Rotating Topological Black Holes

D. Klemm; L. Vanzo; V. Moretti

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Rotating Topological Black Holes

D. Klemm, V. Moretti, L. Vanzo

General Relativity and Quantum Cosmology · Submitted 28 Oct 1997 · 38 references

Pith's verdict: Two explicit rotating AdS metrics, both checkable by substitution; the g>1 case is genuinely new, but its 'rotating black hole' reading is left half-built and the authors say so themselves.

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arXiv IDgr-qc/9710123 (version 2)

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CitationD. Klemm, L. Vanzo, V. Moretti. "Rotating Topological Black Holes." arXiv:gr-qc/9710123 (1997). https://pith.science/paper/F2LNHH54

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Two explicit rotating anti-de Sitter metrics, one toroidal and one with genus g>1 horizon, both of which a competent relativist can verify by plugging into R_{μν} = Λg_{μν}. That is the paper, and it is worth reading for that alone. The g>1 solution was not previously known, and the toroidal one is explicitly distinguished from Lemos-Zanchin rather than being a relabelling of it. Embedding both as limits of Plebanski-Demianski (Eqs. 20-25) is a good move: it gives an external benchmark instead of leaving the reader to trust the analytic continuation. The four-patch construction that makes the toroidal metric smooth on T^2 (Sec IV, the P = λ sin θ / λ cos θ business) is done properly, not hand-waved, and the J ≤ M^{5/3} bound falls out cleanly from a_c in Eq. (48). That part is solid work and I would cite it without hesitation for the toroidal case. The analyticity-domain argument in Sec II (the 'one can easily find piecewise smooth trajectories' paragraph) is a presentational shortcut, not a load-bearing gap. Direct substitution settles the metric, and the Plebanski-Demianski limit is a second independent route. I would not hold the paper up over it. The real soft spot is the g>1 physical interpretation, and the authors are upfront about it: no global axial Killing field, so J is only locally conserved; the integral I in Eq. (42) is left unevaluated ('we were unable to do this, however'); mass is defined only relative to a non-unique background, and the quasi-local energy difference vanishes identically; the surface gravity's reading as a temperature 'remains a little bit obscure.' They also decline to claim uniqueness or collapse formation. So 'rotating black hole' for g>1 currently means a metric with a horizon and a locally defined angular velocity, not a thermodynamic object. That is a fair description of where the field was in 1997, and the paper does not oversell it. One correction to the reader's summary: the classifier record says 'limitations: none found.' That is wrong. The paper states at least six, and they are the interesting part of Sec III. Send it to referees. Ask them to press on whether the g>1 thermodynamics can be pushed further and whether the analyticity step should be replaced by an explicit substitution check in the published version.

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Major objections (3)

[Sec. II, paragraph beginning 'Generally speaking'] The paragraph offers what it calls an independent proof that (10) solves Einstein's equations, but the argument is another analytic-continuation argument: it asserts that the continued parameters lie in the same connected analyticity domain as the real Kerr-de Sitter parameters and that 'one can easily find piecewise smooth trajectories' in IC^7 joining them. The trajectory is not exhibited, and the Plebanski-Demianski route (Eqs. 20-25) involves an infinite rescaling that the authors themselves flag as doubtful ('Anyhow, one could doubt of the procedure as it involves an infinite limit of some parameters'). Since the central claim of the paper is that (10)-(11) solves the field equations, please add a direct substitution check (or state that one has been performed, e.g., with a symbolic algebra system), or exhibit the trajectory. This is a presentational gap rather than a suspected error, but it is the load-bearing verification.
[Sec. II, compactification, Eqs. (15)-(19)] The compactification argument has two gaps. First, the existence proof asserts that a small geodesic polygon has angle sum larger than 2π 'since the metric (15) approaches a flat metric for θ→0'; but flatness gives an angle sum close to 2π, and the sign requires K<0 near θ=0, which is not established (at θ=0 the metric is flat to leading order, dσ^2 ≈ (r^2+a^2)/Ξ(dθ^2+θ^2dφ^2), and K is O(θ^2)). Second, the bootstrap from C^{1,α} to C^{2,α} requires K∈C^{0,α}, which is asserted but not shown: K depends on second derivatives of the metric, which are only known to be bounded. Since the identification is by isometries of a real-analytic local metric, the quotient is smooth by the standard theory of properly discontinuous group actions, and the elliptic detour is unnecessary; please either use that route or supply the two estimates.
[Sec. III.D, Eq. (42) and the paragraph following Eq. (41)] The angular momentum integral I in Eq. (42) is left unevaluated ('we were unable to do this, however'), so J is not actually determined as a function of a and η; and the statement that E(R)-E_0(R)=0 for backgrounds with the same a but different η implies that all solutions with equal a have the same quasi-local energy, which sits uneasily with the mass M in Eq. (41) that does depend on η. The authors acknowledge the background-subtraction problem, but the paper should at least give I for a symmetric fundamental domain (or a bound), and clarify the relation between M and E. This bears directly on whether the g>1 solutions are 'rotating black holes' in a physically meaningful sense, which is the paper's headline claim.

