The Weyl Double Copy for Gravitational Waves

TL;DR

This paper extends the Weyl double copy to radiative vacuum spacetimes of type $N$ by showing $\Psi_{ABCD}=\frac{1}{S}\,\Phi_{(AB}\Phi_{CD)}$ with a degenerate Maxwell spinor $\Phi_{AB}$ and a scalar $S$ that satisfies $\Box S=0$ on the curved background, and that $\Psi_4=\frac{1}{S}\, (\Phi_2)^2$ follows from this construction. For non-twisting radiative spacetimes, the Maxwell field $\Phi_{AB}$ and $S$ also solve their equations on Minkowski spacetime, yielding a Minkowski-space interpretation of the double copy; the results are demonstrated across the three type N subfamilies—Kundt, Robinson–Trautman, and twisting. In Kundt and Robinson–Trautman cases, explicit forms of $S$ and $\Phi_2$ are derived and shown to satisfy the Maxwell and wave equations in the appropriate backgrounds, while twisting type N solutions exhibit a curved double copy with $S$ and $\Phi_{AB}$ depending nontrivially on the geometry and not generally reducing to flat-space fields. A key finding is the non-uniqueness of the Maxwell and scalar data for type N spacetimes, contrasted with the more rigid structure in type D, highlighting functional freedom and potential connections to flat-space scattering amplitudes in the non-twisting sector. Overall, the work broadens the Weyl double copy program to include radiative spacetimes and clarifies when flat-space interpretations of the double copy are possible.

Abstract

We establish the status of the Weyl double copy relation for radiative solutions of the vacuum Einstein equations. We show that all type N vacuum solutions, which describe the radiation region of isolated gravitational systems with appropriate fall-off for the matter fields, admit a degenerate Maxwell field that squares to give the Weyl tensor. This relation defines a scalar that satisfies the wave equation on the background. We show that for non-twisting radiative solutions, the Maxwell field and the scalar also satisfy the Maxwell equation and the wave equation on Minkowski spacetime. Hence, non-twisting solutions have a straightforward double copy interpretation.