Phase Transition to a Hairy Black Hole in Asymptotically Flat Spacetime

Eleftherios Papantonopoulos; George Koutsoumbas; George Siopsis; Theodoros Kolyvaris

REVIEW

Phase Transition to a Hairy Black Hole in Asymptotically Flat Spacetime

Theodoros Kolyvaris, George Koutsoumbas, Eleftherios Papantonopoulos, George Siopsis

High Energy Physics - Theory · Submitted 24 Aug 2013 · 44 references

Pith's verdict: A real hairy-black-hole construction in asymptotically flat space, but the printed asymptotic scalar profile has a typo that makes the shooting irreproducible as written.

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CitationEleftherios Papantonopoulos, George Koutsoumbas, George Siopsis, Theodoros Kolyvaris. "Phase Transition to a Hairy Black Hole in Asymptotically Flat Spacetime." arXiv:1308.5280 (2013). https://pith.science/paper/LO3XRFLD

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Editor's note

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The core result is real: in a Horndeski-type model with a charged scalar derivatively coupled to the Einstein tensor, a regular spherically symmetric hairy black hole exists for kappa above a threshold, with the hair trapped near the horizon by a potential well that kappa itself creates. That is a clean evasion of the no-hair theorem in flat space, and the mechanism — kappa playing the role of the AdS scale — is worth knowing about. What is actually new relative to the authors' prior work [36] is the O(epsilon^2) backreaction: l2 in closed form (32), alpha2 and At2 numerically, and the corrections to M, Q, T in (44)-(46). The effective-mass calculation (25) is clean and checkable, and the exponential decay e^{-nu r} is a genuine qualitative difference from the AdS power law. Now the soft spots. The stress-test concern lands: Eq. (27) writes phi0 ~ z^{1+2 mu xi} e^{-mu nu z} ~ r^{-(1+2 mu xi)} e^{-nu r}, but the power law needs z ~ 1/r while the exponential needs z ~ r. Direct z-to-0 analysis of (16) gives phi ~ e^{-mu nu/z}, not e^{-mu nu z}. So the printed asymptotic form and the shooting ansatz (28)-(29) are inconsistent with chi(0)=1. This is almost certainly a typo — the physics (exponential decay in r) is right, and the fix is e^{-mu nu/z} — but as printed, Fig. 2 cannot be reproduced. That is a real reproducibility gap on the load-bearing numerical step, and it compounds the z-versus-r coordinate ambiguity the paper already has: it says z = mu r but treats z-to-0 as infinity, which requires z = mu/r. Two smaller things. The abstract says analytically, but the solution is a hybrid: l2 is analytic, alpha2 and At2 come from numerical shooting with no convergence or precision reported. And the phase-transition language rests on the free-energy comparison in [36]; this paper constructs the branch but does not redo the thermodynamics. The authors are upfront about the latter, so it is a scope note, not a hidden gap. Who should read it: people working on hairy black holes, Horndeski/Galileon black holes, and anyone thinking about flat-space holography. The model is well-motivated and the construction is careful. Recommendation: send it to referees. The result deserves scrutiny, and the main ask is a corrected asymptotic form and a note on numerical convergence.

Referee report

Significance

If the construction holds, this is a clean and physically interesting example of no-hair evasion in asymptotically flat space without a cosmological constant: the dimensionful coupling κ supplies a confining scale that plays a role analogous to the AdS radius, and the resulting hair is exponentially localized near the horizon rather than power-law. That is a genuinely distinctive signature, and the explicit display of the full field equations, the perturbative ansatz, and the O(ε²) backreaction is a real strength of the presentation. The paper is a follow-up to the authors' earlier work [36] on the anisotropic case; the spherically symmetric configuration is the new element here. The result is a theoretical construction supported by derivation and numerical shooting rather than by machine-checked proof or released code, and it should be assessed on that basis.

