Why is the Weyl double copy local in position space?

TL;DR

This paper explains why the double copy relation is local in position space for certain special spacetimes, linking it to the algebraic properties of three-point amplitudes via a twistorial approach.

Contribution

It demonstrates that for vacuum type-D solutions, the momentum-space, twistor-space, and position-space double copies are equivalent and connected through integral transforms, clarifying the conditions for locality.

Findings

Position-space double copies are related to special algebraic properties of spacetimes.

For vacuum type-D solutions, the double copy is consistent across momentum, twistor, and position spaces.

Locality in position space arises from the structure of three-point amplitudes.

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.