understand the full set of maps between solutions in the Plebanski-Demianski family [7, 8].

Figure 2

We consider the gravitational and electromagnetic impulses imparted on a probe particle 1 via the source particle 2, at leading order in coupling. The effect is captured by the one graviton or photon exchange, respectively.

Figure 3

The Taub-NUT metric admits a double copy structure which relates it to the electromagnetic dyon [6]. We show that the double copy structure relates the standard and dual supertranslation charges to the large electric and magnetic charges of the dyon. This result implies that the duality (1.5) between the standard and dual supertranslation charges is related to the electric-magnetic duality in QED by the double copy structure.