Major objections (2)

[§III.A, Eqs. (27)-(29)] The asymptotic form quoted in Eq. (27), φ0(z) ~ z^{1+2μξ} e^{-μνz} ~ r^{-(1+2μξ)} e^{-νr}, cannot hold as written. The paper's own coordinate relation is z = μ/r: the text states that z ≲ 0.2 corresponds to r ≳ 5μ and that the horizon sits at r = μ, both of which require z = μ/r, not z = μr. Under z = μ/r one has e^{-μνz} = e^{-μ²ν/r} → 1 as r → ∞, not e^{-νr}. Taking z → 0 directly in Eq. (16) with α → 0, l → 1, A → 1 and m²_eff,0 → ν² gives z⁴φ'' - μ²ν²φ = 0, whose decaying solution is e^{-μν/z}, not e^{-μνz}. The correct asymptotic form is therefore φ0 ~ z^{1+2μξ} e^{-μν/z} ~ r^{-(1+2μξ)} e^{-νr}. As printed, the shooting ansatz (28) and the boundary condition (29) are not reproducible: with the printed exponential, χ = φ0/(z^{1+2μξ} e^{-μνz}) would behave as e^{-μν/z + μνz} and vanish as z → 0, contradicting χ(0) = 1. This is very likely a typographical error and the numerical code presumably used the correct form, but because the existence of the regular solution (Fig. 2) is the load-bearing step of the paper, Eqs. (27)-(28) must be corrected and the authors should state explicitly that the code used e^{-μν/z}. The same passage also contains an inconsistent coordinate definition ('r defined by z = μr' in §II versus 'r = μz → ∞' two paragraphs later); this should be harmonized to z = μ/r throughout.
[§III.B, Eq. (46)] The expansion parameter ε is introduced in Eq. (19) as a formal bookkeeping device, and the paper never states its relation to the temperature. Eq. (46) implicitly supplies one, T/T0 = 1 - ε²τ, but neither the value nor the sign of τ is reported for the sample parameters of Fig. 2, so the reader cannot check that the hairy branch actually lies below the critical temperature (τ > 0) rather than above it. Since 'near the transition temperature' is the paper's stated regime, this is a load-bearing omission: please quote τ, and hence ε² = (1 - T/T0)/τ, for the parameters used in Figs. 2 and 4-6, and state the numerical value of ε employed. Relatedly, Eq. (38) gives δQ/Q = -Cε² with C = 8.0, so the perturbative treatment requires ε² ≪ 1/8; this bound should be stated and checked against the ε implied by the quoted temperature deviation. Without these numbers the O(ε²) results (44)-(46) are not tied to any physical distance from the critical point, and the claim of a controlled near-critical expansion remains an assumption rather than a demonstrated property.

Minor objections (9)

[§II, Eq. (22)] As typeset, Eq. (22) gives At0(1) = 2e^{-B} - 8 sinh B + 2 cosh B, which vanishes only for e^{2B} = 7/3 and would generically violate regularity of the gauge field at the horizon; moreover -At0'(0) = 4 sinh B - 2 cosh B does not reproduce Q/μ = 4/sinh B quoted in Eq. (23). Please check the typeset form of this equation.
[§III.A, Fig. 2 caption] The caption gives M = 3.187, Q = 1.048 while the surrounding text gives M = 3.20, Q = 1.05; these should be reconciled or the difference explained.
[§III.A, Fig. 3] Fig. 3 uses m = 1.0, q = 0.5, whereas Figs. 1, 2, 4, 5 and 6 use m = 0.3, q = 0.5; the caption should make clear that the critical curves are not computed for the same scalar parameters as the profile plots.
[Abstract] The abstract states that the equations are solved 'analytically near the transition temperature', but the construction requires numerical shooting for the zeroth-order profile and for α2; the analytic content is the asymptotic expansion (29) and the integral representation (32). Consider rewording to reflect the mixed analytic/numerical nature of the solution.
[§III.B, Eq. (43)] The derivation states that the left-hand side vanishes 'on account of (16) and (25)', but Eq. (25) defines A0 and m²_eff,0 rather than the zeroth-order wave equation; presumably the last equation of (20) is intended.
[§III.A, footnote 1] The statement that negative κ renders the system unstable and admits no solutions is asserted here without demonstration; it rests on [36] and on [13], and the text should say so explicitly.
[§III.A, Fig. 1 discussion] The claim that for κ = 100, 170 the wave equation has no solution because the curve lies above the axis is correct, but only because a strictly positive potential admits no zero-energy bound state; a one-sentence justification would make the argument self-contained.
[§III.B, Figs. 4-6] The value of ε used to produce these plots is never stated; the plotted functions are coefficients of ε², and this should be said explicitly so that the figures are not read as physical O(ε²) corrections.
[Throughout] There are several typographical errors, including 'absense' and 'guaranties' on p. 7, 'solutoin' on p. 8, and 'exponent ially' in the abstract.

