Black holes with unusual topology

L. Vanzo

REVIEW

Black holes with unusual topology

L. Vanzo

General Relativity and Quantum Cosmology · Submitted 5 May 1997 · 41 references

Pith's verdict: A careful, honest thermodynamics paper: the S = A/4 fix for higher-genus AdS black holes is real but rests on a background choice the paper argues for rather than derives.

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arXiv IDgr-qc/9705004 (version 3)

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CitationL. Vanzo. "Black holes with unusual topology." arXiv:gr-qc/9705004 (1997). https://pith.science/paper/ZCK6LMDM

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Editor's note

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The thing to know: the metric class here is not new — Vanzo credits Mann, Brill-Louko-Peldan, and Aminneborg et al. up front — but the thermodynamics is. He assembles the off-shell Euclidean action for arbitrary genus with proper conical-singularity bookkeeping, and shows that the negative-entropy pathology people had been seeing for g > 1 is an artifact of the η0 = 0 subtraction. Switch the reference background to the extreme solution and you get S = A/4 and a non-negative mass spectrum. That is a genuine consistency fix, and it is done carefully: the Ricci decomposition, the horizon roots, the surface gravity, the Euclidean period, and the mass formula all follow standard routes and check out on inspection. The paper is also unusually candid about what it is doing — it says outright that there is freedom in defining the mass, and it flags the genus and the torus conformal class as non-dynamical parameters that ought to have a better origin. That kind of honesty is worth noting; it is not a paper hiding its assumptions. The soft spot is exactly where the stress-test puts it, though I would soften the phrasing. Eq. (1.44) makes the mechanism visible: the unwanted term is killed by the Kronecker factor at r0 = rc, not by any change in the physics. Vanzo's justification is that the extreme Euclidean section is regular for any imaginary-time period, so the background contributes no conical term. That is a regularity criterion, not a derivation from the dynamics, and it is fair to ask whether a naked singularity is a legitimate zero of energy. But in the Euclidean framework of the time this is the standard criterion for a ground state, so 'by fiat' overstates it. The real gap is that no independent Hamiltonian boundary-term argument or first-law check is offered to single out this background. That is a fair criticism, not a fatal one. The self-citation to the same arXiv id is a disclosed correction of a sign error, not circular support. Who should read it: anyone working on AdS black hole thermodynamics or the topology of horizons. It deserves a serious referee — the calculations are checkable and the claim is well-posed.

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Major objections (2)

[Sec. 1.2 and Sec. 1.4, Eq. (1.44)] The central claim of a positive-definite mass spectrum and S = A/4 depends on the choice eta_0 = -l/(3 sqrt 3). The paper acknowledges the freedom in defining the mass (Sec. 1.2: 'there is a certain amount of freedom in defining the mass of the black hole, as this involves the subtraction of a zero point of energy') and argues for this choice on the grounds of Euclidean regularity and the third law. However, the justification is a selection principle, not a derivation from the dynamics. The paper should be more explicit that the positive mass spectrum is a consequence of the subtraction convention, and that with eta_0 = 0 the spectrum includes negative masses and the entropy becomes (A - A_0)/4. The paper does discuss this in Sec. 1.4, but the abstract and the concluding sentence ('we conclude that the topological black holes ... forms a well behaved sequence of positive mass solutions') should be qualified accordingly.
[Sec. 1.4] The ground state of the claimed positive mass spectrum is the extreme solution, which the paper itself says 'is not in fact a black hole' but a naked singularity. Using a naked singularity as the zero of energy raises a cosmic censorship concern that the paper does not address. The paper should discuss whether this is physically acceptable or whether it indicates that the subtraction is unphysical. The analogy with the extreme Reissner-Nordström solution drawn in Sec. 1.4 is not fully convincing, because the extreme RN solution is still a black hole, whereas here the ground state is not.