Minor objections (10)

[Abstract] 'blackholes' should be 'black holes', and 'type-$D$' and 'type D' are used inconsistently.
[Sec. II, Eq. (5)] 'rappresentative' should be 'representative'.
[Sec. II] 'Hoelder' should be 'Hölder', 'analitically' should be 'analytically', and 'indeedly' should be 'indeed'.
[Sec. II, Eq. (19)] The elliptic operator L and the Plebanski-Demianski structure function L (Eqs. 21, 24) share a symbol; please rename one to avoid confusion.
[Sec. III.B] The statement that case (iii,d) 'for a→0 gives the naked singularity discussed in [2,3]' needs clarification: at η=η_- the static solution has a double root at r=ℓ/√3 (an extremal horizon), not a naked singularity, and the naked case is η<η_-.
[Sec. III.C, Eq. (33)] The derivation of κ is not shown; a few intermediate steps, especially the normalization N in Eq. (32), would help the reader check the θ-independence of the result.
[Sec. IV, patch construction] The assignment P=λ sinθ near θ=0,π and P=λ cosθ near θ=π/2,3π/2 should be stated more carefully; as written, both patches appear to be centered at P=0 rather than at the identification points P=±T/2.
[Sec. IV, Eq. (50)] The argument that the horizon is a torus uses conformal flatness, but in two dimensions the scalar curvature is a total divergence for any metric; the conformal-flatness step is unnecessary and the phrasing should be corrected.
[Sec. IV] The Petrov type of the toroidal solution is not shown; since the abstract claims all solutions are type D, please state Ψ2 for (43) or note that it follows from the PD limit.
[References] References [2] and [3] are cited as unpublished or to be published; please update them at proof.

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The paper

The paper claims

The paper constructs exact solutions of Einstein's equations with a negative cosmological constant that describe spinning black holes whose horizons are tori or higher-genus Riemann surfaces, extending the known static topological black-hor

The reading

Static black holes whose event horizons are not spheres but tori or multi-handled surfaces are known to exist once a negative cosmological constant is added to Einstein's equations. Whether such horizons can also *spin* was open: a no-go result ruled out a rotating analogue of the three-dimensional BTZ black hole, so any rotating generalization had to live in a spacetime of non-constant curvature. This paper writes down two such families in closed form. The first is obtained by analytically continuing the Kerr--de Sitter metric (the spinning black hole of de Sitter space) to negative cosmological constant and imaginary rotation, producing black holes whose horizon is a compact Riemann surface of any genus $g>1$; the second is a toroidal black hole read off as a limit of the Plebański--Demiański type-$D$ family. If the construction holds, rotating topological black holes are not forbidden after all, and the toroidal case in particular comes with a clean mass, a clean angular momentum $J = M a$, and a hard upper bound on how fast it can spin.

What carries the argument

Three pieces carry the argument. First, the analytic continuation of Kerr--de Sitter: the authors argue that because the Einstein tensor is an analytic function of the metric components, their inverses and their derivatives, and because the continued parameter values can be joined to the real Kerr--de Sitter values by a path in $mathbb{C}^7$ avoiding the singularities at $Delta_r = 0$ and at the vanishing of the metric determinant, the field equations must continue to hold. Second, the compactification of the $(theta,varphi)$ sector: the induced two-metric on constant-$r$ surfaces is no longer of constant curvature, so the authors identify sides of a geodesic $4g$-gon whose size is fixed by the Gauss--Bonnet requirement $sum_i beta_i = 2pi$, and argue $C^{1,alpha}$ smoothness upgrades to $C^{2,alpha}$ by elliptic regularity. Third, the Plebański--Demiański type-$D$ metric serves as an independent cross-check, since both new solutions arise as limits of it with particular parameter choices.

