Abstract
A characteristic value formulation of the Weyl double copy leads to an asymptotic formulation. We find that the Weyl double copy holds asymptotically in cases where the full solution is algebraically general, using rotating STU supergravity black holes as an example. The asymptotic formulation provides clues regarding the relation between asymptotic symmetries that follows from the double copy. Using the C-metric as an example, we show that a previous interpretation of this gravity solution as a superrotation has a single copy analogue relating the appropriate Liénard-Wiechert potential to a large gauge transformation.
Highlights
Hole spacetimes — e.g. the Kerr solution — could admit a straightforward interpretation as a double copy of a gauge theory solution [5, 6]
The Kerr solution has algebraic type D in the Petrov classification, and it is of Kerr-Schild type, which determines a privileged class of coordinates that can be thought of as those of a flat spacetime; see [7] for other exact vacuum type D solutions and [8] for vacuum type N solutions
The need to worry about gauge choices is absent when dealing only with scattering amplitudes, which are gauge invariant, but an exact relation between explicit classical solutions requires thinking about coordinates
Summary
As described in appendix A, the homomorphism between the Lorentz group and SL(2, C) can be used to write spacetime tensors as spinors [84]. For type D solutions, in general S only satisfies the wave equation in a flat spacetime, rather than the curved background. Having re-expressed the Weyl double copy equation in a null frame, we choose coordinates that will provide a direct relation to a characteristic value formulation of the Einstein equation. Assuming appropriate fall-off conditions for the energy-momentum tensor, there are equations relating the various metric tensor components; see ref. We have assumed an analytic expansion in the metric components This is a consistent assumption from an initial value problem perspective, in the sense that assuming an analytic fall-off for initial data will guarantee that the evolved solution will remain analytic [86]. For type D and for non-twisting type N solutions, the Maxwell field appearing in the Weyl double copy satisfies the Maxwell equation on Minkowski spacetime [7, 8]. The Maxwell scalars must be defined with respect to the curved null frame
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