Classical double copy of black strings in an Anti-de Sitter background

We study the classical double copy for static black string solutions in an Anti–de Sitter (AdS) background. By casting the black string metric into Kerr–Schild form over a cylindrical AdS geometry, we construct the corresponding single and zeroth copies. The single copy describes a gauge field satisfying Maxwell-like equations and sourced by an effective line of color charge, while the zeroth copy is given by a scalar field conformally coupled to the AdS background. We also extend the analysis to charged black strings, identifying the associated modifications in the gauge sector. These results show that the classical double copy consistently applies to extended gravitational objects in curved spacetimes.

💡 Research Summary

The paper investigates the applicability of the classical double copy correspondence to static black string solutions embedded in a four‑dimensional anti‑de Sitter (AdS) spacetime. Starting from the well‑known cylindrical black string metric
(ds^{2}= -F(r)dt^{2}+F^{-1}(r)dr^{2}+r^{2}d\phi^{2}+ \alpha^{2}r^{2}dz^{2})
with (\alpha^{2}=-\Lambda/3) (Λ<0), the authors treat the neutral case where (F(r)=\alpha^{2}r^{2}-\lambda\alpha r) and the charged case where an extra term (\beta^{2}\alpha^{2}r^{2}) appears. The parameter λ encodes the linear mass density of the string, while β encodes its linear charge density.

To expose the double‑copy structure, the metric is rewritten in a generalized Kerr–Schild (KS) form on a cylindrical AdS background:
(g_{\mu\nu}= \bar g_{\mu\nu}+2 H(r) k_{\mu}k_{\nu}).
The background (\bar g_{\mu\nu}) is the pure AdS geometry written in ingoing Eddington–Finkelstein coordinates ((\tau,r,\phi,z)). The null vector is simply (k_{\mu}dx^{\mu}=d\tau) (i.e. (k^{\mu}=(1,0,0,0))). By comparing the full metric with the background, the KS scalar function is identified as (H(r)=\lambda/(2\alpha r)) for the neutral string; the charged case acquires an additional β‑dependent contribution.

Single copy.
Following the standard prescription, the gauge potential is defined as
(A^{a}{\mu}=c^{a} H(r) k{\mu}),
where (c^{a}) is a constant colour vector. In the chosen coordinates the only non‑vanishing component is (A^{a}{\tau}=c^{a}\lambda/(2\alpha r)). The associated field strength has a single component (F^{a}{\tau r}=c^{a}\lambda/(2\alpha r^{2})), describing a purely radial electric field that is uniform along the string axis. Solving the Maxwell equations on the AdS background shows that the field is sourceless for (r\neq0); the singular behaviour at the axis is captured by a distributional current
(J^{a}{\tau}=c^{a}\lambda,\delta^{(2)}(x{\perp})),
with all other components vanishing. This precisely matches the picture of an infinite line of colour charge with linear density proportional to λ, confirming that the single copy of the black string is a cylindrical electrostatic configuration in AdS.

Zeroth copy.
The scalar profile (H(r)) itself serves as the zeroth copy. In maximally symmetric curved backgrounds the scalar does not obey a simple Laplace equation but rather the conformally‑coupled scalar equation
((\bar\nabla^{2}-\bar R/6),\phi=0).
Explicit computation of the covariant Laplacian on the cylindrical AdS background yields (\bar\nabla^{2}H=-\lambda/(\alpha^{2}r)). Using (\bar R=4\Lambda=-12\alpha^{2}) one verifies that the conformal scalar equation is satisfied without any additional source term. Thus the curvature of AdS itself acts as an effective source for the scalar field, and the same function that deforms the metric also solves the scalar field equation.

Charged black string.
When the string carries electric charge, the Einstein–Maxwell system introduces an electromagnetic energy‑momentum tensor. The KS decomposition still works, but the scalar function now contains a β‑dependent piece, and the gauge potential acquires an extra term proportional to β. Consequently the single copy includes not only the line‑charge electric field already described but also a distributed current reflecting the bulk electromagnetic field of the original solution. The authors show that the modified Maxwell equations are still satisfied on the AdS background, confirming that the double‑copy construction extends naturally to the charged case.

Conclusions and outlook.
The work demonstrates three key points: (i) a static cylindrical black string can be cast into a Kerr–Schild form on a curved (AdS) background; (ii) the corresponding single and zeroth copies are respectively a cylindrical electrostatic field sourced by an infinite line of colour charge and a conformally coupled scalar field obeying the appropriate curved‑space equation; (iii) the presence of bulk charge modifies both copies in a controlled way, yet the double‑copy correspondence remains intact. These results broaden the scope of the classical double copy beyond asymptotically flat spacetimes, showing that extended objects with non‑trivial topology (R × S¹) and non‑zero cosmological constant fit naturally into the framework. The paper suggests future extensions to rotating black strings, dynamical (time‑dependent) solutions, higher‑dimensional cylinders, and connections to holographic fluid‑gravity dualities, where the double‑copy language could provide new insights into the gauge‑gravity interplay in curved backgrounds.