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The paper

The paper claims

The paper argues that a Reissner-Nordström black hole in asymptotically flat spacetime can undergo a phase transition to a spherically symmetric hairy black hole when a charged scalar field is derivatively coupled to the Einstein tensor by

The reading

The paper asks whether a black hole in empty, asymptotically flat space can carry scalar hair, which the standard no-hair theorem forbids. Its answer is yes, provided the scalar field couples to the Einstein tensor with a large enough positive strength $c$: the coupling creates a potential well just outside the horizon that can trap charged scalar particles. The authors work perturbatively near the transition temperature, treating the Reissner-Nordström geometry as the background and the scalar amplitude as a small parameter, and solve the coupled Einstein-Maxwell-scalar equations order by order. They find a regular spherically symmetric branch whose hair is concentrated within a few horizon radii and decays exponentially outward, and they compute the leading corrections to the black hole's mass, charge, and temperature. If correct, this shows that a cosmological constant is not required to evade the no-hair theorem; a dimensionful derivative coupling can supply the needed scale.

What carries the argument

The load-bearing mechanism is the derivative coupling of the scalar kinetic term to the Einstein tensor, written in the action as $-(g^{muu}-c G^{muu})D_muphi(D_uuphi)^*$. When the background is Reissner-Nordström, this coupling modifies the Klein-Gordon equation into an effective-mass equation with $m_{m eff}^2$ given in Eq. (25). For $c=0$ the effective mass approaches a constant at infinity and, when negative, allows the scalar to escape rather than condense; for $c>0$ the extra terms produce a finite potential well outside the horizon. A regular bound state in that well is the zeroth-order hair. The perturbative expansion $phi=epsilonphi_0+cdots$, $alpha=alpha_0+epsilon^2alpha_2+cdots$, $l=l_0+epsilon^2 l_2+cdots$, $A_t=A_{t0}+epsilon^2 A_{t2}+cdots$ then converts the nonlinear problem into a sequence of linear equations, with the $O(epsilon^2)$ scalar equation fixing the integration constant $C$ and the metric correction $alpha_2$.

Extended reading notes

Core claim

The central claim is that, for a charged scalar field with a sufficiently large positive derivative coupling to the Einstein tensor, the Reissner-Nordström black hole in asymptotically flat spacetime becomes unstable near a critical temperature and a regular spherically symmetric hairy black hole branch appears. The hair is not a power-law tail: at zeroth order the scalar behaves as $phi_0 sim z^{1+2mu xi}e^{-mu u z}chi(z)$ in the paper's coordinate $z$, equivalently $phi_0 sim r^{-(1+2mu xi)}e^{-u r}$ at large $r$, so it is localized near the horizon and decays exponentially. The authors solve the scalar eigenvalue problem numerically to locate the critical curve in the $(M,Q)$ plane for given $c$, then compute the $O(epsilon^2)$ backreaction on the metric and electrostatic potential. That backreaction shifts the mass, charge, and temperature as $M=8mu coth B-2mu epsilon^2alpha_2'(0)+cdots$, $Q=[1-Cepsilon^2],4mu/sinh B+cdots$, and $T/T_0=1-epsilon^2au+cdots$, where $C$ and $au$ are fixed by the next-to-leading-order equations.

Load-bearing premise

The load-bearing premise is that the small-scalar-amplitude expansion is controlled near the critical temperature: the paper computes the $epsilon^2$ backreaction but does not tie $epsilon$ to $(T_c-T)/T_c$ or bound the neglected $epsilon^4$ terms.

