Maximally symmetric black holes with torsion in Lovelock gravity

B. Cvetkovi\'c; D. Simi\'c

REVIEW

Maximally symmetric black holes with torsion in Lovelock gravity

B. Cvetković, D. Simić

High Energy Physics - Theory · Submitted 7 Oct 2026

Pith's verdict: A legitimate extension of the authors' spherical torsion black hole to planar and hyperbolic horizons, but the central CAS check is unshown and the displayed Riemannian invariants miss k-dependent terms.

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arXiv ID2610.10348 (version 2)

Pith IDK6XU7PAZ, Pith's permanent record of this paper and its review

CitationB. Cvetkovi\'c, D. Simi\'c. "Maximally symmetric black holes with torsion in Lovelock gravity." arXiv:2610.10348 (2026). https://pith.science/paper/K6XU7PAZ

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Editor's note

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Bottom line: this is a real but modest extension of Cvetković–Simić's earlier spherical torsion black hole. The genuinely new content is the planar and hyperbolic branches, the second integration constant C0, the horizon-existence analysis, the vanishing-charge computation, and the near-horizon AdS2×S3/R3/H3 limits. The sector restriction α2 = α1^2/(12α0) is stated openly, and the C0 = 0 spherical limit correctly reduces to Ref. [4]. I spot-checked the horizon conditions (3.21)–(3.25) and the extremal conditions (5.1) against the metric function; they hang together.

The soft spots are two, and they are different in kind. First, as the stress-test note says, the central existence claim is a one-line CAS assertion. No substitution identities or component residuals are displayed for (3.7), (3.12), (3.17) in (2.2)–(2.3), and everything downstream inherits that. For an exact-solutions paper this is the main reproducibility gate. I do not see evidence of error, but I also cannot verify the new branches from the text; a short appendix with residuals or a link to the notebook would settle it.

Second, and more concretely: the Riemannian invariants in Section 3.5 are displayed as k-independent, but the Levi-Civita Ricci scalar for the stated metric is 12C1/r^8 − 24C0/(7r) − 30α0/α1 − 6k/r^2. The last term is absent from (3.26), and the Kretschmann expression presumably has corresponding missing k terms. So for C0 = C1 = 0 and k = ±1 the metric is not regular at r = 0; the claim that the invariants are smooth in that limit is wrong as written. This does not necessarily kill the black hole solutions, but it affects the naked-singularity classification and the stated regularity conclusions.

Minor: the all-zero charges follow cleanly because ρ01 vanishes in this sector, and that part is fine. The zero-entropy remark in Section 6 is asserted rather than derived, but it is not load-bearing for the main construction.

Who this is for: people working on Lovelock gravity with torsion, exact solutions, and near-horizon/topology classification. It deserves a serious referee. The construction is coherent, the extension is real, and the flaws are fixable. I would send it to review with a request for the CAS verification and a correction of the invariants. I would not cite it in its present form.

What would settle it

Run the displayed spherical ansatz — $N = B = \sqrt{C_1/r^6 - (2C_0/7)r - (3\alpha_0/(2\alpha_1))r^2}$ with $A_0 = (6\alpha_0/\alpha_1)r + C_0$, $A_1 = A_2 = A_3 = 0$, $A_4 = \sqrt{1 - (2\alpha_0/\alpha_1)r^2}$ — through an independent fresh tensor-algebra computation of (2.2)–(2.3); a single non-vanishing component, or an obstruction to meeting (3.8) simultaneously, refutes the construction. A cheaper partial check is the torsion equation (2.3), which is linear in the torsion and can be verified by hand once the curvature of the displayed connection is computed. Independently, one can test whether the horizon conditions (3.21)–(3.25) admit a real, positive $r_h$ with a real metric outside, and whether the hyperbolic reality bound (3.20) can be satisfied in the same parameter range.

