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arxiv: 2410.04758 · v1 · pith:4FDND2FFnew · submitted 2024-10-07 · 🌀 gr-qc · math-ph· math.AP· math.MP

Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes

Weihao Zheng This is my paper

Pith reviewed 2026-05-23 20:04 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.APmath.MP
keywords hairy black holesanti-de Sitter spacetimeno-hair conjectureEinstein-Maxwell-Klein-Gordonholographic superconductorsspherically symmetric solutionsbifurcationReissner-Nordström-AdS
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The pith

One-parameter families of hairy black holes bifurcate from Reissner-Nordström-AdS spacetimes in anti-de Sitter space.

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

The paper constructs one-parameter families of static spherically symmetric asymptotically anti-de Sitter black hole solutions with a charged scalar field that satisfy the Einstein-Maxwell-Klein-Gordon equations. Each family branches off a sub-extremal Reissner-Nordström-AdS solution where the scalar field vanishes. For a co-dimensional one set of parameters, either Dirichlet or Neumann boundary conditions can be imposed on the scalar field. This yields a counter-example to a version of the no-hair conjecture with negative cosmological constant and supplies the first rigorous construction of holographic superconductors in the charged scalar case. The nonlinear construction rests on linear growing-mode instabilities established in a companion paper.

Core claim

We construct one-parameter families of static spherically symmetric asymptotically anti-de Sitter black hole solutions (M, g_ε, ϕ_ε) to the Einstein-Maxwell-(charged) Klein-Gordon equations. Each family bifurcates off a sub-extremal Reissner-Nordström-AdS spacetime (M, g_0, ϕ_0 ≡ 0). For a co-dimensional one set of black hole parameters, Dirichlet (respectively Neumann) boundary conditions can be imposed for the scalar field. The construction provides a counter-example to a version of the no-hair conjecture in the context of a negative cosmological constant.

What carries the argument

Nonlinear bifurcation construction that continues linear hair and growing-mode solutions from the Reissner-Nordström-AdS background into the nonlinear regime.

If this is right

  • The solutions provide counter-examples to a version of the no-hair conjecture for asymptotically anti-de Sitter spacetimes.
  • In the charged scalar case the families constitute the first rigorous mathematical examples of holographic superconductors.
  • Dirichlet or Neumann boundary conditions on the scalar field are admissible for a co-dimensional one set of black-hole parameters.
  • The hairy solutions exist as continuous one-parameter families branching from sub-extremal Reissner-Nordström-AdS backgrounds.

Where Pith is reading between the lines

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

  • Instabilities of Reissner-Nordström-AdS may generically lead to new static equilibrium configurations carrying scalar hair.
  • Analogous bifurcation techniques could be applied to other asymptotic boundary conditions or to non-spherically symmetric perturbations.
  • The result tightens the connection between classical black-hole instabilities and condensed-matter phase transitions in the holographic dictionary.

Load-bearing premise

Linear hair and growing-mode solutions exist on the Reissner-Nordström-Anti-de-Sitter black holes, as established in the companion paper.

What would settle it

An explicit calculation or numerical integration showing that the putative bifurcating branch of nonlinear solutions does not exist or fails to satisfy the field equations for the stated range of parameters.

read the original abstract

We construct one-parameter families of static spherically symmetric asymptotically anti-de Sitter black hole solutions $(\mathcal{M},g_{\epsilon},\phi_{\epsilon})$ to the Einstein-Maxwell-(charged) Klein-Gordon equations. Each family bifurcates off a sub-extremal Reissner-Nordstr\"om-AdS spacetime $(\mathcal{M},g_{0},\phi_{0}\equiv0)$. For a co-dimensional one set of black hole parameters, we show that Dirichlet (respectively Neumann) boundary conditions can be imposed for the scalar field. The construction provides a counter-example to a version of the no-hair conjecture in the context of a negative cosmological constant. Our result is based on our companion work [W. Zheng, \emph{Exponentially-growing Mode Instability on the Reissner-Nordstr\"om-Anti-de-Sitter black holes}], in which the existence of linear hair and growing mode solutions have been established. In the charged scalar field case, our result provides the first rigorous mathematical construction of the so-called holographic superconductors, which are of particular significance in high-energy physics.

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 one-parameter families of static spherically symmetric asymptotically AdS black hole solutions (M, g_ε, ϕ_ε) to the Einstein-Maxwell-(charged) Klein-Gordon equations. Each family bifurcates from a sub-extremal Reissner-Nordström-AdS spacetime (M, g_0, ϕ_0 ≡ 0) for a co-dimension-one set of parameters, allowing Dirichlet or Neumann boundary conditions on the scalar. The construction is presented as following from linear hair and growing-mode results in the companion paper by the same author, and is claimed to provide a counter-example to a version of the no-hair conjecture in AdS as well as the first rigorous construction of holographic superconductors.

