REVIEW
Exact black hole solution with a minimally coupled scalar field
High Energy Physics - Theory · Submitted 13 Jun 2004 · 32 references
Pith's verdict: A genuine exact 4D hairy AdS black hole with clean thermodynamics; the phase-transition claim needs one missing paragraph about boundary conditions, not a retraction.
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Referee report
The paper
The paper claims
The paper reports an exact four-dimensional black hole solution of gravity with a minimally coupled, self-interacting scalar field, and argues that below a critical temperature $T_c = 1/(2πl)$ a vacuum black hole spontaneously acquires the
The reading
What carries the argument
Two pieces carry the argument. First, the $\sinh$-squared potential is exactly what makes the field equations integrable for the static, hyperbolic-horizon ansatz, and the conformal-frame map in the Appendix shows it corresponds to a conformally coupled $Ψ^4$ theory, $V ∝ -Ψ^4$, which is the only independent motivation offered for the potential. Second, the Euclidean action calculation: the on-shell action reduces to boundary terms, and the authors show that the divergent piece of the gravitational boundary variation at infinity, $δB_G|_∞$, is cancelled exactly by the scalar contribution $δB_φ|_∞$ — a cancellation that only works because the scalar falls off as $1/r$ rather than the usual $1/r^{3/2+ε}$. That cancellation is what makes the mass, entropy and free energy finite and lets the phase-transition comparison be made at all.
Extended reading notes
Core claim
The central claim is that the field equations of four-dimensional gravity with $Λ = -3/l^2$ plus a minimally coupled scalar with the potential $V(φ) = -3/(4πG l^2) sinh^2(√(4πG/3) φ)$ admit the exact solution $ds^2 = r(r+2Gμ)/(r+Gμ)^2 [ -(r^2/l^2 - (1+Gμ/r)^2) dt^2 + (r^2/l^2 - (1+Gμ/r)^2)^{-1} dr^2 + r^2 dσ^2 ]$ with $φ = √(3/(4πG)) arctanh(Gμ/(r+Gμ))$, where $dσ^2$ is the line element of a compact negative-constant-curvature surface $Σ = H^2/Γ$. The horizon sits at $r_+ = l^2(1 + √(1+4Gμ/l))$, the mass is $M = σμ/(4π)$, and the temperature is $T = (1/(2πl))(2r_+/l - 1)$. Because the scalar cannot be switched off at fixed mass, the theory has two branches of black hole for a given temperature — the scalar-dressed one and the vacuum hyperbolic black hole — and the paper computes the difference of their Euclidean actions to show it changes sign exactly at $T_c = 1/(2πl)$, where both branches meet at the massless, locally AdS configuration.
Load-bearing premise
The whole construction rests on one hand-picked potential, $V(φ) = -3/(4πG l^2) sinh^2(√(4πG/3) φ)$, chosen because it makes the field equations integrable for the metric ansatz, with the conformal-frame map to a $Ψ^4$ theory as the only independent reason to take it seriously.
Editorial extensions
If this is right
- The first law $dM = T dS$ holds with $M = σμ/(4π)$ and $S = σ R(r_+)^2/(4G)$, so the scalar-dressed hole has a consistent thermodynamic description despite the scalar reaching the asymptotic region.
- The specific heat $C = (σ l^2/(4G))(2r_+/l - 1)$ is positive throughout the allowed mass range, so the dressed black hole can always sit in equilibrium with a heat bath.
- The free-energy difference between the dressed and vacuum branches changes sign at $T_c = 1/(2πl)$, giving a second-order transition with order parameter $|tanh(√(4πG/3) φ(r_+))| = (T_c - T)/(T_c + T)$ for $T < T_c$ and zero above.
- In the conformal frame the same solution becomes a black hole with a conformally coupled scalar and, for negative mass, three horizons, which the authors flag as a richer causal structure to be analysed elsewhere.
- Because the scalar back-reacts all the way to infinity, the asymptotic geometry deviates from the bare black hole at order $1/ρ^4$, yet the conserved charges remain finite.
Reading between the lines
- The transition is stated purely in thermodynamic (free-energy) terms; whether a dynamical decay or dressing actually proceeds is not computed, so a reader should treat 'spontaneous dressing up' as a statement about which saddle dominates the Euclidean path integral, not about a demonstrated time-dependent process.
- The potential is picked for integrability, and it is unbounded below; if one insists on a top-down or symmetry origin for the scalar sector, the solution survives as a solution but loses its claim to being a physically motivated model rather than a solvable toy.
- The $1/r$ falloff of the scalar and the exact cancellation it engineers suggest a broader checkable pattern: any asymptotically AdS solution with a scalar saturating the Breitenlohner-Freedman bound should require the same matter-plus-gravity boundary-term cancellation to yield finite charges.
- The ferromagnetic analogy invites a direct test on the dual side: a boundary CFT dual to this hyperbolic black hole should show a matching order parameter with the same $(T_c - T)/(T_c + T)$ shape near the transition.
Original abstract
An exact four-dimensional black hole solution of gravity with a minimally coupled self-interacting scalar field is reported. The event horizon is a surface of negative constant curvature enclosing the curvature singularity at the origin, and the scalar field is regular everywhere outside the origin. This solution is an asymptotically locally AdS spacetime. The strong energy condition is satisfied on and outside the event horizon. The thermodynamical analysis shows the existence of a critical temperature, below which a black hole in vacuum undergoes a spontaneous dressing up with a nontrivial scalar field in a process reminiscent of ferromagnetism.
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.
