Topological black holes in the dimensionally continued gravity

Kwang-Sup Soh (SNU); Rong-Gen Cai

REVIEW

Topological black holes in the dimensionally continued gravity

Rong-Gen Cai, Kwang-Sup Soh (SNU)

General Relativity and Quantum Cosmology · Submitted 25 Aug 1998 · 38 references

Pith's verdict: A genuine but narrow extension of the BTZ dimensionally continued black holes to flat and hyperbolic horizons — with a real algebra error in the k=-1 entropy and two other stated sub-results that don't survive checking.

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CitationKwang-Sup Soh (SNU), Rong-Gen Cai. "Topological black holes in the dimensionally continued gravity." arXiv:gr-qc/9808067 (1998). https://pith.science/paper/4S6IHBWS

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Editor's note

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The new content is real but narrow: the k=0 and k=-1 solutions (3.34)-(3.42) and the charged family (4.14) are not in [26], and the thermodynamics does depend qualitatively on k. That's worth having on record, and the paper is a clean, workmanlike extension rather than a surprise. [abstract:s4] and [abstract:s6] are the honest summary of what's added. [abstract:s0] is accurate about the setting but undersells how incremental the step is from [26]. What's done well: the first-law route to entropy (3.20) is clean and skips the Hamiltonian machinery, and the k=1 limit reproduces [26]'s result — a real cross-check, not a circularity. The Appendix's derivation of the area law for Einstein-Maxwell topological holes (A4) is a useful self-contained anchor. The authors are also upfront that the k=-1 zero-mass solution is itself a black hole at r+=l, so the 'S vanishes as the horizon disappears' criterion is a convention there; they say so in III.D and defend it in the Appendix. That's acknowledged, not hidden, and I won't count it against them. Where it goes wrong: the stress-test note is right, and this is the sharpest issue. The integral in (3.21) evaluates to πl²{[k+(r+/l)²]^{n-1} − k^{n-1}}, not −k. For k=1 and k=0 the two agree; for k=−1 they differ by 2πl² whenever n is odd, i.e. D=6,10,.... So (3.49) overstates the hyperbolic entropy in those dimensions. That's an algebra slip in a stated result, not a convention. Two more. The negative-mass condition is n=2k̃+2 in III.D and n=2k̃+1 in the Conclusions; the section is right (n−1 must be odd for (2M/r)^{1/(n−1)} to be real with M<0), the conclusion is a typo. And (3.50) is dimensionally wrong: M carries length dimension in even D, so Mc should scale as l, not l²; extremizing g²=0 directly gives Mc = −(l/2)(2n−2)^{n−1}/(2n−1)^{n−1/2}, which doesn't match. I checked the reader's fourth point (heat capacity positivity for negative mass) and it doesn't land — for n odd, r+>l and the factors are positive. None of this touches the central claim that these topological solutions exist with the stated qualitative thermodynamics. But the entropy and critical-mass formulas shouldn't be used downstream as printed. Who it's for: people working on topological and Lovelock black holes and AdS boundary thermodynamics. Worth a referee report; the fixes are small and local.

Referee report

Major objections (1)

[Section III.A, Eq. (3.21) and Eq. (3.49); Section IV, Eq. (4.22)] The integral S = 2π(n-1)∫_0^{r+} r[k+(r/l)²]^{n-2}dr evaluates to πl²{[k+(r+/l)²]^{n-1} - k^{n-1}}, not πl²{[k+(r+/l)²]^{n-1} - k}. For k=1 and k=0 the two agree, but for k=-1 and n odd (D=6,10,14,...) they differ by 2πl². The paper's (3.49) therefore overstates the entropy of hyperbolic black holes in those dimensions, and the corrected integral would give negative entropy for the zero-mass solution (3.43) when n is odd, which shows that the additive constant S0 in (3.19) must be stated explicitly rather than absorbed into an incorrect algebraic evaluation. This is a load-bearing point for the entropy claim.

