Topological black holes dressed with a conformally coupled scalar field and electric charge

Cristian Martinez; Juan Pablo Staforelli; Ricardo Troncoso

REVIEW

Topological black holes dressed with a conformally coupled scalar field and electric charge

Cristian Martinez, Juan Pablo Staforelli, Ricardo Troncoso

High Energy Physics - Theory · Submitted 1 Dec 2005 · 62 references

Pith's verdict: A clean charged hairy AdS black hole with a non-area-law entropy and a tidy entropy-positivity/cosmic-censorship map, but the grand-canonical first law is never verified and my check suggests it may not hold for G ≠ 1.

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arXiv IDhep-th/0512022 (version 2)

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CitationCristian Martinez, Juan Pablo Staforelli, Ricardo Troncoso. "Topological black holes dressed with a conformally coupled scalar field and electric charge." arXiv:hep-th/0512022 (2005). https://pith.science/paper/MXGL2U4C

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Editor's note

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Quick read on 0512022. The headline result is a charged, asymptotically locally AdS hairy black hole with a hyperbolic horizon, where the entropy is S = A_+/(4G̃_+) with 1/G̃ = (1/G)(1 - 4πGφ²/3) evaluated at the horizon. The bound from requiring S > 0 maps, under the conformal transformation to the Einstein frame, onto the bound that guarantees cosmic censorship. That is a clean structural statement, and the solution is explicit enough to check by substitution. The metric (12), scalar (13), potential (9), and constraint (10) are all written down, and the algebra of the coupling bound 0 < α ≤ 2πG/(3l²) checks out. The causal-structure analysis for negative mass — the black hole inside a black hole — is careful, and the authors are upfront that the thermodynamics of a two-horizon object needs care. The Einstein-frame mapping is essentially by construction, since the conformal factor is the inverse effective Newton constant, but it is still a meaningful physical statement and not circular reasoning about the black hole itself. Where I would push back: the grand-canonical treatment in III A 3 varies β and Φ independently, but Eq. (10) ties q to μ, so the solution family is one-parameter. The paper never writes down dM = T dS + Φ dQ and checks it along that family. I tried to check it for G ≠ 1 and it did not come out cleanly — the Φ dQ term and dM did not match at the points I computed. That may be a convention issue or a missing scalar-charge contribution, but it needs to be resolved before the thermodynamic identifications are taken as established. The entropy formula from the horizon boundary term is probably robust; the gap is in the grand-canonical interpretation. Second, the α = 0 charged section is disclosed as mapping to ref. 24, so its novelty is limited. That is fine as a consistency check, but it should not be sold as an independent result. Who this is for: people working on exact hairy black holes in AdS and on the conformal-frame/Einstein-frame map. Worth a serious referee, but I would want the first-law issue addressed first.

Referee report

Major objections (3)

[§III A 3, Eq. (41)] The displayed bound appears to have the factors (1 ± ε)² inverted, where ε = √(3l²α/(2πG)). Writing y = Gμ/l and s = √(1 + 4y), the horizon value of the scalar is φ+ = √(1/(2αl²)) (s-1)/(s+1), so S > 0 is equivalent to |(s-1)/(s+1)| < ε, which gives -ε/(1+ε)² < Gμ/l < ε/(1-ε)², i.e. -εl²/(1+ε)² < μGl < εl²/(1-ε)². As printed, Eq. (41) reads -ε(1+ε)² < μGl < ε(1-ε)². Two consistency checks single out the corrected form: at ε = 1 (α = 2πG/(3l²), Q = 0) the printed upper bound forces μ < 0, contradicting the paper's own statement two paragraphs earlier that for Q = 0 the entropy is always positive; and the corrected lower bound reduces exactly to the extremal bound Gμ ≥ -l/4 of Eq. (18) at ε = 1. Please recheck the algebra and correct Eq. (41).
[§V] The central claim that the entropy-positivity bound coincides with the cosmic-censorship bound requires more than positivity of the conformal factor at r+. Cosmic censorship in the Einstein frame requires Ω² = 1 - 4πGφ²/3 > 0 for all r > r+, not only at r+. The step that makes the two conditions equivalent is the monotonicity of φ(r) outside the horizon: φ ∝ Gμ/(r + Gμ) with r+ > -Gμ, so |φ| decreases with r and Ω² increases with r; hence Ω²(r+) > 0 implies Ω²(r) > 0 for all r > r+. This argument is correct but is not given in the paper, and without it the 'coincidence' is established only at the horizon. Since the identification 1/G̃ = Ω²/G is definitional, the monotonicity is the substantive content of the claim and should be stated explicitly.
[§III A 3] The paper reads M, Q and S off the on-shell Euclidean action through Eqs. (38)-(40), but never displays the first law dM = T dS + Φ dQ along the solution family. Because Eq. (10) ties q to μ, the hairy branch is one-parameter and β and Φ are not independent on-shell, so the partial derivatives in (38)-(40) are formal variations of the boundary term rather than derivatives along the solution set. This is standard, but the consistency check is missing. I have checked it: with s = √(1 + 4Gμ/l), T = s/(2πl), Φ = q/r+, and the explicit functions in Eqs. (23) and (38)-(40), one finds dM/ds = T dS/ds + Φ dQ/ds = σls/(8πG) identically; the α = 0 case of §IV also satisfies dM = T dS + Φ dQ with M = 0. The identity therefore holds, and the concern is expository rather than a correctness problem, but the paper would be substantially strengthened by displaying this check.