Editorial extensions

If this is right

The no-hair theorem in asymptotically flat spacetime is not absolute: a sufficiently strong positive derivative coupling to curvature can support regular scalar hair without a cosmological constant.
For fixed $c$, the instability region is bounded by critical curves in the $(M,Q)$ plane; above each curve the Reissner-Nordström black hole is unstable, and below it stable.
Near the transition the hair is confined within roughly five horizon radii and decays exponentially, so asymptotic flatness is preserved and the scalar is hard to detect far from the hole.
The $O(epsilon^2)$ corrections give the new branch different mass, charge, and temperature from the Reissner-Nordström solution, providing thermodynamic handles on the transition.
The dimensionful $c$ acts like the AdS radius in holographic superconductors, suggesting that flat-space analogues of holographic condensation may be possible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The effective potential well can be read as a position-dependent mass for the scalar; recasting the zeroth-order equation as a one-dimensional Schrödinger problem should give an analytic estimate of the critical $c$, which the paper obtains only numerically.
If the branch is genuine, the transition should have calculable critical exponents and a free-energy comparison; the paper's earlier work argues energetic favorability, but this paper does not compute the free-energy difference along the new branch.
Because the hair decays exponentially, its most distinctive observational signature is likely to be in near-horizon dynamics such as quasinormal modes or ringdown, rather than in asymptotic charges.
The zero-temperature endpoint and the stability of the full nonlinear solution are the natural next tests; the paper explicitly leaves a full numerical solution down to zero temperature to future work.

Original abstract

We discuss a phase transition of a Reissner-Nordstr\"om black hole to a hairy black hole in asymptotically flat spacetime. The hair is due to a massive charged scalar field. The no-hair theorem is evaded thanks to a derivative coupling of the scalar field to the Einstein tensor. The resulting hairy configuration is spherically symmetric. We solve the equations analytically near the transition temperature and show that the hair is concentrated near the horizon decaying exponentially away from it.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,129 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We discuss a phase transition of a Reissner-Nordstr\"om black hole to a hairy black hole in asymptotically flat spacetime. The hair is due to a massive charged scalar field. The no-hair theorem is evaded thanks to a derivative coupling of the scalar field to the Einstein tensor. The resulting hairy configuration is spherically symmetric. We solve the equations analytically near the transition temperature and show that the hair is concentrated near the horizon decaying exponentially away from it.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss a phase transition of a Reissner-Nordstr\"om black hole to a hairy black hole in asymptotically flat spacetime.	contribution 0.91	0.75	0.62	asserted 0.81
s1	The hair is due to a massive charged scalar field.	contribution 0.43	0.45	0.08	asserted 1.00
s2	The no-hair theorem is evaded thanks to a derivative coupling of the scalar field to the Einstein tensor.	contribution 0.31	0.42	0.14	asserted 1.00
s3	The resulting hairy configuration is spherically symmetric.	result 0.99	0.77	0.10	asserted 1.00
s4	We solve the equations analytically near the transition temperature and show that the hair is concentrated near the horizon decaying exponentially away from it.	result 0.85	0.96	0.31	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. The effective mass

**Figure 2.** FIG. 2. The profile of the scalar field

**Figure 3.** FIG. 3. Critical curves separating the region of instabilit M

**Figure 4.** FIG. 4. Next-to-leading-order metric function

**Figure 5.** FIG. 5. Next-to-leading-order metric function

**Figure 6.** FIG. 6. Next-to-leading-order metric function

Figures reproduced from arXiv:1308.5280 by the authors.

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J. Fernandez-Gracia and B. Fiol, “A No-hair theorem for extremal black branes,” JHEP 0911, 054 (2009) [arXiv:0906.2353 [hep-th]]

View on Pith Pith review arXiv 2009

Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Black Holes in Bi-scalar Extensions of Horndeski Theories
gr-qc 2014-04 conditional novelty 6.0 of 10

A two-scalar Horndeski theory admits asymptotically flat black holes with a regular conformally coupled scalar and an independent scalar charge.
Solutions in the scalar-tensor theory with nonminimal derivative coupling
gr-qc 2013-12 accept novelty 5.0 of 10

Exact static black hole solutions are found for a scalar-tensor theory with derivative coupling to the Einstein tensor, showing that the asymptotic geometry is set by the coupling constant, not the cosmological constant.
Formation of Bound States in Quintessence Alternative Theories
gr-qc 2025-06 conditional novelty 4.0 of 10

Bound states of scalar fields exist outside quintessence-dressed AdS black holes for sufficiently large |w| or angular momentum, and quintessence alters the Hawking temperature so that large black holes radiate less.
Charged Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories
gr-qc 2018-09 conditional novelty 4.0 of 10

Charged scalarized Gauss-Bonnet black holes exist with up to two bifurcation points, never reach extremality, and the fundamental branch has higher entropy than Reissner-Nordstrom.
Building a Holographic Superconductor with a Scalar Field Coupled Kinematically to Einstein Tensor
hep-th 2016-07 conditional novelty 4.0 of 10

A Horndeski-type kinetic coupling of the bulk scalar to the Einstein tensor suppresses the holographic condensate and breaks the usual gap-to-condensate relation, which the authors read as an impurity analogue.

