Kerr-Schild Double Copy of the Randall-Sundrum Black String

Abstract: We construct the Kerr-Schild classical double copy of the black string in the Randall-Sundrum II model, deriving the single and zeroth copies, and verifying the associated field equations. The single copy gauge field is independent of the holographic coordinate and satisfies a sourceless Maxwell equation on the curved background, in direct analogy with the Coulomb field of the Schwarzschild double copy. The zeroth copy scalar obeys a modified Klein-Gordon equation with a first-order derivative term along the extra dimension; a field redefinition yields a standard Klein-Gordon equation with effective mass $m² = 12/l<sup>2$, induced by the warp factor. We further show that an alternative Kerr-Schild splitting, gravitationally equivalent to the canonical one, produces a physically inequivalent double copy: the gauge field is supported by a conserved but delocalized bulk current, and the zeroth copy satisfies a massless equation that carries no imprint of the warped extra dimension.

• The paper extends the classical double copy paradigm by mapping RSII black string solutions to gauge and scalar fields in a warped AdS₅ geometry.
• It employs the Kerr–Schild formalism to capture how the warp factor influences the localization and dynamics of the single and zeroth copies.
• The study demonstrates that ambiguity in the Kerr–Schild splitting critically impacts physical interpretations and guides proper double copy realizations.

Kerr–Schild Double Copy of the Randall–Sundrum Black String: An Expert Analysis

Introduction and Theoretical Context

This work conducts a detailed analysis of the Kerr–Schild double copy construction for black string solutions in the five-dimensional Randall–Sundrum II (RSII) braneworld scenario. Central to this effort is the extension of the classical double copy paradigm, originally rooted in the Bern–Carrasco–Johansson (BCJ) color-kinematics duality, to curved backgrounds featuring warped extra dimensions and exact, analytically tractable gravitational solutions (2604.05447).

The RSII model provides an anti-de Sitter (AdS$_5$) bulk geometry with gravity localized on a four-dimensional brane via a non-compact, warped extra dimension, and the associated black string solution is a direct extension of the four-dimensional Schwarzschild metric along this extra dimension [Chamblin et al., hep-th/9909205]. The double copy program, and specifically the Kerr–Schild formalism, is particularly amenable to this context, enabling the systematic association of exact gauge field and scalar field configurations (“single” and “zeroth” copies, respectively) to gravitational backgrounds.

A key motivation is understanding how the presence of a warped fifth dimension manifests in the single and zeroth copies, and how the ambiguity inherent in the Kerr–Schild ansatz impacts the physical content of the double copy in higher-dimensional warped spacetime.

Kerr–Schild Structure and Double Copy Formalism in AdS$_5$ Bulk

The AdS$_5$ bulk metric in the RSII model adopts a conformal coordinate system:

$$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$

with brane at $z = l$ and the $z$ coordinate extending to infinity. The black string solution is straightforwardly constructed by replacing the Minkowski sector with the four-dimensional Schwarzschild solution, preserving the underlying conformal structure.

Recasting the black string in ingoing Eddington–Finkelstein coordinates, the metric is expressed in canonical Kerr–Schild form:

$g_{MN} = \bar{g}_{MN} + \phi\,k_M k_N,$

where $\bar{g}_{MN}$ is the AdS$_5$ background, $\phi = \frac{2M}{r} \frac{z^2}{l^2}$, and $_5$0 for Eddington–Finkelstein retarded time $_5$1. This precisely mirrors the classical double copy construction for Schwarzschild in four dimensions [Monteiro et al., (Monteiro et al., 2014)], but incorporates the explicit dependence on the holographic coordinate via the AdS warp factor.

Single Copy: Maxwell Solutions and Bulk Localization

The canonical single copy gauge field, $_5$2, reduces to

$_5$3

with all components independent of the AdS extra dimension $_5$4. The field strength has nonzero component only $_5$5. The associated equations of motion, derived by contraction of the linearized Einstein equations with an appropriate Killing vector, yield a sourceless Maxwell equation on the AdS$_5$6 background:

$_5$7

where $_5$8 is the AdS$_5$9 covariant derivative and the only source is a point-like charge at $_5$0. Notably, the solution exhibits no support outside the $_5$1 singularity and does not encode $_5$2-dependent bulk structure beyond the overall metric.