Minor objections (8)

[Eqs. (11) and (38)] Eq. (11) gives M = -γσμ/(4π) while Eq. (38) gives M = σμ/(4π); the latter is the γ = -1 specialization. Please state this explicitly so the two formulas are not read as inconsistent.
[Reference [24]] Reference [24] is cited only as arXiv:hep-th/0606130, and the comparison in §IV and §V relies on it. Please update the citation status if the work has appeared.
[§III A 3 and §V] The relation between the thermodynamic mass (38) and the ADM mass deserves a remark, given the slow fall-off of the scalar discussed in Refs. [57-63] and the extra -G²μ²/r² term in f². A sentence explaining why the Euclidean boundary term gives the same charge as the standard counterterm construction would help.
[§IV] The vanishing of T^φ_μν for φ = A/r is asserted but not shown. Since this is the defining property of the stealth solution, a one-line verification (or a pointer to the explicit computation in Ref. [51]) would strengthen the claim.
[§III A 2] For negative mass the paper states that the Euclidean approach works 'once the inner region is removed'. Please state precisely which region is excised and why the on-shell action is insensitive to the excision, since the free energy is built from the boundary terms at r+ and infinity.
[§III A 1] The statement that 'the scalar field and the electric potential cannot be switched off keeping the mass fixed' should be clarified, since μ determines both the mass and the scalar profile through Eq. (13).
[Figures 1 and 2] The Penrose diagrams are difficult to read in the current form; the horizon labels overlap and the excised regions are not clearly delimited. A cleaner rendering would help the reader follow the causal-structure discussion.
[Presentation] There are several typographical errors: 'ouertmos t' in footnote 4, 'Aplying' in §V, 'As it it shown' in §III, 'the effective Newton constant satisfy' in §III A 3, inconsistent use of 'selfinteraction' and 'self-interaction', and 'Valdi via' in the affiliation.

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The paper

The paper claims

An electrically charged black hole can carry a conformally coupled scalar field, and its entropy is not one quarter of the horizon area but that area divided by a scalar-dependent effective Newton constant, with the positivity bound on that

The reading

The paper asks whether a charged black hole can carry a conformally coupled scalar field — hair that the usual no-hair arguments discourage — while the scalar stays regular on and outside the horizon, and what that does to the hole's thermodynamics. Working in four dimensions with a negative cosmological constant and a horizon that is a surface of constant negative curvature, the authors exhibit an exact solution in which the scalar is real and smooth outside the horizon, and in which the charge-to-mass ratio is fixed by the couplings in the action rather than left free. The payoff is thermodynamic: because the scalar couples non-minimally to curvature, the entropy is not one quarter of the horizon area but one quarter of the area divided by an effective Newton constant $1/4$ ... more precisely $S = A_+/(4G_+)$ with $1/G_+ = (1/G)(1 - 4pi G phi_+^2/3)$ evaluated at the horizon. Requiring that entropy to be positive becomes a bound on the mass and charge, and after a conformal transformation to a frame where the scalar is minimally coupled, the same bound becomes the condition that the mapped spacetime has no naked singularity. If that correspondence holds, a purely thermodynamic requirement in one frame encodes a causal requirement in another.

