REVIEW
Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons
General Relativity and Quantum Cosmology · Submitted 7 May 1997 · 61 references
Pith's verdict: A clean, careful extension of the area law to toroidal and higher-genus horizons; the finite-boundary derivation is the solid core, and the only real soft spot is the disclosed k=-1 subtraction-scheme dependence.
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Referee report
The paper
The paper claims
The area law survives on toroidal and higher-genus horizons.
The reading
What carries the argument
The load-bearing device is the Brown-York quasilocal energy at a finite spherical-section boundary, $U(R_B) = -(R_B^2 V/4π)(√{F(R_B)}/R_B + 4π ε_0(R_B))$, interpreted as the thermodynamic internal energy. Because the local Hawking temperature vanishes at infinity, the first law is first written at finite $R_B$, where both sides are finite and nonvanishing; multiplying by $√{F(R_B)}$ and taking $R_B → ∞$ then produces the infinite-space first law $d(VM/4π) = T_∞ d(¼ V R_h^2) + φ de$ with the renormalized temperature $T_∞ = F'(R_h)/4π$ and the Killing-time potential $φ = Q/R_h$. The second device is the Hamiltonian reduction of the Euclidean action to the single functions $I^*_{gc}(R_h,q)$ and $I^*_c(R_h)$, whose unique stationary points reproduce the classical black holes and whose Gaussian weight supplies the stability statements.
Extended reading notes
Core claim
The paper establishes that the thermodynamics of Einstein-Maxwell spacetimes with a negative cosmological constant goes through essentially unchanged when the two-sphere is replaced by a closed surface $Σ_k$ of constant curvature $k=0$ or $k=-1$. Horizons exist for masses above a critical value $M_{crit}(Q)$ given in closed form, and when they do the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity whose horizon bifurcation two-surface has the topology of $Σ_k$ rather than $S^2$. Using the Brown-York quasilocal energy $U(R_B)$ at a finite boundary $R_B$ as the internal thermodynamic energy, and the local Hawking temperature $T_H(R_B)=F'(R_h)/[4π√{F(R_B)}]$, the variation $dU = T_H d(¼ V R_h^2) + ildeΦ de$ yields the entropy $S = ¼ A_h$, one quarter of the horizon area, independent of horizon topology. In the large-box limit the heat capacities at fixed ADM charge, at fixed redshifted horizon-boundary potential, and at fixed boundary-infinity potential are positive definite for $k=0$ and $k=-1$, whereas for $k=1$ all three are indefinite. With the boundary taken strictly to infinity, the reduced Euclidean actions $I^*_{gc}$ and $I^*_c$ each have exactly one stationary point, a global minimum, so the grand canonical and canonical ensembles are well defined and dominated by a unique classical solution, with the area law recovered from $S=(1-β_∞∂_{β_∞})ln Z$.
Load-bearing premise
The first law fixes the entropy only up to an additive constant, and the paper chooses that constant so the entropy comes out as exactly one quarter of the horizon area; the absolute area law is therefore a consistent convention rather than something the first law alone forces.
Editorial extensions
If this is right
- The theorems that force spherical horizon topology in asymptotically flat spacetimes do not carry over once the cosmological constant is negative, and the paper exhibits explicit black holes with toroidal, Klein-bottle, and higher-genus horizons.
- Entropy is one quarter of the horizon area for every closed horizon topology in this family, so the Bekenstein-Hawking coefficient is not an artifact of spherical symmetry.
- For $k=0$ and $k=-1$ the heat capacities $C_Q$, $C_{ildeΦ_B}$ and $C_{Φ_B}$ are positive definite in the large-box limit, unlike the $k=1$ Reissner-Nordström-anti-de Sitter case where they change sign in a calculable region of parameter space.
- With the boundary at infinity, both the canonical and grand canonical ensembles are convergent and dominated by a unique classical solution, so the saddle-point thermodynamics of these black holes is well posed.
- Degenerate-horizon solutions are not black holes here: future null infinity is a single connected component whose past is the whole spacetime, in contrast with the extreme Reissner-Nordström solution.
- Black holes with negative-curvature horizons can carry either sign of ADM mass, with $M_{crit}$ negative when $|Q|<ℓ/2$, so positive mass is not required for a black hole interpretation in this family.
Reading between the lines
- The entropy is fixed only up to an additive function of the cosmological constant and the topology of $Σ_k$; a microscopic state count, or a dual description, would have to decide whether a genus-dependent constant appears, and that is where the real content of "topology-independent area law" would be tested.
- Because the $k=-1$ black holes can have negative ADM mass while remaining thermodynamically stable in these ensembles, the familiar link between positive energy and thermal stability does not hold in this family.
