The classical double copy in maximally symmetric spacetimes

TL;DR

This work extends the classical double copy from flat spacetime to curved, maximally symmetric backgrounds by employing Kerr-Schild decompositions with (A)dS bases. It constructs and analyzes explicit single and zeroth copies for a broad set of solutions, including (A)dS-Schwarzschild, Kerr-(A)dS, black strings/branes, various AdS wave geometries, and the BTZ black hole, detailing the localized sources and equations of motion for the copies. A key finding is that stationary and time-dependent copies obey different curvature-coupled equations, with conformal coupling appearing in the scalar copy at four dimensions and curvature-induced mass terms affecting the gauge copy in wave cases. The results provide a practical framework for deriving Einstein–Maxwell systems from neutral curved backgrounds and highlight ambiguities in Kerr-Schild decompositions, suggesting several avenues for extension and deeper connections to holography and BCJ-type structures.

Abstract

The classical double copy procedure relates classical asymptotically-flat gravitational field solutions to Yang-Mills and scalar field solutions living in Minkowski space. In this paper we extend this correspondence to maximally symmetric curved spacetimes. We consider asymptotically (A)dS spacetimes in Kerr-Schild form and construct the corresponding single and zeroth copies. In order to clarify the interpretation of these copies, we study several examples including (A)dS-Schwarzschild, (A)dS-Kerr, black strings, black branes, and waves, paying particular attention to the source terms. We find that the single and zeroth copies of stationary solutions satisfy different equations than those of wave solutions. We also consider how to obtain Einstein-Maxwell solutions using this procedure. Finally, we derive the classical single and zeroth copy of the BTZ black hole.

Figures (2)

• Figure 1: Kundt waves in de Sitter space. We can consider dS as a four-dimensional hyperboloid embedded in a flat 5d space with coordinates $Z^a$. This figure shows portions of de Sitter covered by the coordinates in Eq. \ref{['wdS']} at different values of the time $Z^0$ with $Z^4=0$. The portions of 2-spheres correspond to different snapshots in time and the semi-circles on them are the wavefronts of constant $u$. The gravitational, gauge, and scalar waves all have wavefronts of this shape. For more details, see Griffiths:2009dfaGriffiths:2003bk.
• Figure 2: Planar shock wave in AdS. The source travels on a null geodesic at fixed $u=0$, $x=x_0$, and $y=0$. The gravitational, gauge, and scalar shock waves all have this structure.