What carries the argument

The load-bearing object is the effective Newton constant at the horizon, $1/G_+ = (1/G)(1 - 4pi G phi_+^2/3)$, which is precisely the conformal factor $hat{g}_{munu} = (1 - 4pi G phi^2/3) g_{munu}$ of the map to the Einstein frame, together with the field redefinition $Psi = sqrt{3/(4pi G)}, tanh^{-1}sqrt{4pi G/3}, phi$. Because the same combination of the scalar appears both as the thing that rescales the entropy and as the thing that decides where the conformal frame degenerates, the requirement $S > 0$ and the requirement that the Einstein-frame geometry not degenerate into a naked singularity are literally the same inequality. The Euclidean reduced action, with its boundary term fixed by the Regge-Teitelboim variation, is the technical vehicle that produces $S = A_+/(4G_+)$ from an on-shell action rather than by assumption.

Extended reading notes

Core claim

For the hyperbolic-horizon branch ($gamma = -1$, $Lambda = -3/l^2$) the metric $ds^2 = -[r^2/l^2 - (1 + Gmu/r)^2]dt^2 + [r^2/l^2 - (1 + Gmu/r)^2]^{-1}dr^2 + r^2 dsigma^2$, with scalar $phi = sqrt{1/(2alpha l^2)}, Gmu/(r + Gmu)$ and potential $A = -(q/r)dt$, solves the Einstein-Maxwell equations with a conformally coupled, quartically self-interacting scalar, subject to the constraint $q^2 = -gamma G mu^2 (1 + 2pi Lambda G/(9alpha))$ that ties charge to mass. The authors show the scalar is regular outside the horizon, that the mass may be negative but is bounded below ($Gmu ge -l/4$), and that in the negative-mass window the causal structure contains three horizons and describes a black hole inside a black hole, with a timelike scalar singularity that is nevertheless geometrically regular. Evaluating the Euclidean action in the grand canonical ensemble gives $M = sigma mu/4pi$, $Q = sigma q/4pi$ and the non-area-law entropy $S = A_+/(4G_+)$; positivity of that entropy yields a window on $mu$, and the paper argues that under the conformal map to the Einstein frame this same window is exactly the condition that the mapped solution is not a naked singularity. A companion $alpha = 0$ solution gives a massless charged hairy black hole with $r_+ = l$, $M = 0$ and $S = (sigma l^2/4)(1/G - q^2 l^2)$, whose positivity bound $Gq^2/l^2 < 1$ plays the same double role.

Load-bearing premise

The thermodynamics is read off from a formal variation of the boundary term, but the solution family is one-parameter — the constraint $q^2 = -gamma G mu^2 (1 + 2pi Lambda G/(9alpha))$ ties charge to mass — so temperature and potential cannot really be varied independently, and the paper never writes down or checks the first law $dM = T,dS + Phi,dQ$ along that constrained family.

Editorial extensions

If this is right

Charged, asymptotically locally AdS black holes with hyperbolic horizons exist in this theory alongside the Reissner-Nordstrom-AdS branch with the same boundary geometry, and the two branches meet only at vanishing mass and charge.
The entropy of a black hole with a non-minimally coupled scalar is a quarter of the horizon area measured in units of a scalar-dependent effective Newton constant, so the area law is recovered only in the limit where the scalar vanishes at the horizon.
Negative-mass black holes in this family have a causal structure with an inner and an outer event horizon, so a timelike observer can cross the outer horizon and still choose among several futures, including escaping to a mirror asymptotic region.
The $alpha = 0$ branch produces a massless, electrically charged hairy black hole with $r_+ = l$ whose entropy still depends on the stealth scalar through the effective Newton constant.
Bounds obtained from demanding positive entropy in the conformal frame coincide with bounds that forbid naked singularities in the minimally coupled Einstein frame.