Extended reading notes

Core claim

The central claim is that Eqs. (10)--(11) and Eqs. (43)--(46) both solve Einstein's equations with $R_{ij} = (2/3)|Lambda| ... $ — more precisely $R_{ij} = 2|Lambda| g_{ij}$ with $|Lambda| = -3/ell^{-2}$ — and both describe uncharged rotating black holes with non-spherical horizons. The genus $g>1$ metric comes from applying the continuation $t -> i t$, $r -> i r$, $theta -> i theta$, $eta -> -i eta$, $a -> i a$ to Kerr--de Sitter, and reduces to the known static topological black hole when $a=0$. Its Weyl tensor has a single non-vanishing tetrad component, $Psi_2 = -2eta(r + i a cosh theta)^{-2}$, so the solution is Petrov type $D$ and the Kerr--de Sitter curvature singularity at $rho^2=0$ disappears, since $r^2 + a^2 cosh^2 theta$ never vanishes; the manifold extends through $r=0$ and necessarily contains closed timelike curves. The toroidal metric (43) instead keeps a genuine curvature singularity at $rho^2 = 0$, retains a global rotational Killing field, and has horizons at the roots of $Delta_r = a^2 - 2 m r + ell^{-2} r^4$, which exist only while $a$ stays below $a_c = sqrt{3}(m/2)^{2/3} ell^{1/3}$.

Load-bearing premise

The genus $g>1$ solution is a solution only if the analytically continued Kerr--de Sitter parameters really lie in the same connected analyticity domain as the real ones, and the paper asserts that connecting path exists (saying one can 'easily find piecewise smooth trajectories') rather than exhibiting it, while its own Plebański--Demiański cross-check passes through an infinite-parameter limit the authors themselves flag as doubtful.

Editorial extensions

If this is right

Rotating black holes with toroidal and higher-genus horizons exist as exact solutions, so the absence of a rotating BTZ analogue does not generalize to four dimensions with a negative cosmological constant.
The toroidal solution carries a conserved mass $M = m T / 2pi$ and angular momentum $J = M a$, with $a$ capped at $a_c$, so the angular momentum of such a black hole is bounded by a power of its mass ($J$ scales at most as $M^{5/3}$).
For genus $g>1$ the horizon admits no global rotational Killing field, so the angular momentum is only conserved relative to a chosen set of observers at infinity, and the first law of black hole thermodynamics needs reformulating in a quasi-local language.
In one extreme case the static configuration is a naked singularity, but an arbitrarily small rotation turns it into a black hole, which the authors read as support for cosmic censorship.
The surface gravity of the genus $g>1$ solution, Eq. (33), comes out constant over the horizon even without rotational symmetry, so its interpretation as a Hawking temperature is left unclear.

Reading between the lines

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

Because the higher-genus solutions have no global axial symmetry, they behave less like spinning black holes and more like stationary solitons of anti-de Sitter gravity; testing whether they are linearly stable under perturbations would separate the two readings.
The bound $a <= a_c$ on the toroidal solution is the four-dimensional analogue of the Kerr extremality bound, so one could ask whether $J_{max} prop M^{5/3}$ survives in the charged Lemos--Zanchin toroidal solution and whether it matches a thermodynamic stability criterion.
The analytic-continuation argument is really a claim about connected components of a parameter space minus a discriminant locus; a direct symbolic evaluation of the Einstein tensor for Eqs. (10)--(11) would settle it without any complex-path reasoning.
Since the genus-$g$ horizon lacks a global Killing field, the natural next step is a quasi-local Brown--York computation on a finite boundary rather than an asymptotic one, which is exactly the programme the authors say they were unable to complete.

Original abstract

A class of metrics solving Einstein's equations with negative cosmological constant and representing rotating, topological black holes is presented. All such solutions are in the Petrov type-$D$ class, and can be obtained from the most general metric known in this class by acting with suitably chosen discrete groups of isometries. First, by analytical continuation of the Kerr-de Sitter metric, a solution describing uncharged, rotating black holes whose event horizon is a Riemann surface of arbitrary genus $g > 1$, is obtained. Then a solution representing a rotating, uncharged toroidal black hole is also presented. The higher genus black holes appear to be quite exotic objects, they lack global axial symmetry and have an intricate causal structure. The toroidal blackholes appear to be simpler, they have rotational symmetry and the amount of rotation they can have is bounded by some power of the mass.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,931 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.47 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.61 · compares with prior results 0.29 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

A class of metrics solving Einstein's equations with negative cosmological constant and representing rotating, topological black holes is presented. All such solutions are in the Petrov type-$D$ class, and can be obtained from the most general metric known in this class by acting with suitably chosen discrete groups of isometries. First, by analytical continuation of the Kerr-de Sitter metric, a solution describing uncharged, rotating black holes whose event horizon is a Riemann surface of arbitrary genus $g > 1$, is obtained. Then a solution representing a rotating, uncharged toroidal black hole is also presented. The higher genus black holes appear to be quite exotic objects, they lack global axial symmetry and have an intricate causal structure. The toroidal blackholes appear to be simpler, they have rotational symmetry and the amount of rotation they can have is bounded by some power of the mass.