Key result: The single copy is not dynamically localized to the brane, and its $_5$3-independence reflects the direct absorption of the warp factor by the Kerr–Schild parametrization. This underscores that gauge field localization in RSII requires non-gravitational mechanisms, in contrast to the localization of gravity [Davoudiasl et al., hep-ph/9911262; Pomarol, hep-ph/9911294].

Zeroth Copy: Modified Scalar Field Dynamics

The zeroth copy, extracted via a further contraction, is the scalar $_5$4. The resulting equation of motion for $_5$5 is a modified Klein–Gordon equation with an explicit first-order derivative in $_5$6:

$_5$7

indicating direct coupling to the extra dimension via the AdS warp factor. A field redefinition

$_5$8

removes the first-order term and recasts the equation as a standard Klein–Gordon equation with effective mass $_5$9:

$$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$0

Key implication: The physically relevant, normalizable scalar mode $$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$1 localizes near the brane, in analogy with gravitational zero modes in the RSII scenario. The mass parameter induces a holographic CFT dual interpretation with conformal dimension $$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$2 [Maldacena, hep-th/9711200].

Physical Ambiguity: Alternative Kerr–Schild Splittings

The paper carefully investigates the ambiguity in the Kerr–Schild ansatz, parameterized by local rescalings $$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$3, $$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$4, which leave the metric unchanged but affect the single and zeroth copies. An explicit alternative splitting is constructed:

$$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$5

The corresponding single copy $$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$6 and field strengths now carry explicit $$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$7-dependence, but the gauge field is no longer sourced solely at $$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$8. Instead, the Maxwell equations are supported by a conserved, delocalized bulk current:

$$ds^2 = \frac{l^2}{z^2} \left(-dt^2 + dr^2 + r^2 d\Omega_2^2 + dz^2\right),$$9

The associated zeroth copy $z = l$0 is independent of $z = l$1, and satisfies the five-dimensional massless Klein–Gordon equation, carrying no imprint of the warped extra dimension.

This establishes that, although the gravitational solution remains unchanged, the physics encoded in the single and zeroth copies depends crucially on the Kerr–Schild realization. The criterion of absence of delocalized sources in the single copy favors the canonical splitting as the physically meaningful double copy realization [Carrillo-González et al., (Carrillo-Gonzalez et al., 2017)]. The alternative splitting fails to reproduce the expected localization and holographic structure.

Implications, Relationship to Holography, and Future Perspectives

The results concretely demonstrate that the Kerr–Schild double copy in warped extra dimensional backgrounds is sensitive to both the geometric and physical features of the bulk. The explicit warping and presence of the extra dimension manifests in scalar field dynamics and normalizability, with a direct mapping to CFT operator dimensions via AdS/CFT [Maldacena, hep-th/9711200].

The constructed formalism paves the way for analyzing more intricate braneworld black object solutions, such as rotating or charged black holes [Aliev & Gumrukcuoglu, hep-th/0502223; Neves & Molina, (Neves et al., 2012)], with possible connections to tidal charge and higher curvature corrections. The results have implications for understanding how extra-dimensional features propagate through the double copy into effective gauge and scalar field theory descriptions.

The impact of classical instabilities (Gregory–Laflamme, hep-th/9301052) and their potential double copy signatures is also highlighted as a promising avenue. Furthermore, the relation of these results to Weyl double copy constructions in higher dimensions and their holographic boundary CFT correspondence remains to be elucidated [Alkac et al., (Alkac et al., 2021); Luna et al., (Luna et al., 2018)].

Conclusion

This study provides a comprehensive extension of the Kerr–Schild classical double copy formalism to five-dimensional warped geometries, specifically the RSII black string, demonstrating that physical properties such as source localization and effective mass parameters in the single and zeroth copies are tightly constrained by the choice of Kerr–Schild decomposition (2604.05447). The work clarifies the encoding of bulk extra-dimensional structure in double copy constructions and establishes a set of physicality criteria necessary for a correct field-theoretic interpretation. These insights inform further explorations of double copy correspondences in warped geometries, holographic systems, and higher-dimensional gravity.