arxiv: 2605.04145 · v1 · submitted 2026-05-05 · ✦ hep-th · gr-qc

Recognition: 3 theorem links

· Lean Theorem

When AdS₃ Grows Hair: Boson Stars, Black Holes, and Double-Trace Deformations

David Sola Gil, Jorge E. Santos, Oscar J. C. Dias

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:04 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords AdS3 gravityboson starshairy black holesdouble-trace deformationsBTZ black holesinstability endpointmicrocanonical ensemblephase diagram

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The pith

Global AdS3 with double-trace deformation below a critical value becomes unstable to a zero-frequency boson star that serves as the true ground state.

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

The paper constructs regular boson stars and both axisymmetric and non-axisymmetric hairy black holes in three-dimensional Einstein gravity coupled to a massive complex scalar field subject to double-trace boundary conditions. It demonstrates that when the deformation parameter lies below the AdS threshold, empty global AdS3 is linearly unstable and its nonlinear endpoint is a static boson star whose total energy is lower than that of AdS3. Hairy black holes branch from the BTZ family precisely at the onset of each azimuthal instability. In the microcanonical ensemble the hairy black holes always carry higher entropy than BTZ black holes at the same mass and angular momentum, so they dominate the thermodynamics whenever they exist.

Core claim

When the double-trace parameter satisfies κ less than κ_AdS, global AdS3 becomes unstable and its nonlinear endpoint is a zero-frequency boson star with energy below that of AdS3, thereby providing the true ground state of the theory. Axisymmetric and non-axisymmetric hairy black holes bifurcate from the BTZ family at the corresponding double-trace instability onset. In the microcanonical ensemble hairy black holes always carry greater entropy than BTZ at fixed mass and angular momentum and thus dominate whenever they exist. In the singular extremal limit axisymmetric black holes saturate a generalised minimum-energy theorem under double-trace boundary conditions.

What carries the argument

The double-trace deformation parameter κ that sets modified boundary conditions for the scalar field, allowing families of regular, horizonless boson stars and hairy black holes to exist as nonlinear solutions.

Load-bearing premise

The instability of global AdS3 evolves to the static zero-frequency boson star rather than to a different time-dependent or singular configuration.

What would settle it

A direct numerical time evolution of perturbed global AdS3 initial data under the double-trace conditions that either settles to the static boson star solution or evolves to a different final state.

read the original abstract

We analyse three-dimensional Einstein gravity coupled to a massive complex scalar field with double-trace boundary conditions. Using high-precision spectral methods, we construct regular AdS$_3$ boson stars together with axisymmetric and non-axisymmetric hairy black holes. For each azimuthal number $m$, the hairy black holes bifurcate from the BTZ family at the corresponding double-trace instability onset. When the double-trace parameter satisfies $\kappa < \kappa_{\rm AdS}$, global AdS$_3$ becomes unstable and we identify its nonlinear endpoint as a zero-frequency boson star with energy below that of AdS$_3$, thereby providing the true ground state of the theory. In the microcanonical ensemble, hairy black holes always carry greater entropy than BTZ at fixed mass and angular momentum, and thus dominate whenever they exist. With notable exceptions, typically hairy black holes do not dominate the canonical nor the grand-canonical ensembles. We further show that, in the singular extremal limit, axisymmetric black holes saturate a generalised minimum-energy theorem under double-trace boundary conditions. These results yield the full nonlinear phase diagram of AdS$_3$ gravity with double-trace deformations.

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.