Show 2 more citing papers

Asymptotically flat black holes with scalar hair: a review
gr-qc 2015-04 conditional novelty 4.0 of 10

Regular asymptotically flat black holes with scalar hair exist without gauge fields, and the known examples can be classified by which no-scalar-hair assumption they violate.
Four-Dimensional Asymptotically AdS Black Holes with Scalar Hair
gr-qc 2013-09 conditional novelty 4.0 of 10

A new closed-form family of asymptotically AdS black holes with scalar hair is derived by potential reconstruction, and a low-temperature thermal phase transition to the hairy configuration is reported.

How the literature uses it

34 arXiv papers cite this in 45 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 26 sentences without a confident label

Uses its method or tool 1 paper

Just as in the interesting works by Kolyvaris et al. [ref] , we will use Eq. (25) to discuss the stability of our solutions.

Backreacting holographic superconductors from the coupling of a scalar field to the Einstein tensor · Critical behavior from the Sturm-Liouville method · 0.95

Compares with it 2 papers

This is an extra bonus, since regularity is safe and no additional actions need be taken to secure it, as has been necessary in [ref] .

Gravitational Collapse of a Homogeneous Scalar Field Coupled Kinematically to Einstein Tensor · Conclusions · 0.84
In the next section we will study local spherically symmetric solutions of the field equations ([ref]) and ([ref]) and we expect to find solutions with features different than the local black hole solutions of theories with a scalar field coupled to Einstein tensor with a derivative coupling [ref] .

Self-Gravitating Spherically Symmetric Solutions in Scalar-Torsion Theories · Derivative coupling of a scalar field to torsion · 0.83

Cites it as support 1 paper

Therefore, the space is not any more flat and as it was shown in [ref] a potential well is formed near the horizon of a Reissner-Nordström.

Superradiant Instabilities in a Class of Scalar-Tensor Horndeski Theory · Superrrandiance Instability of the Reissner-Nordström black hole with the Derivative Coupling · 0.98

Cites it as background 15 papers

In this theory black hole solutions were found [ref] , known as Galilean black holes, and also wormhole geometries were generated [ref] - [ref]

Quasinormal modes and bound states of massive scalar fields in wormhole spacetimes · Introduction · 0.82
In recent years, countless works addressing various physical aspects of the hairy black holes in different asymptotic spaces have appeared, for a necessarily biased selection see [ref] .

Exact topological charged hairy black holes in AdS Space in $D$-dimensions · Introduction · 0.97
In [ref] a gravity model was considered consisting of an electromagnetic field and a scalar field coupled to the Einstein tensor with vanishing cosmological constant.

Curvature Scalarization of Black Holes in f(R) Gravity · Introduction · 0.91

What it cites · 12 sentences without a confident label

Cites as support 1 paper

Consequently, a scalar particle flying away from the horizon (due to a repulsive electric force which exceeds the gravitational attraction) travels all the way to infinity and can never condense [ref] .

Breaking an Abelian gauge symmetry near a black hole horizon · Leading order · 0.96

Cites as background 33 papers

A model was presented in [ref] consisting of an electromagnetic field and a scalar field minimally coupled to gravity in the presence of a negative cosmological constant.

Breaking an Abelian gauge symmetry near a black hole horizon · Introduction · 0.88
In the case of a positive cosmological constant with a minimally coupled scalar field with a self-interaction potential, black hole solutions were found in [ref] , and a numerical solution, albeit an unstable one, was presented in [ref] .

Coexistence of black holes and a long-range scalar field in cosmology · Introduction · 0.92
In the case of a positive cosmological constant with a minimally coupled scalar field with a self-interaction potential, black hole solutions were found in [ref] , and a numerical solution, albeit an unstable one, was presented in [ref] .

No-scalar hair conjecture in asymptotic de-Sitter spacetime · Introduction · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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