- The paper's closing note records that a different subtraction procedure produces a $k=-1$ entropy shifted by a constant, which suggests the interesting open question is not the coefficient $1/4$ but whether entropy in anti-de Sitter is defined only up to topological data.
- The non-closed choices for $Σ_k$ — cylinder, Möbius band, surfaces with finitely many infinities — are set aside because ADM mass and charge diverge there; a regulated treatment could extend the stability analysis to horizons of infinite area.
Original abstract
We examine counterparts of the Reissner-Nordstrom-anti-de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface Sigma of constant negative or zero curvature. When horizons exist, the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not S^2. Maximal analytic extensions of the solutions are given. The local Hawking temperature is found. When Sigma is closed, we derive the first law of thermodynamics using a Brown-York type quasilocal energy at a finite boundary, and we identify the entropy as one quarter of the horizon area, independent of the horizon topology. The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution.
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
We examine counterparts of the Reissner-Nordstrom-anti-de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface Sigma of constant negative or zero curvature. When horizons exist, the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not S^2. Maximal analytic extensions of the solutions are given. The local Hawking temperature is found. When Sigma is closed, we derive the first law of thermodynamics using a Brown-York type quasilocal energy at a finite boundary, and we identify the entropy as one quarter of the horizon area, independent of the horizon topology. The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We examine counterparts of the Reissner-Nordstrom-anti-de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface Sigma of constant negative or zero curvature. | contribution 0.97 | 0.84 | 0.05 | asserted 0.94 |
| s1 | When horizons exist, the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not S^2. | result 0.69 | 0.30 | 0.09 | asserted 0.99 |
| s2 | Maximal analytic extensions of the solutions are given. | contribution 0.76 | 0.92 | 0.03 | asserted 1.00 |
| s3 | The local Hawking temperature is found. | result 0.90 | 0.95 | 0.04 | asserted 1.00 |
| s4 | When Sigma is closed, we derive the first law of thermodynamics using a Brown-York type quasilocal energy at a finite boundary, and we identify the entropy as one quarter of the horizon area, independent of the horizon topology. | result 0.89 | 0.96 | 0.35 | asserted 1.00 |
| s5 | The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. | result 1.00 | 0.93 | 0.12 | asserted 1.00 |
| s6 | With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. | method 0.71 | 0.90 | 0.10 | asserted 0.79 |
| s7 | Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution. | result 1.00 | 0.93 | 0.33 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
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Figures
Figures reproduced from arXiv:gr-qc/9705012 by the authors.
References
How the literature uses it
51 arXiv papers cite this in 74 sentences
Papers citing it · 42 sentences without a confident label
Uses its method or tool 2 papers
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Also, the Hawking temperature can be derived from the Euclidean regularity methods [ref] in our considered metric by[The same result can be obtained via the normalized temperature definition [ref] [equation] which is independent of the normalization of the horizon generator.] [equation] which is infinite on the horizon and has the finite non-vanishing value at infinity, given by [equation]
New five-dimensional Bianchi type magnetically charged hairy topological black hole solutions in string theory · Thermodynamic properties of the topological black hole solutions · 0.97
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One possible regularization procedure is that of [ref] , where we subtract the ([ref]) solution with $k<0$ and $μ=Q=0$.
Thermodynamics of the Maxwell-Gauss-Bonnet anti-de Sitter Black Hole with Higher Derivative Gauge Corrections · Free Energy · 0.96
Compares with it 1 paper
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This is in analogy with the topological black hole solutions where the constant curvature $k$ of horizon appears in the mass [ref] .] [equation] [equation]
New five-dimensional Bianchi type magnetically charged hairy topological black hole solutions in string theory · Thermodynamic properties of the topological black hole solutions · 0.85
Cites it as background 26 papers
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The general Schwarzschild-anti-de Sitter metric ansatz we will study is [equation] where $dΣ_k^2$ is a metric on a locally homogeneous three-dimensional surface of constant curvature $k$ ($k=+1$, $k=0$, and $k=-1$ correspond to three-dimensional sphere, flat, and hyperboloid, respectively) [ref] .
Probing extra dimension through gravitational wave observations of compact binaries and their electromagnetic counterparts · Null geodesics in five-dimensional AdS space-time · 0.87
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An alternative to a spherical AdS black hole would be to choose one with a hyperbolic horizon [ref] .
Low-lying quasinormal modes of topological AdS black holes and hydrodynamics · Introduction · 0.82
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The thermodynamics of BTZ black holes and their stability under appropriate boundary conditions have been investigated in [ref] .
Black holes with toroidal horizons in (d +1)-dimensional space-time · Introduction · 0.81
Discussion
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