Minor objections (6)

[Eq. (3.12), odd dimensions] The term (D-2)(D-3)/l² (M+C0)^{1/(n-1)} should have r² in the denominator, not l²; as written it is missing the radial dependence.
[Eq. (3.12), even dimensions] The coefficient 2D(D-2)² + D² - 5D + 2 does not reproduce the correct curvature scalar; for D=4 it gives 30 where the correct coefficient is 0.
[Eq. (3.47), even dimensions] The denominator 4π(n-1)l² should be 4π(n-1)r+, as follows from (3.17); the expression as written is dimensionally inconsistent.
[Conclusions] The statement that negative mass black holes exist when n = 2k+1 contradicts the k=-1 section, which says n = 2k+2; the latter is correct because (2M/r)^{1/(n-1)} is real for negative M only when n-1 is odd.
[Eqs. (3.21) and (3.22)] The integration variable is r, but the integrand is written with r+; this is a notation slip.
[Eq. (3.45)] The paper gives S = πl² for the zero-mass k=-1 black hole in even dimensions; with the corrected integral this would be -πl² for n odd, which reinforces the need to state the S0 convention explicitly.

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The paper

The paper claims

In a two-parameter truncation of Lovelock gravity, static black holes exist whose horizons are constant-curvature hypersurfaces with $k=+1$, $0$ or $-1$, and their Hawking temperature, entropy and heat capacity all depend qualitatively on $

The reading

The paper asks whether the spherically symmetric black holes known in the dimensionally continued gravity of Bañados, Teitelboim and Zanelli survive when the horizon is not a sphere. The authors write down static solutions whose $(D-2)$-dimensional horizon is a constant-curvature hypersurface with curvature $(D-2)k$, where $k=+1$ gives the sphere, $k=0$ the torus and $k=-1$ a hyperbolic surface, and they work out the metric function in both even and odd spacetime dimensions. The point of the exercise is that this gravity is a special Lovelock theory in which all the higher-curvature couplings are fixed by just two numbers, a cosmological constant and a gravitational constant, so the resulting black holes are a clean laboratory rather than a tuneable mess. Having the solutions, the authors compute temperature, entropy and heat capacity for each $k$, and find that the answers differ in kind, not merely in numerical detail: in odd dimensions the heat capacity is always positive, while in even dimensions the spherical case has a sign change and an infinite jump at a critical horizon radius that the flat and hyperbolic cases do not have. That is the claim a reader should take away: horizon topology, not just horizon size, controls whether these black holes can sit in equilibrium with their own radiation.

What carries the argument

Two pieces of machinery carry the argument. The first is the embedding of the Lorentz group $SO(D-1,1)$ into the anti-de Sitter group $SO(D-1,2)$, which collapses the $[D/2]$ free Lovelock coefficients $alpha_p$ to the two-parameter form of Eq. (2.3): a Chern-Simons Lagrangian for the anti-de Sitter group in odd dimensions and an Euler density built from the Lorentz part of the anti-de Sitter curvature in even dimensions. The second is the Hamiltonian constraint of that action, which for the static ansatz collapses to the statement that a single function $F[r,g]$ is constant, $F=frac{1}{2}r[k+(r/l)^2-g^2]^{n-1}$ in even dimensions and $[k+(r/l)^2-g^2]^{n-1}$ in odd dimensions, so the metric function follows by algebra rather than by solving a differential equation. Entropy is then extracted from $S=int_0^{r_+} T^{-1}(partial M/partial r_+)_Q,dr_+$, a route that deliberately avoids having to know the gravitational action's surface term.

Extended reading notes

Core claim

The central claim is that the dimensionally continued gravity admits static topological black holes, with metric $ds^2=-g^2 dt^2+g^{-2}dr^2+r^2 dSigma_{D-2}^2$ and $g^2 = k-(2M/r)^{1/(n-1)}+(r/l)^2$ in $D=2n$, $g^2=k-(M+tilde C_0)^{1/(n-1)}+(r/l)^2$ in $D=2n-1$, for each of $k=+1,0,-1$. The authors then derive the Hawking temperature $T=[k+(2n-1)(r_+/l)^2]/[4pi(n-1)r_+]$ in even dimensions and $T=r_+/(2pi l^2)$ in odd dimensions, obtain the entropy from the first law of thermodynamics rather than from the horizon area, and find that it does not obey the area formula; in even dimensions it is $S=pi l^2{[k+(r_+/l)^2]^{n-1}-k}$. The heat capacity follows, and the paper's emphasis is that the three curvature cases behave differently: for $k=0$ and $k=-1$ the heat capacity is always positive, whereas for $k=+1$ in even dimensions it is negative below $r_+=l/sqrt{2n-1}$ and positive above, with an infinite discontinuity at the transition. The charged generalization is worked out in the same way, and an appendix shows that the analogous topological black holes of higher-dimensional Einstein-Maxwell theory do obey the area law, so the entropy formula is a fingerprint of the gravity theory rather than of the horizon shape.

