Why is the Weyl double copy local in position space?

Abstract

The double copy relates momentum-space scattering amplitudes in gauge and gravity theories. It has also been extended to classical solutions, where in some cases an exact double copy can be formulated directly in terms of products of fields in position space. This is seemingly at odds with the momentum-space origins of the double copy, and the question of why exact double copies are possible in position space — and when this form will break — has remained largely unanswered. In this paper, we provide an answer to this question, using a recently developed twistorial formulation of the double copy. We show that for certain vacuum type-D solutions, the momentum-space, twistor-space and position-space double copies amount to the same thing, and are directly related by integral transforms. Locality in position space is ultimately a consequence of the very special form of momentum-space three-point amplitudes, and we thus confirm suspicions that local position-space double copies are possible only for highly algebraically-special spacetimes.

Aligned fields double copy to Kerr-NUT-(A)dS

“…In the Weyl double copy, a single copy field strength and zeroth copy scalar are proposed which combine to form the Weyl tensor directly. Following the first proposal for Petrov type D spacetimes in [39], the Weyl double copy was extended to other Petrov types [40] and has found justification from amplitudes directly [41][42][43][44][45][46][47][48][49]. Aside from the spinor formalism (extended to five dimensions in [7]), we additionally verify our five-dimensional copy via the Weyl doubling formula of [50].…”

“…bumpy black holes, black rings, etc) may require significant extension. Indeed, the argument for Weyl locality in position space given in [48] indicates that non-algebraically special spacetimes cannot have a local Weyl single copy, and spacetimes possessing a principal tensor are always algebraically special. Nevertheless, we hope that our work here may provide an approach to make further spacetimes at least perturbatively accessible.…”

Einstein-Maxwell theory and the Weyl double copy

“…Finally, we have studied an elementary extension of the original Weyl double copy, and begun to gather evidence of a double-copy prescription for the Ricci spinor. To better understand both double-copy structures on a more fundamental level, it would be interesting to explore the twistorial foundations of the sourced Weyl double copy, utilizing and extending [21,22,36,38,[67][68][69].…”

A spinorial double copy for ${\cal N}=0$ supergravity

“…Unlike the 4d Weyl double copy, the Cotton double copy was found to only work for solutions of Petrov type N. Reference [83] confirmed this by applying similar twistor methods to the four-dimensional case of ref. [76], but now in a three-dimensional context. The appropriate twistor space in this case is called minitwistor space, and an appropriate generalisation of the Penrose transform must be used due to the presence of the topological mass.…”

“…The aim of this paper is to apply the twistor methods of refs. [76,78,83] to the starting point of ref. [90].…”