Minor objections (10)

[Abstract] The abstract states that the genus g>1 black hole has 'no other free parameters apart from the mass and the charge', but the paper treats only uncharged solutions; the metric (1.4) contains no charge parameter. Either include charge in the analysis or remove it from the abstract.
[References] Reference [19] is listed as gr-qc/9705004, which is this paper's own arXiv identifier. This appears to be a self-citation error; the reference should be corrected to the intended earlier work.
[Eq. (1.8)] The expression for r_+ has a typesetting error: the denominator and numerator of the cube-root terms are garbled. The formula should be checked and corrected.
[Eq. (1.30)] The mass formula has a typesetting error: the bracket structure and the placement of the -l^2 term are unclear. The formula should be checked and corrected.
[Eq. (1.31)] The mass formula for the torus is garbled; it should read M = |Im tau| 8 pi^2 l^4 T^3 / 27.
[Eq. (1.38)] The action formula has a typesetting error in the bracket structure and the placement of the r_0 terms. The formula should be checked and corrected.
[Sec. 1.1] The statement 'the allowed range of negative mass being M > -l/(3 sqrt 3)' is missing the (g-1) factor; the correct bound is M > -l(g-1)/(3 sqrt 3). Similarly, 'the mass is bounded by a number of order l' in Sec. 1.2 should read order l(g-1).
[Abstract] The abstract says matter in anti-de Sitter space can violate 'the weak energy condition', but the topology theorems cited in the introduction assume the dominant energy condition. The paper should be precise about which energy condition is relevant.
[Sec. 1.1] The statement that the genus g>1 black hole has 'no other free parameters apart from the mass and the charge' is inconsistent with the uncharged metric (1.4); the only continuous parameters are eta (mass) and l (cosmological constant), plus the discrete genus g.
[Sec. 1.4] The comparison with the extreme Reissner-Nordström solution is a bit loose; the analogy could be clarified, particularly regarding why the extreme RN solution is a black hole while the extreme topological solution is a naked singularity.

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The paper

The paper claims

Black holes in anti-de Sitter space can have horizons shaped like tori or higher-genus Riemann surfaces, and their thermodynamics becomes well behaved — positive mass, entropy equal to one quarter the horizon area — once the zero of energy,

The reading

Einstein's equations with a negative cosmological constant do not force a black hole horizon to be a sphere. Vanzo shows that the standard static ansatz $ds^2=-V dt^2+V^{-1}dr^2+r^2 dsigma^2$ still solves the equations when the two-surface carrying $dsigma^2$ has constant curvature of either sign, so the horizon can be a torus or a Riemann surface of any genus $g>1$. Each such solution has a horizon radius, a surface gravity, and a temperature fixed by the period of the Euclidean continuation, and a mass defined by Hamiltonian subtraction against a reference solution of the same topology. The paper's real target is a bookkeeping problem: with the natural subtraction (the zero-mass, locally anti-de Sitter solution) the higher-genus entropy picks up a negative topological term and goes negative at low temperature, which is nonsense. Vanzo argues that subtracting instead the extreme solution — the one whose Euclidean section is regular at any period — removes the term and leaves $S=A/4$ with a non-negative mass spectrum.

What carries the argument

The load-bearing object is the lapse function $V=kappa-k'/r+r^2/ell^2$, which satisfies $R_{ab}=Lambda g_{ab}$ with $Lambda=-3ell^{-2}$ for any constant-curvature two-metric $sigma_{ij}$ with $R_{ij}=kappa,sigma_{ij}$. Rescaling sets $kappa=-1$ for $g>1$ and $kappa=0$ for the torus, giving $V=-1-2eta/r+r^2/ell^2$ and $V=-2eta/r+r^2/ell^2$ respectively, with the horizon at the positive root $r_+$. The temperature comes from requiring the Euclidean section to close smoothly at $r_+$: $beta_+=4piell^2 r_+/(3r_+^2-ell^2)$ for $g>1$ and $beta_+=4piell^2/(3r_+)$ for the torus. The mass comes from the Hamiltonian boundary term $-frac{1}{8pi}int_{S_g}sqrt{V}(Theta-Theta_0),r^2sqrt{sigma},d^2x$ evaluated at large radius, which forces the background to carry the same genus as the solution. The entropy comes from the off-shell Euclidean action with the Gauss-Bonnet conical-defect contribution $frac{A}{16pi}frac{beta_+}{beta}(beta_+-beta)$ for both solution and background.

Extended reading notes

Core claim

The central claim is that the whole family of topological anti-de Sitter black holes, from the torus up through every genus, forms a thermodynamically consistent sequence once the reference background used to define energy is chosen to be the extreme solution with $eta_0=-ell/(3sqrt{3})$ (equivalently $r_0=r_c=ell/sqrt{3}$). The mass formula $M=(eta-eta_0)(g-1)+eta|mathrm{Im},tau|,delta(g,1)/(4pi)$ ties the mass to the horizon geometry, the first law $dM=kappa,dA/8pi$ holds for every genus, and the off-shell Euclidean action $I=beta M-A/4+[1-delta(r_0,r_c)]A_0/4$ reproduces both the mean energy and the entropy. With the extreme background the indicator factor $[1-delta(r_0,r_c)]$ switches off, the conical contribution of the background disappears, and the entropy reduces to exactly one quarter of the horizon area, $S=A/4-pi r_0^2(g-1)[1-delta(r_0,r_c)]$. The author reads this as evidence that the negative-mass and negative-entropy branches found under the other subtraction were an artifact of the wrong zero of energy, and that the third law of thermodynamics lines up with the third law of black hole mechanics: the zero-temperature state is the extreme solution, which is a naked singularity rather than a black hole.