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	A class of metrics solving Einstein's equations with negative cosmological constant and representing rotating, topological black holes is presented.	contribution 1.00	0.91	0.53	asserted 1.00
s1	All such solutions are in the Petrov type-$D$ class, and can be obtained from the most general metric known in this class by acting with suitably chosen discrete groups of isometries.	result 0.69	0.60	0.22	asserted 1.00
s2	First, by analytical continuation of the Kerr-de Sitter metric, a solution describing uncharged, rotating black holes whose event horizon is a Riemann surface of arbitrary genus $g > 1$, is obtained.	contribution 0.68	0.91	0.19	asserted 1.00
s3	Then a solution representing a rotating, uncharged toroidal black hole is also presented.	contribution 0.83	0.89	0.08	asserted 1.00
s4	The higher genus black holes appear to be quite exotic objects, they lack global axial symmetry and have an intricate causal structure.	interpretation 0.71	0.70	0.11	hedged possibility 0.94
s5	The toroidal blackholes appear to be simpler, they have rotational symmetry and the amount of rotation they can have is bounded by some power of the mass.	result 0.61	0.77	0.10	hedged possibility 0.77

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

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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relations to other papers

levels run

P abstract

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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

Figures

**Figure 5.** FIG. 5. Penrose-Carter diagram for the toroidal blackhole in the extreme case where

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References

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How the literature uses it

35 arXiv papers cite this in 66 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 · 47 sentences without a confident label

Contests it 2 papers

The black hole is “rotating" in the $ζ$ direction, which however does not have a circular topology[In [ref] , both planar coordinates are compactified to circles, but this will give rise to a discontinuity [see the erratum to that work] in the metric unless the time coordinate is also involved in the compactification, which would lead to closed timelike worldlines.

Fragile Black Holes and an Angular Momentum Cutoff in Peripheral Heavy Ion Collisions · Section · contest 0.74
However, it was realised (see the errata of [ref] ) that when rotation is present, the coordinate identifications that turn the hyperbolic surface into a Riemann surface of higher genus cannot be performed consistently.

Black holes with bottle-shaped horizons · Introduction · contest 0.81

Builds on it 6 papers

The fact that J$^*$ computes the ADM angular momentum of the KMV black hole discussed below follows from the calculations in [ref] , suitably adapted from the compactified case studied there to the planar case we need here.].

Fragile Black Holes and an Angular Momentum Cutoff in Peripheral Heavy Ion Collisions · Section · 0.95
The first of these solutions (to the Einstein equations with a negative cosmological constant) is the metric we have called the “KMV$_0$ metric”, obtained in the zero-charge case by Klemm, Moretti, and Vanzo [ref] ; with the addition of electric charge, we call these the “QKMV$_0$ metrics”: [equation] where the coordinates and $L$ are as in equation ([ref]), and where [equation]

Angular Momentum in QGP Holography · Section · 0.93
These dyonic metrics generalize the “KMV$_0$” metrics given by Klemm, Moretti, and Vanzo [ref] as the first examples of “rotating” planar black holes; hence we call them the “$ℓ$dyKMV$_0$” metrics.

Field Theories Without a Holographic Dual · Section · 0.80

Uses its method or tool 1 paper

(This can be shown either by adapting the methods explained in [ref] , or by a simple heuristic procedure explained in Appendix B.) The physical interpretation of $ℓ$ is less clear, but it can be revealed by the following argument.

Inverse Magnetic/Shear Catalysis · Section · 0.85

Compares with it 1 paper

For example, in $d=4$, the metric ([ref]) will describe a naked singularity for sufficiently small $m$, a black hole with an outer and an inner horizon for sufficiently large $m$, and an extremal black hole for the critical value $m=8√(3)l/9$; however a $d=4$ rotating topological black hole [ref] (in the $a→l$ limit) will have a double-horizon causal structure for any value of $m>0$.

Ultraspinning limits and rotating hyperboloid membranes · Rotating topological black holes · 0.87

Cites it as support 1 paper

This was confirmed by Klemm et al. in Section IV.B of [ref] .

Angular Momentum in QGP Holography · Section · 0.99

Cites it as background 8 papers

A black hole, with topologically planar event horizon —$$ the boundary is in fact globally conformally flat —$$ having a metric inducing a shearing motion at infinity, was first given by Klemm et al. [ref] ; we have called it the “KMV$_0$ metric”.

Inverse Magnetic/Shear Catalysis · Section · 0.83
For the details of the structure inside the event horizon, and for the Penrose diagram, we again refer the reader to [ref] .

Fragile Black Holes and an Angular Momentum Cutoff in Peripheral Heavy Ion Collisions · Section · 0.94
Rotating topological black holes are shown in [ref] .

Complexity growth for topological black holes by holographic method · Rotating topological black holes · 0.81

Discussion

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