REVIEW
Conformally dressed black hole in 2+1 dimensions
General Relativity and Quantum Cosmology · Submitted 9 Apr 1996 · 28 references
Pith's verdict: A short, checkable exact solution with a 2/3 entropy factor; the real open questions are novelty against Chan's preprint and the slow scalar fall-off.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
An exact 2+1-dimensional black hole dressed by a conformal scalar field, whose entropy comes out two-thirds of the area law while still obeying the first law of thermodynamics.
The reading
What carries the argument
The load-bearing device is the conformal coupling $xi_D = (D-2)/(4(D-1))$, which in three dimensions makes the scalar stress tensor traceless and thereby collapses the trace of the Einstein equation into the single statement of constant scalar curvature, $R = -6l^{-2}$. That reduction is what makes the radial $r$-$r$ equation a closed constraint, $0 = (Psi')^2 - (1/8)(Psi^2)''$, whose general solution is $Psi = A/sqrt{r+B}$; the $v$-$v$ equation then ties $a$, $b$ and $A$ to $B$ and $kappa$, leaving one integration constant. The second device is the Hamiltonian action with a boundary term split into an infinity piece and a horizon piece; the infinity piece contributes $beta,delta(3r_+^2/(32l^2))$ and carries no scalar contribution, while the horizon piece, evaluated with $delta zeta = 2delta r_+/(9r_+)$ and the regularity condition $beta F'|_{r_+} = 4pi$, contributes $-pi,delta r_+/3$. Together these fix both the energy and the entropy prefactor.
Extended reading notes
Core claim
The paper reports an exact black hole solution of three-dimensional Einstein gravity with a negative cosmological constant conformally coupled to a massless scalar field. With the conformal coupling fixed at $D=3$, the scalar stress tensor is traceless, which forces the geometry to have constant scalar curvature $R=-6l^{-2}$; solving the remaining radial constraint gives $F(r) = (r+B)^2(r-2B)/(r l^2)$ together with $Psi(r) = sqrt{8B/kappa}/(r+B)$, with the horizon at $r_+=2B$. The authors show the Kretschmann scalar diverges only at the origin, that the scalar field is regular everywhere, that the solution is asymptotically anti-de Sitter with no term linear in $r$ (so the asymptotic symmetry group is the conformal one), and that the constant $B$ is not an independent charge: it is tied to the mass, so the scalar produces no new hair. Requiring the Euclidean continuation to be free of a conical singularity at the horizon fixes the temperature $T = 9r_+/(16pi l^2)$. Evaluating the Euclidean action on the classical solution, using the Hamiltonian form of the action with boundary terms at infinity and at the horizon, gives $I_E = beta, 3r_+^2/(32l^2) - pi r_+/3$, from which $M = 3r_+^2/(32l^2)$ and $S = pi r_+/3$ — two-thirds of $A/4$ — and these satisfy $dM = T,dS$. The authors also note that the same static solution follows from a stationary spherically symmetric ansatz, a Birkhoff-type statement for this system.
Load-bearing premise
The entropy result rests on the horizon boundary term of the Euclidean action, whose evaluation from the intermediate expression $delta B_E(r_+) = beta[delta F/9 + F' delta r_+/36]$ down to $-pi,delta r_+/3$ is compressed in the text and assumes the reduced minisuperspace Hamiltonian with these boundary conditions captures the full scalar contribution at the horizon.
Editorial extensions
If this is right
- The entropy of this black hole is $S = pi r_+/3 = (2/3)(A/4)$, so the quarter-area law is not universal in three dimensions once conformally coupled matter is present.
- The first law $dM = T,dS$ holds with $M = 3r_+^2/(32l^2)$ and $T = 9r_+/(16pi l^2)$, so the thermodynamic description is internally consistent even with the modified entropy.
- The scalar field carries no independent conserved charge: $B$ is fixed by the mass, so the solution is not a hairy black hole in the strict sense, and it is asymptotically indistinguishable from the matter-free case.
- The solution is nonperturbative in the scalar field, so it cannot be reached by linearizing around the matter-free black hole; any perturbative treatment would miss it.
- Assuming stationarity and spherical symmetry, the field equations force the same static solution, giving a Birkhoff-type theorem for this matter system.
Reading between the lines
- The $2/3$ factor is a clean target for an independent entropy calculation — for instance a Wald-type or conical-defect computation that never passes through the minisuperspace reduction — and agreement or disagreement there would be a sharp test of the whole framework.
- Because the deviation is rational rather than logarithmic or state-dependent, this solution is a useful calibration case for proposals that correct the area law by treating entropy as a functional of the matter sector rather than of geometry alone.
- The authors' closing note that a recently reported family of scalar-tensor solutions might contain this one as a special case is a real open question about novelty; checking whether the $B$-parametrized metric sits inside that family would settle it either way.
- The stability question the authors defer — circularly symmetric linear perturbations — is the natural next step: an unstable solution would still be a valid counterexample to the area law, but a stable one would make the $2/3$ factor a physical prediction rather than a curiosity.