Load-bearing premise

The argument rests on treating the extreme solution — the one whose Euclidean section stays regular at any imaginary-time period — as the physically correct zero of energy, a choice the paper defends by regularity and by the disappearance of a negative entropy term rather than by deriving it from the dynamics; the author concedes there is freedom in where the zero of energy is placed.

Editorial extensions

If this is right

For every genus $g$, the first law $dM=kappa,dA/8pi$ holds in vacuum, and the mass is a monotonically increasing function of temperature, so the specific heat is positive and the canonical ensemble is stable.
The toroidal black hole has $r_+=(8piell^2M/|mathrm{Im},tau|)^{1/3}$, so its size is set by the mass and the torus's conformal class rather than by the cosmological constant; small toroidal black holes can exist at any mass.
Higher-genus black holes have no small-mass limit: their minimum size is of order $ell$, so they are cosmological objects that could only have existed when the cosmological constant was large.
The density of states grows as $rho(E)simexp(C_1E^{2/3})$ with exponent fixed by $A/4$, matching the behaviour of a massless gas in two spatial dimensions.
The genus, and for the torus the Teichmüller parameter $tau$, enter the metric as free non-dynamical parameters, so the spectrum contains disconnected sectors labelled by horizon topology.

Reading between the lines

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

If the entropy really is $A/4$ independent of subtraction, then the negative-entropy branch is a diagnostic of a bad reference frame rather than a physical instability, which suggests a general rule: any candidate zero of energy whose Euclidean section is not regular at arbitrary period should be suspect in anti-de Sitter thermodynamics.
The genus label behaving as a fixed external parameter is the sharpest loose end; a path-integral treatment that sums over horizon topologies would turn it into a dynamical variable and could in principle allow transitions between genus sectors, which the paper explicitly says it cannot rule out.
The same subtraction ambiguity should show up in the charged topological black holes, where an inner horizon gives a second natural candidate for the zero of energy; checking whether the extreme charged solution plays the same regularizing role would test whether the mechanism is about extremality rather than about this particular uncharged family.
Because the toroidal solution survives in a nearly flat universe while higher-genus ones do not, the observable content of this family is concentrated in the $g=1$ case, which makes the torus's conformal class the one free parameter worth asking about physically.

Original abstract

The Einstein's equations with a negative cosmological constant admit solutions which are asymptotically anti-de Sitter space. Matter fields in anti-de Sitter space can be in stable equilibrium even if the potential energy is unbounded from below, violating the weak energy condition. Hence there is no fundamental reason that black hole's horizons should have spherical topology. In anti-de Sitter space the Einstein's equations admit black hole solutions where the horizon can be a Riemann surface with genus $g$. The case $g=0$ is the asymptotically anti-de Sitter black hole first studied by Hawking-Page, which has spherical topology. The genus one black hole has a new free parameter entering the metric, the conformal class to which the torus belongs. The genus $g>1$ black hole has no other free parameters apart from the mass and the charge. All such black holes exhibits a natural temperature which is identified as the period of the Euclidean continuation and there is a mass formula connecting the mass with the surface gravity and the horizon area of the black hole. The Euclidean action and entropy are computed and used to argue that the mass spectrum of states is positive definite.

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.

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What the abstract itself states, as probabilities: claims first 0.14 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.29 · compares with prior results 0.61 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

The Einstein's equations with a negative cosmological constant admit solutions which are asymptotically anti-de Sitter space. Matter fields in anti-de Sitter space can be in stable equilibrium even if the potential energy is unbounded from below, violating the weak energy condition. Hence there is no fundamental reason that black hole's horizons should have spherical topology. In anti-de Sitter space the Einstein's equations admit black hole solutions where the horizon can be a Riemann surface with genus $g$. The case $g=0$ is the asymptotically anti-de Sitter black hole first studied by Hawking-Page, which has spherical topology. The genus one black hole has a new free parameter entering the metric, the conformal class to which the torus belongs. The genus $g>1$ black hole has no other free parameters apart from the mass and the charge. All such black holes exhibits a natural temperature which is identified as the period of the Euclidean continuation and there is a mass formula connecting the mass with the surface gravity and the horizon area of the black hole. The Euclidean action and entropy are computed and used to argue that the mass spectrum of states is positive definite.