The abstract, sentence by sentence
An exact four-dimensional black hole solution of gravity with a minimally coupled self-interacting scalar field is reported. The event horizon is a surface of negative constant curvature enclosing the curvature singularity at the origin, and the scalar field is regular everywhere outside the origin. This solution is an asymptotically locally AdS spacetime. The strong energy condition is satisfied on and outside the event horizon. The thermodynamical analysis shows the existence of a critical temperature, below which a black hole in vacuum undergoes a spontaneous dressing up with a nontrivial scalar field in a process reminiscent of ferromagnetism.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An exact four-dimensional black hole solution of gravity with a minimally coupled self-interacting scalar field is reported. | contribution 0.94 | 0.92 | 0.79 | asserted 1.00 |
| s1 | The event horizon is a surface of negative constant curvature enclosing the curvature singularity at the origin, and the scalar field is regular everywhere outside the origin. | result 0.99 | 0.86 | 0.18 | asserted 1.00 |
| s2 | This solution is an asymptotically locally AdS spacetime. | result 0.97 | 0.83 | 0.15 | asserted 1.00 |
| s3 | The strong energy condition is satisfied on and outside the event horizon. | result 1.00 | 0.84 | 0.09 | asserted 1.00 |
| s4 | The thermodynamical analysis shows the existence of a critical temperature, below which a black hole in vacuum undergoes a spontaneous dressing up with a nontrivial scalar field in a process reminiscent of ferromagnetism. | result 0.95 | 0.91 | 0.43 | asserted 1.00 |
What was examined, and what was not
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References
How later papers used it
85 later papers cite it
4 later papers by other authors are listed below: 1 built on it and 3 used its method or tool.
Builds on it 1 of 2 papers with this label
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The hair are of the secondary kind and the solution generalizes the one obtained in [ref] to the case of non-minimal coupling and reduces to it for appropriate values of the parameters characterizing the model.
Super-entropic black hole with Immirzi hair (2020) · Topological hairy black hole · 0.98
Uses its method or tool 3 of 6 papers with this label
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Used the euclidean action formulation of Martinez et al. to consider black brane solutions as a thermodynamical system.
Hyperscaling violation for scalar black branes in arbitrary dimensions (2012) · 1.00 · paraphrased: the licence does not allow quoting
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Used the Euclidean action formalism of Martinez et al. to derive an expression for the total mass.
Infrared Behavior of Scalar Condensates in Effective Holographic Theories (2013) · 1.00 · paraphrased: the licence does not allow quoting
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Used the euclidean action formalism of Martinez et al. to set up the thermodynamics of its black brane solutions.
Phase transition and hyperscaling violation for scalar Black Branes (2012) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
79 arXiv papers cite this in 127 sentences
Papers citing it · 61 sentences without a confident label
Builds on it 1 paper
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The black hole solutions we construct are variations of the solutions found in [ref] and recently in [ref] .
Interactions resolve state-dependence in a toy-model of AdS black holes · Dual description of hairy AdS black holes · 0.96
Uses its method or tool 5 papers
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In this section, we discuss black hole thermodynamics by using the canonical Hamilton formulation [ref] .
Thermodynamics of Black Holes in Horava-Lifshitz Gravity · Black Hole Thermodynamics · 1.00
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We will use the euclidean action formulation of Martinez et al. [ref] .
Hyperscaling violation for scalar black branes in arbitrary dimensions · Thermodynamics and phase transitions · 1.00
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Consequently, we will calculate the variation of the boundary term $δℬ$ and through this we will obtain $ℬ$ and, hence, $ℐ_E$ [ref] .
Thermodynamically massless Simpson-Visser black holes · Thermodynamics of the Simpson-Visser spacetime · 0.94
Uses its data or values 1 paper
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The temperature, entropy and mass of the black hole are given respectively by [ref] T=1/ π ( r_+/l - 1/2 ) , S_MTZ=σl^2/4G ( 2r_+/l-1 ) , M_MTZ=σr_+/4 πG ( r_+/l - 1 ) .
Exact Gravity Dual of a Gapless Superconductor · Black Hole with Scalar Hair · 0.95
Compares with it 8 papers
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This kind of solutions are conformally related to the solutions found by Martinez et al [ref] working in the Einstein frame.
Thermodynamical properties of hairy black holes in n spacetimes dimensions · The $n=4$ solutions · 0.89
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This critical temperature occurs at a point where the black hole mass flips sign, as in the case of the MTZ black hole [ref] .
A New Class of Exact Hairy Black Hole Solutions · Introduction · 1.00
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Notice that the solution in Eq. [ref] is continuously connected to that of Refs. [ref] in the limit $λ→1$.
Asymptotically AdS black hole with a conformally-coupled scalar field in the first-order formalism of gravity · Torsional black holes dressed with scalar fields · 0.88
Cites it as support 1 paper
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In addition, we have observed from free energy that the second-order phase transition may occur between NBTZ and CBTZ [ref] .
Thermodynamics of Einstein-Born-Infeld black holes in three dimensions · Summary · 0.98
Cites it as background 35 papers
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Recently, a black hole solution with a minimally coupled self-interacting scalar was found in AdS$_4$ spacetimes [ref] .
Phase transition for black holes with scalar hair and topological black holes · Introduction · 0.93
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On later, it was proposed that a transition between black hole with scalar hair (Martinez-Troncoso-Zanelli black hole [ref] ) and a topological black hole is possible to occur as a second-order phase transition in asymptotically AdS$_4$ spacetimes [ref] .
Phase transitions for the Lifshitz black holes · Introduction · 0.83
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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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