Abstract
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 BMS 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.
Highlights
A solution-generating technique known in the literature as the complex coordinate transformation provides a set of maps between different solutions of general relativity
The double copy structure of the three point amplitude implies that the phase shift is doubled, and we show that the resulting minimal coupling reproduces the impulse of a test particle moving in the Taub-NUT background, as computed from the classical geodesic equations
III we show that this complex transformation rotates the standard and dual supertranslation charges into each other, and we interpret the transformation as a duality operation
Summary
A solution-generating technique known in the literature as the complex coordinate transformation provides a set of maps between different solutions of general relativity. As was shown by one of the authors [6], the Taub-NUT metric admits a double copy structure whose “square root” is precisely the electromagnetic dyon This suggests that the exponentiated phase shift once again is related to some complex coordinate transformation which can be identified as some form of a gravitational electric-magnetic duality. The double copy structure of the three point amplitude implies that the phase shift is doubled, and we show that the resulting minimal coupling reproduces the impulse of a test particle moving in the Taub-NUT background, as computed from the classical geodesic equations. V with a discussion followed by Appendixes reviewing the standard complex coordinates transformation algorithm and some technical computational details
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