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	The Einstein's equations with a negative cosmological constant admit solutions which are asymptotically anti-de Sitter space.	background 1.00	0.06	0.04	asserted 0.99
s1	Matter fields in anti-de Sitter space can be in stable equilibrium even if the potential energy is unbounded from below, violating the weak energy condition.	background 0.87	0.13	0.07	asserted 0.86
s2	Hence there is no fundamental reason that black hole's horizons should have spherical topology.	interpretation 0.79	0.16	0.47	asserted 0.94
s3	In anti-de Sitter space the Einstein's equations admit black hole solutions where the horizon can be a Riemann surface with genus $g$.	result 0.59	0.48	0.47	asserted 0.98
s4	The case $g=0$ is the asymptotically anti-de Sitter black hole first studied by Hawking-Page, which has spherical topology.	background 0.98	0.06	0.06	asserted 1.00
s5	The genus one black hole has a new free parameter entering the metric, the conformal class to which the torus belongs.	result 0.88	0.63	0.06	asserted 1.00
s6	The genus $g>1$ black hole has no other free parameters apart from the mass and the charge.	result 0.93	0.66	0.10	asserted 1.00
s7	All such black holes exhibits a natural temperature which is identified as the period of the Euclidean continuation and there is a mass formula connecting the mass with the surface gravity and the horizon area of the black hole.	result 0.95	0.68	0.17	asserted 1.00
s8	The Euclidean action and entropy are computed and used to argue that the mass spectrum of states is positive definite.	result 0.84	0.85	0.21	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v3

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

References

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71 later papers cite it

Evidence: substantial. 71 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 110 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 1 used its method or tool and 2 cited it as support.

Uses its method or tool 1 paper

Used the standard ADM result, as in this paper, to compute the mass.

Black hole and BTZ-black string in the Einstein-SU(2) Skyrme model (2018) · 0.96 · paraphrased: the licence does not allow quoting

Cites it as support 2 papers

Said that this supports the consideration of some of these extremal solutions as groundstates, citing this paper.

Cosmic censorship in Lovelock theory (2013) · 0.98 · paraphrased: the licence does not allow quoting
Said that this supports the consideration of some of these extremal solutions as groundstates, citing this paper.

Lovelock gravity, black holes and holography (2015) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

65 arXiv papers cite this in 99 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 · 51 sentences without a confident label

Builds on it 1 paper

Les trous noirs topologiques rencontr�s � la sous-section [ref] en dimension $D=4$ [ref] se g�n�ralisent facilement en dimension $D ≥4$ [ref] .

Black hole solutions of modified gravity theories · La solution de Tangherlini et ses versions topologiques · 0.87

Uses its data or values 1 paper

The Hawking temperature of the MTBH is equal to [ref] – [ref] [equation] and from the QNF ([ref]) of the massless Dirac field we find [equation] We see that the first expression in formulas ([ref]) saturates the inequality ([ref]) and that the second expression does not satisfy the previously mentioned inequality.

Quasinormal frequencies of the Dirac field in the massless topological black hole · Hod's bound · 0.89

Compares with it 1 paper

We also note from ([ref]) and ([ref]) that the mass becomes negative at zero temperature [equation] This is somewhat similar to the AdS-Schwarzschild with negative curvature horizon [ref] .

Dissipative force on an external quark in heavy quark cloud · Introduction · 0.87

Cites it as support 1 paper

This supports the consideration of some of these extremal solutions as groundstates [ref] .

Cosmic censorship in Lovelock theory · The puzzle of multi horizon black holes · 0.98

Cites it as background 38 papers

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
These black holes are known as topological black holes [ref] .

Hawking Radiation via Gravitational Anomalies in Non-spherical Topologies · Mode Analysis of the Wave Equation of a Scalar Field in the Background of a TBH · 0.91
For the uncharged topological black holes of Einstein's theory the first law of the thermodynamics reads [ref] – [ref] [equation] where $M$ stands for the mass of the black hole, $κ$ denotes the surface gravity of the event horizon, and $A$ stands for the horizon area.

Entropy spectrum of the D-dimensional massless topological black hole · Introduction · 0.82

Discussion

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