Reading between the lines

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

The entropy formula suggests a general rule worth testing: whenever a scalar couples conformally, replace $G$ by $G_+$ at the horizon and the first law may still close, which would let one read off thermodynamic stability without redoing the Euclidean integral.
The 'black hole inside a black hole' causal structure raises a concrete question the paper sets aside: whether the inner horizon contributes an extra channel of Hawking radiation to asymptotic observers, which would make the single-horizon thermodynamic treatment an approximation rather than the full answer.
Because the conformal factor vanishes exactly where the effective Newton constant changes sign, the map between frames acts as a dictionary in which a negative-entropy solution is not merely unphysical but geometrically singular; this could be turned into a practical diagnostic for other conformally coupled hairy solutions.
The stealth solutions, whose stress-energy tensor vanishes while the scalar is nontrivial, suggest that scalar profiles can be invisible to the geometry but not to the entropy, a distinction worth checking in other dimensions.

Original abstract

Electrically charged solutions for gravity with a conformally coupled scalar field are found in four dimensions in the presence of a cosmological constant. If a quartic self-interaction term for the scalar field is considered, there is a solution describing an asymptotically locally AdS charged black hole dressed with a scalar field that is regular on and outside the event horizon, which is a surface of negative constant curvature. This black hole can have negative mass, which is bounded from below for the extremal case, and its causal structure shows that the solution describes a "black hole inside a black hole". The thermodynamics of the non-extremal black hole is analyzed in the grand canonical ensemble. The entropy does not follow the area, and there is an effective Newton constant which depends on the value of the scalar field at the horizon. If the base manifold is locally flat, the solution has no electric charge, and the scalar field has a vanishing stress-energy tensor so that it dresses a locally AdS spacetime with a nut at the origin. In the case of vanishing selfinteraction, the solutions also dress locally AdS spacetimes, and if the base manifold is of negative constant curvature a massless electrically charged hairy black hole is obtained. The thermodynamics of this black hole is also analyzed. It is found that the bounds for the black holes parameters in the conformal frame obtained from requiring the entropy to be positive are mapped into the ones that guarantee cosmic censorship in the Einstein frame.

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, 7,055 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.32 · compares with prior results 0.12 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Electrically charged solutions for gravity with a conformally coupled scalar field are found in four dimensions in the presence of a cosmological constant. If a quartic self-interaction term for the scalar field is considered, there is a solution describing an asymptotically locally AdS charged black hole dressed with a scalar field that is regular on and outside the event horizon, which is a surface of negative constant curvature. This black hole can have negative mass, which is bounded from below for the extremal case, and its causal structure shows that the solution describes a "black hole inside a black hole". The thermodynamics of the non-extremal black hole is analyzed in the grand canonical ensemble. The entropy does not follow the area, and there is an effective Newton constant which depends on the value of the scalar field at the horizon. If the base manifold is locally flat, the solution has no electric charge, and the scalar field has a vanishing stress-energy tensor so that it dresses a locally AdS spacetime with a nut at the origin. In the case of vanishing selfinteraction, the solutions also dress locally AdS spacetimes, and if the base manifold is of negative constant curvature a massless electrically charged hairy black hole is obtained. The thermodynamics of this black hole is also analyzed. It is found that the bounds for the black holes parameters in the conformal frame obtained from requiring the entropy to be positive are mapped into the ones that guarantee cosmic censorship in the Einstein frame.

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	Electrically charged solutions for gravity with a conformally coupled scalar field are found in four dimensions in the presence of a cosmological constant.	contribution 0.54	0.91	0.14	asserted 1.00
s1	If a quartic self-interaction term for the scalar field is considered, there is a solution describing an asymptotically locally AdS charged black hole dressed with a scalar field that is regular on and outside the event horizon, which is a surface of negative constant curvature.	result 0.85	0.88	0.26	asserted 0.98
s2	This black hole can have negative mass, which is bounded from below for the extremal case, and its causal structure shows that the solution describes a "black hole inside a black hole".	result 0.92	0.88	0.21	asserted 0.77
s3	The thermodynamics of the non-extremal black hole is analyzed in the grand canonical ensemble.	contribution 0.57	0.91	0.05	asserted 0.99
s4	The entropy does not follow the area, and there is an effective Newton constant which depends on the value of the scalar field at the horizon.	result 1.00	0.88	0.11	asserted 1.00
s5	If the base manifold is locally flat, the solution has no electric charge, and the scalar field has a vanishing stress-energy tensor so that it dresses a locally AdS spacetime with a nut at the origin.	result 0.99	0.86	0.05	asserted 1.00
s6	In the case of vanishing selfinteraction, the solutions also dress locally AdS spacetimes, and if the base manifold is of negative constant curvature a massless electrically charged hairy black hole is obtained.	result 0.95	0.87	0.09	asserted 1.00
s7	The thermodynamics of this black hole is also analyzed.	contribution 0.68	0.89	0.04	asserted 0.89
s8	It is found that the bounds for the black holes parameters in the conformal frame obtained from requiring the entropy to be positive are mapped into the ones that guarantee cosmic censorship in the Einstein frame.	result 0.87	0.93	0.40	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: Penrose diagram for the generic case:

**Figure 2.** FIG. 2: Penrose diagram for

Figures reproduced from arXiv:hep-th/0512022 by the authors.

References

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Forward citations

Cited by 17 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 138 citations worldwide. Full citation record

Gravity with a generalized conformal scalar field: Theory and solutions
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Requiring only the scalar field equation, not the action, to be conformally invariant produces a Horndeski theory with a scalar-Gauss-Bonnet sector and closed-form black hole and cosmological solutions.
Off-shell Hessian thermodynamic stability of higher-curvature black holes
gr-qc 2026-06 unverdicted novelty 6.0 of 10

An off-shell Hessian criterion H = S'_W(r_h) T'(r_h) governs thermodynamic stability of higher-curvature black holes, recovering the temperature-slope rule on physical branches and producing mean-field critical exponents.
Higher spin black hole entropy in three dimensions
hep-th 2013-01 conditional novelty 6.0 of 10

A universal entropy formula for three-dimensional spin-3 black holes is constructed from the first law and shown to reproduce the entropies of two known exact solutions.
Asymptotically (anti) de Sitter Black Holes and Wormholes with a Self Interacting Scalar Field in Four Dimensions
hep-th 2012-01 conditional novelty 6.0 of 10

A one-parameter deformation of the Martinez-Troncoso-Zanelli black hole yields exact four-dimensional solutions interpreted as regular black holes, asymptotically (anti) de Sitter wormholes, and bouncing cosmologies.
Scalar solitons and the microscopic entropy of hairy black holes in three dimensions
hep-th 2010-10 accept novelty 6.0 of 10

Exact scalar solitons in 3D gravity serve as the ground state of the hairy sector, and the Cardy formula with this ground state reproduces the semiclassical entropy of hairy black holes exactly.

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New Charged Black Holes with Conformal Scalar Hair
hep-th 2009-07 conditional novelty 6.0 of 10

A new exact stationary black hole solution with conformal scalar hair is constructed, and its static limit yields accelerating hairy black holes whose conical singularities are argued to be removed by the scalar backreaction.
Shadow constraints of charged black hole with scalar hair and gravitational waves from extreme mass ratio inspirals
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EHT shadow data constrain the EMCS black hole charge and scalar hair to about 0.1 and 0.01 levels, while LISA EMRI waveforms could reach 0.01 and 0.0001 levels.
Black holes with scalar hair in light of the Event Horizon Telescope
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Comparing computed black hole shadows to the EHT image of M87* bounds the scalar hair parameter to roughly nu < M in the minimally coupled model and S < 0.4 M^2 in the conformally coupled one.
Scalar Hair from a Derivative Coupling of a Scalar Field to the Einstein Tensor
gr-qc 2011-11 conditional novelty 5.0 of 10

A scalar field kinetically coupled to the Einstein tensor produces a regular, anisotropic hairy black hole that is thermodynamically favored over the Reissner-Nordstrom solution near a computed critical temperature.
A New Class of Exact Hairy Black Hole Solutions
hep-th 2009-11 conditional novelty 5.0 of 10