Original abstract
A three dimensional black hole solution of Einstein equations with negative cosmological constant coupled to a conformal scalar field is given. The solution is static, circularly symmetric, asymptotically anti-de Sitter and nonperturbative in the conformal field. The curvature tensor is singular at the origin while the scalar field is regular everywhere. The condition that the Euclidean geometry be regular at the horizon fixes the temperature to be $T=\frac{9\, r_+}{16\pi l^2}$. Using the Hamiltonian formulation including boundary terms of the Euclidean action, the entropy is found to be $\frac{2}{3}$ of the standard value ($\frac{1}{4} A$), and in agreement with the first law of thermodynamics.
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
A three dimensional black hole solution of Einstein equations with negative cosmological constant coupled to a conformal scalar field is given. The solution is static, circularly symmetric, asymptotically anti-de Sitter and nonperturbative in the conformal field. The curvature tensor is singular at the origin while the scalar field is regular everywhere. The condition that the Euclidean geometry be regular at the horizon fixes the temperature to be $T=\frac{9\, r_+}{16\pi l^2}$. Using the Hamiltonian formulation including boundary terms of the Euclidean action, the entropy is found to be $\frac{2}{3}$ of the standard value ($\frac{1}{4} A$), and in agreement with the first law of thermodynamics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A three dimensional black hole solution of Einstein equations with negative cosmological constant coupled to a conformal scalar field is given. | contribution 0.97 | 0.89 | 0.46 | asserted 1.00 |
| s1 | The solution is static, circularly symmetric, asymptotically anti-de Sitter and nonperturbative in the conformal field. | result 0.75 | 0.71 | 0.16 | asserted 1.00 |
| s2 | The curvature tensor is singular at the origin while the scalar field is regular everywhere. | result 1.00 | 0.81 | 0.08 | asserted 1.00 |
| s3 | The condition that the Euclidean geometry be regular at the horizon fixes the temperature to be $T=\frac{9\, r_+}{16\pi l^2}$. | result 0.97 | 0.86 | 0.10 | asserted 1.00 |
| s4 | Using the Hamiltonian formulation including boundary terms of the Euclidean action, the entropy is found to be $\frac{2}{3}$ of the standard value ($\frac{1}{4} A$), and in agreement with the first law of thermodynamics. | result 1.00 | 0.95 | 0.36 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
32 arXiv papers cite this in 42 sentences
Papers citing it · 27 sentences without a confident label
Uses its method or tool 1 paper
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One might try to search for a saddle contributing [ref] by putting a normal mode scalar profile on top of the threshold BTZ and solve the backreaction perturbatively in $G_N$ (in a method similar to [ref] ), hoping that the backreaction resums.
What's the Matter with 3D Gravity? · Discussion · 0.93
Compares with it 3 papers
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By equation ([ref]), the metric function ([ref]) becomes [equation] which can be reduced to the solutions in [ref] and [ref] when $U(r)=Λ=-ℓ^-2$ and $δ=1$.
General Black Hole Solutions in (2+1)-dimensions with a Scalar Field Non-Minimally Coupled to Gravity · General solution with a scalar field non-minimally coupled to Einstein's gravity · 0.95
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This stands in stark contrast to the cases of i) the static BTZ black hole of GR, which is geodesically incomplete [ref] , and the rotating BTZ black hole which has a standard Cauchy horizon, being prone to mass inflation instability [ref] ; and ii) most other $2+1$-dimensional black holes, where modifications of GR generally introduce curvature singularities or non-extremal Cauchy horizons [ref] .
Regular BTZ black holes from an infinite tower of corrections · Introduction · 0.98
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Curiously, one also obtains the same form of the metric, although with a reduced set of parameters, when coupling a classical conformal massless scalar field to the BTZ black hole [ref] ; we discuss this case more in Sec. [ref] (see also [ref] ).
Singularities in 2D and 3D quantum black holes · Inside quantum black holes · 0.90
Cites it as background 9 papers
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These solutions correspond to a BTZ black hole conformally dressed with a scalar field [ref] .
Black Holes on Thin 3-branes of Codimension-2 and their Extension into the Bulk · Introduction · 0.86
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On the other hand, including electromagnetic and scalar fields also leads to new solutions, such as those found for the Einstein-Maxwell [ref] , Einstein-Maxwell-dilaton [ref] and Maxwell-Brans-Dicke [ref] systems, as well as for minimally and non-minimally coupled scalars [ref] , including theories obtained from certain three-dimensional limits of Lovelock theories [ref] .
On regular charged black holes in three dimensions · Introduction · 0.95
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In (2+1)-dimensions black holes with scalar field non-minimally coupled to gravity are known to exist [ref] .
General Black Hole Solutions in (2+1)-dimensions with a Scalar Field Non-Minimally Coupled to Gravity · Introduction · 0.91
Discussion
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