The Kerr-Schild double copy in curved spacetime

TL;DR

Addresses extending the classical double copy to curved spacetime using Kerr-Schild metrics, proving that gravitons defined on curved backgrounds can be obtained as double copies of gauge fields on corresponding backgrounds with $g_{μν}=\bar{g}_{μν}+\tilde{h}_{μν}$ and $h_{μν}= (κ/2) φ k_μ k_ν$. It identifies two curved-space double-copy paradigms: type A, linking a gauge field on a nontrivial background to a graviton on a related background, and type B, relating a graviton on a curved background to a gauge field on the same background; both are illustrated via exact Kerr-Schild examples including Schwarzschild, de Sitter, and conformally flat spacetimes. A key result is that type A admits a well-defined zeroth copy to a biadjoint scalar field, connecting to flat-space amplitude structures, while type B generally lacks a clean zeroth copy. The findings broaden the double copy's scope toward cosmological applications and motivate further work on non-Kerr-Schild backgrounds, investigation of type B applicability, and potential higher-point amplitude extensions.

Abstract

The double copy is a much-studied relationship between scattering amplitudes in gauge and gravity theories, that has subsequently been extended to classical field solutions. In nearly all previous examples, the graviton field is defined around Minkowski space. Recently, it has been suggested that one may set up a double copy for gravitons defined around a non-trivial background. We investigate this idea from the point of view of the classical double copy. First, we use Kerr-Schild spacetimes to construct graviton solutions in curved space, as double copies of gauge fields on non-zero gauge backgrounds. Next, we find that we can reinterpret such cases in terms of a graviton on a non-Minkowski background, whose single copy is a gauge field in the same background spacetime. The latter type of double copy persists even when the background is not of Kerr-Schild form, and we provide examples involving conformally flat metrics. Our results will be useful in extending the remit of the double copy, including to possible cosmological applications.

Figures (2)

• Figure 1: Two possible interpretations of a double copy in curved space: in type A, a gauge field has a non-trivial background field $\bar{A}_\mu^a$ in Minkowski space, and copies to a graviton defined on a curved background $\bar{g}_{\mu\nu}$, where $\bar{g}_{\mu\nu}$ and $\bar{A}_\mu^a$ are themselves related by a double copy relationship. In type B, a gauge field on a non-dynamical curved background $\bar{g}_{\mu\nu}$ double copies to a graviton defined around the same background.
• Figure 2: Generalisation of the type A double copy of figure \ref{['fig:curvedmap']} to include the zeroth copy, which relates the gauge field defined with a non-trivial background to similar solutions in a biadjoint scalar theory.