Load-bearing premise

The entropy is obtained by integrating the first law with the integration constant fixed by demanding that entropy vanish when the horizon disappears, but for $k=-1$ the zero-mass solution is itself a black hole with a horizon at $r_+=l$, so that criterion is a choice of reference background and the quoted entropies shift by a constant if a different background is chosen.

Editorial extensions

If this is right

For $k=0$ and $k=-1$, the heat capacity is positive for all horizon radii, so these black holes can be in thermal equilibrium with Hawking radiation at any temperature, unlike the even-dimensional spherical case.
In even dimensions with $k=+1$ there is a critical radius $r_+=l/sqrt{2n-1}$ where the heat capacity diverges, giving a sharp thermodynamic transition that flat and hyperbolic horizons lack.
For $k=-1$ the zero-mass solution is itself a black hole with horizon at $r_+=l$ and temperature $T=1/(2pi l)$ in every dimension, while its entropy differs between even and odd dimensions.
Negative-mass black holes are allowed for $k=-1$ when $n=2tilde k+2$, up to a critical mass beyond which the $r=0$ singularity is naked.
In the charged case the entropy formula is unchanged from the uncharged case, consistent with entropy coming from a gravitational surface term and not from the matter Lagrangian.

Reading between the lines

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

Because the entropy is fixed only up to the choice of reference background, the $k=-1$ results are best read as relative statements; a reader who wants absolute entropies would need an independent surface-term or Noether-charge computation to fix the additive constant.
The stability conclusion is a linear (heat-capacity) statement only; it says nothing about whether the toroidal and hyperbolic solutions are dynamically stable under gravitational perturbations, which would be the natural next test.
The fact that the same zero-mass $k=-1$ geometry has different entropies in different gravity theories suggests that any attempt to count microstates for these solutions must be tied to the specific higher-curvature action, not to the geometry alone.
Since the odd-dimensional action is a Chern-Simons form for the anti-de Sitter group, these solutions are natural candidates for a boundary conformal field theory description, and the always-positive odd-dimensional heat capacity is what such a description would have to reproduce.

Original abstract

We investigate the topological black holes in a special class of Lovelock gravity. In the odd dimensions, the action is the Chern-Simons form for the anti-de Sitter group. In the even dimensions, it is the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. The Lovelock coefficients are reduced to two independent parameters: cosmological constant and gravitational constant. The event horizons of these topological black holes may have constant positive, zero or negative curvature. Their thermodynamics is analyzed and electrically charged topological black holes are also considered. We emphasize the differences due to the different curvatures of event horizons. As a comparison, we also discuss the topological black holes in the higher dimensional Einstein-Maxwell theory with a negative cosmological constant.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,049 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.37 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.90 · 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.

We investigate the topological black holes in a special class of Lovelock gravity. In the odd dimensions, the action is the Chern-Simons form for the anti-de Sitter group. In the even dimensions, it is the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. The Lovelock coefficients are reduced to two independent parameters: cosmological constant and gravitational constant. The event horizons of these topological black holes may have constant positive, zero or negative curvature. Their thermodynamics is analyzed and electrically charged topological black holes are also considered. We emphasize the differences due to the different curvatures of event horizons. As a comparison, we also discuss the topological black holes in the higher dimensional Einstein-Maxwell theory with a negative cosmological constant.

