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

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 $m^2 = 12/l^2$, 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.