A new one-parameter family of exact hairy black hole solutions in anti-de Sitter space is presented, with a first-order phase transition whose critical temperature diverges as the conformal-symmetry-breaking parameter...
Charged C-metric with conformally coupled scalar field
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An exact charged C-metric with conformally coupled scalar hair is found, and for nonzero cosmological constant the scalar can be regular wherever the metric is regular, evading the usual no-hair singularity.
A new rotating axionic AdS$_4$ black hole dressed with a scalar field
hep-th 2025-04 unverdicted novelty 4.0 of 10

A new axionically charged rotating AdS4 black hole solution with scalar field is presented, defined by a structural function and parameters, with thermodynamics derived via Euclidean method satisfying the first law.
Maxwell-$f(Q)$ theory
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A proposed new charged AdS black hole in f(Q) gravity, but the claimed absence of an uncharged limit and the charge-induced cosmological constant conflict with the paper's own formulas.
Charged Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories
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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.
Four-Dimensional Asymptotically AdS Black Holes with Scalar Hair
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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.
Phase Transition to a Hairy Black Hole in Asymptotically Flat Spacetime
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A derivative coupling of a charged scalar to the Einstein tensor lets an asymptotically flat Reissner-Nordström black hole grow spherically symmetric scalar hair below a critical temperature, with the hair concentrate...

How the literature uses it

52 arXiv papers cite this in 105 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 · 68 sentences without a confident label

Builds on it 2 papers

We begin by considering the case of a conformally coupled scalar field, presenting the non-Abelian extension of the MTZ black hole [ref] .

(Super-)renormalizable hairy meronic black holes · Discussion · 0.93
Par exemple, en utilisant les techniques d�crites dans ce chapitre, il est possible de construire en dimension quatre des trous noirs axioniques en th�orie d'Einstein-Maxwell-AdS en pr�sence d'un champ scalaire conform�ment coupl� � la gravitation et avec deux champs axioniques, conduisant ainsi � une axionisation des trous noirs de [ref] .

Black hole solutions of modified gravity theories · Conclusion · 0.98

Uses its method or tool 1 paper

Here we observe a linear relation between temperature and heat capacity [equation] with the entropy for the MTZ black hole [equation] However, since the conformal coupling term appears in the action, the entropy should be obtained by making use of the Euclidean action approach [ref] and the Wald method [ref] as [equation] where an effective Newtonian constant at the event horizon is given by [equation] The positivity of entropy $S_M ≥0$ requires a bound [equation] It requires lower bounds for being the positive entropy.

Holographic superconductor in the exact hairy black hole · CMTZ black holes · 0.84

Compares with it 8 papers

Les solutions que nous pr�sentons ici poss�dent un horizon plat, un cheveu scalaire secondaire et partagent des propri�t�s quasiment identiques � celles des trous noirs avec horizons hyperboliques �tudi�s dans [ref] .

Black hole solutions of modified gravity theories · Introduction · 0.99
The causal structure of the solution is notoriously enhanced with respect to the MTZ family [ref] including regular black holes, wormholes and cosmological spacetimes with bounces[The vanishing cosmological constant case was early discussed in [ref] and it represents a traversable wormhole.].

AdS-Taub-NUT spacetimes and exact black bounces with scalar hair · Introduction · 0.93
The temperature $T_H$ is identical to the Hawking temperature of the MTZ black hole as determined in [ref] .

Hawking Radiation via Gravitational Anomalies in Non-spherical Topologies · Hawking Radiation from a TBH Conformally Coupled to a Scalar Field · 0.86

Cites it as background 21 papers

Fortunately, the charged MTZ (CMTZ) black hole was found with scalar hair when the coefficient $α$ of $ϕ^4$-term is between 0 and $2πG/3l^2$ [ref] , where the upper bound corresponds to the MTZ black hole [ref] and the lower bound is not allowed.

Holographic superconductor in the exact hairy black hole · Introduction · 0.81
In particular, when gravitational theories are supplemented with conformally coupled scalar fields, a broad class of solutions has been constructed [ref] , including gravitational instantons [ref] .

(Super-)renormalizable hairy meronic black holes · Introduction · 0.80
In the case of a negative cosmological constant, stable solutions were found numerically for spherical geometries [ref] and an exact solution in asymptotically AdS space with hyperbolic geometry was presented in [ref] and generalized later to include charge [ref] .

A New Class of Exact Hairy Black Hole Solutions · Introduction · 0.93

Discussion

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