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arxiv: 2604.05447 · v1 · submitted 2026-04-07 · ✦ hep-th · gr-qc

Recognition: 2 theorem links

· Lean Theorem

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

Jes\'us A. Rodr\'iguez
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:06 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Kerr-Schild double copyRandall-Sundrum modelblack stringwarped extra dimensionsclassical double copyMaxwell equationKlein-Gordon equation
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The pith

The Randall-Sundrum II black string admits a Kerr-Schild double copy in which the single-copy gauge field is independent of the extra dimension and satisfies a sourceless Maxwell equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the classical double copy of the black string solution in the Randall-Sundrum II model using the Kerr-Schild ansatz. It derives the single-copy gauge field and zeroth-copy scalar, then verifies that both satisfy their respective field equations on the warped background. The single copy remains unchanged along the holographic coordinate, in direct analogy with the Coulomb field of the Schwarzschild solution. The zeroth copy obeys a modified Klein-Gordon equation that, after a field redefinition, becomes a standard massive equation with effective mass squared equal to 12 over l squared induced by the warp factor. An alternative but gravitationally equivalent Kerr-Schild splitting produces an inequivalent double copy whose gauge field is sourced by a conserved yet delocalized bulk current and whose scalar remains massless.

Core claim

The authors apply the Kerr-Schild ansatz directly to the Randall-Sundrum II black string metric and obtain its classical double copy. The resulting single copy is a gauge field independent of the holographic coordinate that satisfies a sourceless Maxwell equation on the curved background. The zeroth copy is a scalar obeying a modified Klein-Gordon equation containing a first-order derivative term along the extra dimension; a redefinition converts it to an ordinary massive Klein-Gordon equation with m squared equals 12 over l squared. They further exhibit an alternative Kerr-Schild decomposition that is gravitationally equivalent yet yields a physically different double copy: the gauge field,

What carries the argument

The Kerr-Schild ansatz applied to the RS II black string metric, which splits the geometry into a background term plus a null vector squared term and thereby maps the solution to a single-copy gauge field and zeroth-copy scalar.

If this is right

  • The single-copy gauge field is independent of the holographic coordinate and obeys a sourceless Maxwell equation.
  • The zeroth-copy scalar acquires an effective mass m squared equals 12 over l squared from the warp factor after a field redefinition.
  • An alternative but gravitationally equivalent Kerr-Schild splitting produces a gauge field sourced by a conserved delocalized bulk current and a massless scalar.
  • The field equations of both copies can be verified directly on the curved warped geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The independence of the single copy from the holographic direction suggests that double-copy constructions may preserve certain symmetries even when the gravitational solution lives in a warped bulk.
  • The effective mass induced in the zeroth copy by the warp factor offers a concrete example of how extra-dimensional geometry can appear as mass terms in the lower-dimensional copy fields.

Load-bearing premise

The Kerr-Schild ansatz and the classical double copy map can be applied directly to the warped RS II black string without additional corrections arising from the extra dimension or the brane.

What would settle it

An explicit computation showing that the single-copy gauge field depends on the holographic coordinate or fails to satisfy the sourceless Maxwell equation on the warped background would disprove the central claim.

read the original 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 $m^2 = 12/l^2$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper constructs 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 warped background, in analogy with the Schwarzschild Coulomb field. The zeroth copy scalar obeys a modified Klein-Gordon equation with a first-order derivative term; a field redefinition yields a standard Klein-Gordon equation with effective mass m² = 12/l² induced by the warp factor. An alternative but gravitationally equivalent Kerr-Schild splitting is shown to produce a physically inequivalent double copy featuring a delocalized bulk current and a massless zeroth copy.