Desk Editor's Note private letter to a colleague

This paper numerically fills out the nonlinear phase diagram for AdS3 gravity with double-trace boundary conditions, constructing boson stars and hairy black holes while comparing ensembles, though the endpoint claim rests on static solutions.

read the letter

The paper maps out the nonlinear phase diagram for AdS3 with double-trace boundary conditions. They construct regular boson stars and hairy black holes, both axisymmetric and non-axisymmetric, and compare them across ensembles. What stands out is the use of high-precision spectral methods to build these solutions. The hairy black holes bifurcate from the BTZ family at the onset of the double-trace instability for each azimuthal number. When the deformation parameter is below the critical value, global AdS3 is unstable, and they point to a zero-frequency boson star as the endpoint because it has lower energy. In the microcanonical ensemble, the hairy black holes always have more entropy than BTZ at the same mass and angular momentum, so they dominate there. In other ensembles, it's more mixed. This fills in the nonlinear side of the story. Earlier papers likely covered the linear instability, but here we get the actual solutions and the full diagram, including how they compare in different thermodynamic ensembles. They also verify a generalized minimum-energy theorem in the extremal limit. The numerics appear solid for what they do. The main limitation is that the endpoint identification relies on the static boson star existing with lower energy and being regular. Without explicit time-dependent evolution of the instability under these boundary conditions, it's not shown that this is where the system goes rather than to a different configuration. That's the inference from the static picture. If the full paper includes any dynamical evidence or strong arguments against alternatives, that would help. This is for researchers in three-dimensional gravity and holography, especially those looking at relevant deformations and their effects on stability and thermodynamics. Anyone building phase diagrams in AdS gravity will find the constructions and comparisons useful. It deserves a serious referee because the numerical work is nontrivial and the results are concrete. I would send it to peer review. The constructions are worth checking in detail, and the ensemble analysis adds value.

Referee Report

2 major / 2 minor

Summary. The paper studies 3D Einstein gravity coupled to a massive complex scalar with double-trace boundary conditions. Using high-precision spectral methods, it constructs regular boson stars and axisymmetric/non-axisymmetric hairy black holes. For each m, hairy BHs bifurcate from BTZ at the double-trace instability threshold. When κ < κ_AdS, global AdS3 is unstable; its nonlinear endpoint is identified as a zero-frequency boson star of lower energy, taken as the true ground state. Hairy BHs dominate the microcanonical ensemble at fixed M,J but typically do not dominate canonical or grand-canonical ensembles. In the singular extremal limit, axisymmetric solutions saturate a generalized minimum-energy theorem. The work yields the full nonlinear phase diagram.

Significance. If the central claims hold, the manuscript supplies the complete nonlinear phase structure of AdS3 gravity under double-trace deformations, including the ground state and ensemble dominance. The high-precision spectral constructions, explicit bifurcation from BTZ, and energy comparisons constitute a clear technical advance; the minimum-energy theorem in the extremal limit is a further strength.

major comments (2)

[Abstract / phase-diagram discussion] Abstract and the section presenting the phase diagram: the claim that the zero-frequency boson star constitutes the nonlinear endpoint of the AdS3 instability (for κ < κ_AdS) rests on the existence of the static solution, its lower energy relative to AdS3, and the linear instability threshold, but is not supported by explicit time-dependent evolution of the Einstein-scalar system under the double-trace boundary conditions. This inference is load-bearing for the central claim about the true ground state.
[Numerical methods / boson-star constructions] Section on numerical constructions: while high-precision spectral methods are used, the manuscript does not report explicit convergence tests, error bars, or direct comparisons against known analytic limits (e.g., the massless or κ=0 cases) in the main text or appendices; such checks are necessary to underwrite the reported bifurcation points and energy ordering.

minor comments (2)

[Introduction] Notation for the double-trace parameter κ and the critical value κ_AdS should be defined once in the introduction with an explicit equation reference rather than only in the abstract.
[Figures] Figure captions for the phase diagrams should include the precise values of m and κ ranges shown, together with a brief statement of the numerical resolution employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for recognizing the technical contributions of our work on the nonlinear phase structure of AdS3 gravity with double-trace deformations. We address each major comment below and outline the revisions we will implement to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract / phase-diagram discussion] Abstract and the section presenting the phase diagram: the claim that the zero-frequency boson star constitutes the nonlinear endpoint of the AdS3 instability (for κ < κ_AdS) rests on the existence of the static solution, its lower energy relative to AdS3, and the linear instability threshold, but is not supported by explicit time-dependent evolution of the Einstein-scalar system under the double-trace boundary conditions. This inference is load-bearing for the central claim about the true ground state.