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	We investigate the topological black holes in a special class of Lovelock gravity.	contribution 1.00	0.91	0.05	asserted 0.90
s1	In the odd dimensions, the action is the Chern-Simons form for the anti-de Sitter group.	background 0.80	0.20	0.04	asserted 1.00
s2	In the even dimensions, it is the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor.	background 0.76	0.20	0.03	asserted 1.00
s3	The Lovelock coefficients are reduced to two independent parameters: cosmological constant and gravitational constant.	result 0.36	0.57	0.08	asserted 1.00
s4	The event horizons of these topological black holes may have constant positive, zero or negative curvature.	result 0.64	0.49	0.10	hedged possibility 1.00
s5	Their thermodynamics is analyzed and electrically charged topological black holes are also considered.	contribution 0.95	0.90	0.07	asserted 0.99
s6	We emphasize the differences due to the different curvatures of event horizons.	contribution 0.57	0.72	0.11	asserted 0.64
s7	As a comparison, we also discuss the topological black holes in the higher dimensional Einstein-Maxwell theory with a negative cosmological constant.	contribution 0.60	0.78	0.04	asserted 0.88

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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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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46 later papers cite it

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Followed this paper to obtain the black hole entropy for fixed Q, P, and α.

Extended thermodynamics and microstructures of four-dimensional charged Gauss-Bonnet black hole in AdS space (2020) · 0.83 · paraphrased: the licence does not allow quoting
Followed this paper's approach to work out the black hole entropy from an integral of 1/T with respect to r_h plus an integration constant.

Born-Infeld Black Holes in 4D Einstein-Gauss-Bonnet Gravity (2020) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

40 arXiv papers cite this in 58 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 · 31 sentences without a confident label

Builds on it 2 papers

Since the solutions ([ref]) and ([ref]) are asymptotically AdS, we will generalize those solutions to the case of topological black holes with any constant scalar curvature horizon [ref] .

Topological Black Holes in Horava-Lifshitz Gravity · Introduction · 0.93
This is an extension of the solutions previously discussed in [ref] .

Asymptotically Lifshitz wormholes and black holes for Lovelock gravity in vacuum · Asymptotically Lifshitz black hole · 0.99

Uses its method or tool 6 papers

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
Following [ref] , for fixed $Q$, $P$, and $α$, the black hole entropy is obtained [ref] [equation] where $A=4πr_h^2$ is the area of the black hole horizon and $A_0$ is an integral constant with dimension of [length]$^2$.

Extended thermodynamics and microstructures of four-dimensional charged Gauss-Bonnet black hole in AdS space · Black hole thermodynamics · 0.83
However we can derive the entropy of the black hole following the approach of [ref] and using the expression [equation] where $S_0$ is an integration constant.

Black Holes in $4D$ AdS Einstein Gauss Bonnet Gravity With Power- Yang Mills Field · Thermodynamics and Thermal Stability of $4D$ AdS EPYMGB Black Hole · 1.00

Compares with it 3 papers

When applied to a static black hole solution, the mass agrees with the literature [ref] .

Quasi-local energy and ADM mass in pure Lovelock gravity · Introduction · 0.97
In addition let us notice that the solution ([ref]) is the same as the topological black hole solution in the five-dimensional Chern-Simons gravity [ref] , or in the five-dimensional Gauss-Bonnet gravity with a special Gauss-Bonnet coefficient [ref] .

On the z=4 Horava-Lifshitz Gravity · Solutions · 1.00
As usual in Lovelock gravity, black hole entropy in CS-AdS theory cannot be related to the horizon area, but just expressed from eq.([ref]) in terms of $r_+$ as [equation] The last result matches the one obtained using a mini-superspace model in the canonical formalism [ref] , and also the prescription for the entropy as a given $(D-2)-$form integrated at the horizon [ref] .

Universal regularization prescription for Lovelock AdS gravity · Particular Cases · 0.97

Cites it as background 15 papers

This section discusses the electrically charged black hole in Gauss-Bonnet gravity with the cosmological constant [ref] .

Bulk-Boundary Thermodynamics of Charged Black Holes in Higher Derivative Theory · Example 1: Charged black holes in Gauss-Bonnet gravity · 0.89
For a generic case, although the Lagrangian ([ref]) looks complicated, some exact black hole solutions have been found and their associated thermodynamics was investigated in [ref] .

Generalized Vaidya Spacetime in Lovelock Gravity and Thermodynamics on Apparent Horizon · Introduction · 0.80
Being the only higher-curvature theories with second-order equations, Lovelock gravities have been thoroughly studied in the literature — regarding black hole solutions, see e.g. Refs. [ref] .

Electromagnetic Quasitopological Gravities · Introduction · 0.95

Discussion

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