Significance. If the verifications are complete, this extends the classical double copy to warped extra-dimensional geometries and demonstrates how the RS warp factor imprints an effective mass on the zeroth copy. The explicit construction of two inequivalent double copies from equivalent metrics is a useful observation on the non-uniqueness of the map in such backgrounds, potentially relevant for holographic or braneworld applications of the double copy.

major comments (2)
  1. [§4 (field equation verification)] The verification that the single copy satisfies the sourceless Maxwell equation on the 5D warped background (stated in the abstract and presumably detailed in the section deriving the single copy) is central to the analogy with the Schwarzschild case, yet the manuscript provides no explicit intermediate steps, component calculations, or checks for extra-dimensional source terms arising from the warp factor. This makes it difficult to confirm the claim without hidden assumptions.
  2. [Discussion of alternative splitting] The demonstration of an alternative Kerr-Schild splitting that is gravitationally equivalent but yields a gauge field with delocalized bulk current and a massless zeroth copy (final section) indicates sensitivity to the decomposition choice. This raises a load-bearing question for the central claim: additional criteria must be provided to justify preferring the canonical splitting as the physically relevant double copy in the warped RS II background, rather than treating the map as uniquely determined by the metric.
minor comments (1)
  1. [Notation] The notation for the holographic coordinate and the AdS radius l should be checked for consistency across all equations and text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of its significance, and constructive major comments. We address each point below and will incorporate revisions to improve clarity and address the concerns raised.

read point-by-point responses

  1. Referee: [§4 (field equation verification)] The verification that the single copy satisfies the sourceless Maxwell equation on the 5D warped background (stated in the abstract and presumably detailed in the section deriving the single copy) is central to the analogy with the Schwarzschild case, yet the manuscript provides no explicit intermediate steps, component calculations, or checks for extra-dimensional source terms arising from the warp factor. This makes it difficult to confirm the claim without hidden assumptions.

    Authors: We agree that explicit intermediate steps are necessary for full transparency. In the revised manuscript we will add a new subsection (or appendix) containing the component-by-component verification of the sourceless Maxwell equation in the 5D warped geometry. This will explicitly demonstrate that all warp-factor contributions cancel for the chosen single-copy gauge field, with no residual extra-dimensional source terms. The calculations have already been performed and confirm the abstract claim. revision: yes

  2. Referee: [Discussion of alternative splitting] The demonstration of an alternative Kerr-Schild splitting that is gravitationally equivalent but yields a gauge field with delocalized bulk current and a massless zeroth copy (final section) indicates sensitivity to the decomposition choice. This raises a load-bearing question for the central claim: additional criteria must be provided to justify preferring the canonical splitting as the physically relevant double copy in the warped RS II background, rather than treating the map as uniquely determined by the metric.

    Authors: The referee correctly identifies that the paper highlights the non-uniqueness of the Kerr-Schild decomposition. To strengthen the discussion, we will expand the final section to articulate explicit criteria for preferring the canonical splitting: it yields a sourceless single copy obeying the homogeneous Maxwell equations (directly analogous to the Schwarzschild case) and a zeroth copy that carries the warp-factor imprint via an effective mass. The alternative splitting, while gravitationally equivalent, introduces a delocalized bulk current; we will argue that this makes it less natural for the double-copy map in the RS II context. We will present this as a physically motivated preference rather than a claim of uniqueness. revision: yes

Circularity Check

0 steps flagged

Kerr-Schild double copy for RS II black string is self-contained with no circular steps

full rationale

The paper begins from the known Randall-Sundrum II black string metric (an established solution) and applies the standard Kerr-Schild ansatz to decompose it into single and zeroth copies. The single-copy gauge field is obtained by direct identification and shown to satisfy the sourceless Maxwell equation on the warped background by explicit substitution and verification. The zeroth-copy scalar equation is derived from the same decomposition; the first-order derivative term is removed by a field redefinition whose resulting effective mass m² = 12/l² follows algebraically from the warp factor in the input metric. An alternative splitting is also exhibited, confirming that the construction does not rely on a unique or self-referential choice. No parameter is fitted to data, no result is renamed as a prediction, and no load-bearing premise reduces to a self-citation or prior ansatz of the authors. The derivation chain is therefore independent of its target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard Kerr-Schild ansatz applied to the known RS II black string metric and the classical double copy dictionary; no new free parameters are introduced beyond the model warp scale l, and no new entities are postulated.

axioms (2)
  • domain assumption The Kerr-Schild form can be used for the RS II black string metric
    Invoked in the opening construction statement of the abstract.
  • domain assumption The classical double copy map extends to this curved, warped background
    Underlying the derivation of single and zeroth copies and the verification of their equations.

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Reference graph

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