Authors: We agree that our identification of the zero-frequency boson star as the nonlinear endpoint is an inference drawn from the existence of a static solution with lower energy than global AdS3, its bifurcation structure matching the linear instability, and the absence of other static competitors. Explicit time-dependent simulations of the Einstein-scalar system with the double-trace boundary conditions would provide direct dynamical confirmation, but such evolutions are computationally intensive and lie beyond the scope of the present work, which is devoted to the construction and thermodynamic analysis of static solutions. We will revise the abstract and the phase-diagram discussion to qualify the statement as a well-supported conjecture based on the static and linear evidence, while emphasizing that the core results—the construction of the boson stars and hairy black holes, their bifurcation from BTZ, and the microcanonical dominance—remain unaffected. revision: partial
Referee: [Numerical methods / boson-star constructions] Section on numerical constructions: while high-precision spectral methods are used, the manuscript does not report explicit convergence tests, error bars, or direct comparisons against known analytic limits (e.g., the massless or κ=0 cases) in the main text or appendices; such checks are necessary to underwrite the reported bifurcation points and energy ordering.

Authors: We thank the referee for this observation. Although the spectral method is designed for high accuracy, we acknowledge that the main text and appendices currently lack explicit documentation of convergence with respect to the number of Chebyshev modes, residual norms, and direct comparisons to the massless scalar limit or the κ=0 case (where analytic boson-star solutions are available). We will add a new subsection in the numerical methods section together with an appendix containing these tests, including tables of error estimates, convergence rates, and validation against known limits. This will directly support the reported bifurcation thresholds and energy comparisons. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions are independent numerical solutions

full rationale

The paper solves the Einstein-scalar system with double-trace boundary conditions via high-precision spectral methods to construct static boson stars and hairy black holes. Linear instability thresholds are computed separately and used only to locate bifurcations; energies are compared directly from the solutions. No step equates a reported prediction to a fitted parameter by construction, nor does any load-bearing claim reduce to a self-citation chain or ansatz smuggled from prior work. The derivation remains self-contained against the field equations and boundary conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Einstein equations in three dimensions with negative cosmological constant and on the assumption that regular, asymptotically AdS solutions exist for the chosen boundary conditions; no new free parameters are fitted and no new entities are postulated.

axioms (2)

standard math Einstein gravity in 3D with negative cosmological constant coupled to a massive complex scalar
Standard setup invoked throughout the abstract for the field equations.
domain assumption Existence and regularity of static axisymmetric and non-axisymmetric solutions under double-trace boundary conditions
Required for the numerical construction of boson stars and hairy black holes.

pith-pipeline@v0.9.0 · 5515 in / 1333 out tokens · 33573 ms · 2026-05-08T18:04:23.841355+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

Foundation/RealityFromDistinction (spacetime emergence: signature (1,3)) reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We analyse three-dimensional Einstein gravity coupled to a massive complex scalar field with double-trace boundary conditions. ... we identify its nonlinear endpoint as a zero-frequency boson star with energy below that of AdS3, thereby providing the true ground state of the theory.
Cost.FunctionalEquation / AlphaCoordinateFixation washburn_uniqueness_aczel / J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

β = κ α, where κ is, a priori, a real constant. These boundary conditions correspond holographically to double-trace deformations of the dual CFT operator.
Constants (parameter-free chain) constants_from_phi_ladder unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

For concreteness, we focus on the case µ²L² = −15/16 ... within the range −1 < µ²L² < 0 where double-trace boundary conditions of the form (2.22) are allowed.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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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