Significance. If the linear spectral input holds and the bifurcation applies, the result would be significant: it supplies the first rigorous mathematical existence proof for hairy RN-AdS black holes with charged scalar hair and for holographic superconductors, which are of interest in AdS/CFT and high-energy physics. The use of bifurcation from linear modes to nonlinear solutions is a standard technique that, when the kernel and transversality conditions are verified, yields falsifiable predictions for the existence of such families.

major comments (2)
  1. [Abstract and Introduction] Abstract and Introduction: The central existence claim for the nonlinear families rests entirely on the linear instability and growing-mode results of the companion paper [W. Zheng] without any re-derivation, citation of specific spectral data (kernel dimension, simplicity of the zero eigenvalue), or verification of the transversality condition needed for a Crandall-Rabinowitz-type bifurcation theorem. This dependence is load-bearing for the stated one-parameter families and the boundary-condition claims.
  2. [Main construction (reliance on companion)] The manuscript asserts that the families remain asymptotically AdS and regular at the horizon under the chosen co-dimension-one parameter slice, but provides no explicit description or proof of how the bifurcation preserves these properties once the linear mode is given; all such steps are deferred to the companion without independent checks here.
minor comments (1)
  1. [Abstract] The abstract contains inline LaTeX commands (e.g., $g_ε$, ϕ_ε) that should be rendered consistently in the published version for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. Below we respond point-by-point to the major comments, clarifying the division of labor between the companion paper and the present manuscript while indicating where revisions will strengthen the exposition.

read point-by-point responses

  1. Referee: [Abstract and Introduction] Abstract and Introduction: The central existence claim for the nonlinear families rests entirely on the linear instability and growing-mode results of the companion paper [W. Zheng] without any re-derivation, citation of specific spectral data (kernel dimension, simplicity of the zero eigenvalue), or verification of the transversality condition needed for a Crandall-Rabinowitz-type bifurcation theorem. This dependence is load-bearing for the stated one-parameter families and the boundary-condition claims.

    Authors: The companion paper establishes the linear spectral data required by the Crandall-Rabinowitz theorem: a simple zero eigenvalue whose kernel is one-dimensional, together with the transversality (non-degeneracy) condition on the parameter slice. The present manuscript invokes the theorem on that basis and cites the companion for these facts. To make the dependence fully explicit, we will add direct references in the revised abstract and introduction to the precise statements (e.g., the theorem asserting simplicity of the zero eigenvalue and the lemma verifying transversality) from the companion work. revision: yes

  2. Referee: [Main construction (reliance on companion)] The manuscript asserts that the families remain asymptotically AdS and regular at the horizon under the chosen co-dimension-one parameter slice, but provides no explicit description or proof of how the bifurcation preserves these properties once the linear mode is given; all such steps are deferred to the companion without independent checks here.

    Authors: The bifurcation is performed in Banach spaces whose norms encode both the asymptotically AdS decay and regularity at the horizon. Because the linear mode belongs to these spaces and the nonlinear branch is obtained by a small perturbation in the same topology, the asymptotic and regularity properties are preserved by construction. A concise paragraph outlining this functional-analytic setting and its compatibility with the boundary conditions will be inserted in the construction section; the detailed verification of the function spaces themselves remains in the companion paper, which supplies the linear analysis. revision: partial

Circularity Check

1 steps flagged

Nonlinear bifurcation of hairy AdS black holes rests on linear mode existence from same-author companion paper

specific steps
  1. self citation load bearing [Abstract]
    "Our result is based on our companion work [W. Zheng, Exponentially-growing Mode Instability on the Reissner-Nordström-Anti-de-Sitter black holes], in which the existence of linear hair and growing mode solutions have been established."

    The nonlinear existence of the bifurcating families (M, g_ε, ϕ_ε) is asserted to follow from applying a bifurcation theorem at a simple zero eigenvalue. The required one-dimensional kernel and transversality condition are imported wholesale from the companion paper by the same author; the present manuscript provides no re-derivation or independent confirmation of these linear spectral facts under the stated boundary conditions.

full rationale

The paper's central claim is the existence of one-parameter families of nonlinear hairy black hole solutions bifurcating from RN-AdS via a static linear mode. It explicitly states that this construction is based on the companion paper's establishment of linear hair and growing modes, which supply the kernel and (presumably) transversality condition for the bifurcation theorem. No independent re-derivation or verification of these spectral inputs appears in the present text. This makes the load-bearing step a self-citation whose validity is not re-checked here, though the nonlinear analysis itself may contain independent content. The result is therefore partially dependent on the self-cited linear work rather than fully self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction depends on linear analysis from the companion paper and standard existence theorems for nonlinear PDEs; no free parameters or invented entities are explicitly introduced in the abstract beyond the bifurcation parameter ε.

axioms (1)
  • domain assumption Existence of linear hair and growing mode solutions on RN-AdS from companion paper
    Invoked to justify the bifurcation construction for the nonlinear solutions.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We construct one-parameter families of static spherically symmetric asymptotically anti-de Sitter black hole solutions (M,g_ε,ϕ_ε) to the Einstein-Maxwell-(charged) Klein-Gordon equations. Each family bifurcates off a sub-extremal Reissner–Nordström-AdS spacetime (M,g_0,ϕ_0≡0).

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Our result is based on our companion work [W. Zheng, Exponentially-growing Mode Instability on the Reissner–Nordström-Anti-de-Sitter black holes], in which the existence of linear hair and growing mode solutions have been established.

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matches
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extends
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uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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