Referee report

Significance

If the central substitution identity is correct, the paper gives a clean two-parameter family of torsion black holes with all three maximally symmetric horizon geometries in 5D Lovelock gravity, including planar and hyperbolic branches that are not covered by the earlier spherical C0 = 0 benchmark. The vanishing of all conserved charges is unusual and well-motivated by the special sector, and the regular near-horizon limits explicitly address the torsion-induced subtlety raised in [25]. The authors are also transparent about the main scope restrictions: the solutions exist only in the sector (3.8), and the hyperbolic reality condition (3.20)/(5.13) is stated. The main weakness is reproducibility: the existence claim is the load-bearing step, and it is currently condensed into a single sentence with no residual equations, script, or algebraic appendix. The horizon algebra and extremality conditions I checked are internally consistent with the displayed metric function, which gives some confidence that the CAS step is plausible, but that confidence is not proof.

Major objections (1)

[§3.1-3.3, Eqs. (3.7), (3.12), (3.17)] The central existence claim is not reproducible from the manuscript. Each of the three solution displays is introduced only by 'Solving the field equations with the help of software packages [21] gives', with no substituted component equations from (2.2)-(2.3) and no verification script. This matters because the C0-dependent planar and hyperbolic branches are new: the published C0 = 0 spherical solution [4] supports the limiting subfamily but cannot certify the second constant C0 or the k = 0, -1 branches. Everything downstream — the invariants (3.26)-(3.30), the zero charges (4.6), and the near-horizon limits (5.7)-(5.14) — inherits this unverified identity. Please provide, for each symmetry class, the independent residual equations of (2.2)-(2.3) after substitution, show that they are identically satisfied using (3.8), and give the computer algebra scripts or a supplementary notebook. A single exhibited nontrivial residual for each k would be sufficient; without this, the existence statement is an assertion rather than a verifiable result.

Minor objections (4)

[§3.1 and §3.3] There are several typos and broken phrases, such as 'for large values ofraddition constraint must hold' after Eq. (3.9), which should read 'an additional constraint must hold'.
[References] Reference [16] is malformed: 'Phys. Rev. D 76044021 (2007)' should contain the actual volume/page/article number; 'Phys. Rev. D 77024002' in [17] has the same problem.
[§5.2, Eq. (5.10)] The planar near-horizon spin-connection components are stated without derivation. Since the pre-limit A2 is k-independent and the displayed components contain r_h, a one-line calculation connecting A2 in (3.12) to the expressions in (5.10) would remove ambiguity about the displayed r_h/ℓ factors.
[§6] The sentence that the on-shell action vanishes and hence the entropy vanishes is not backed by a computation in the text. Either include the short evaluation of the action on the solutions or soften the statement, since entropy was not otherwise defined or calculated in the paper.

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The paper

The paper claims

Five-dimensional Lovelock gravity admits massless black holes whose torsion, not the metric, carries the horizon topology.

The reading

The paper's aim is to show that five-dimensional Lovelock gravity — a higher-curvature theory of gravity whose field equations stay second-order — admits static black holes whose torsion is intrinsic to the solution rather than an added decoration. For each of the three maximally symmetric horizon geometries (spherical, planar, hyperbolic) the authors present a solution with two free constants, a single shared metric function, and a spin connection whose nontrivial components depend on the horizon topology. They further claim that all conserved charges vanish, so torsion black holes need not carry mass, and that the extremal limits are regular near-horizon geometries $\mathrm{AdS}_2 \times S^3$, $\mathrm{AdS}_2 \times \mathbb{R}^3$, and $\mathrm{AdS}_2 \times H^3$ that keep their torsion. A reader should care because this places torsion, not just curvature, inside the catalogue of exact black hole solutions of a well-controlled higher-curvature theory.

What carries the argument

The load-bearing objects are the first-order dynamical variables of Lovelock gravity — the vielbein $e^i$ and the spin connection $\omega^{ij}$ — with the action (2.1) built from the three dimensionally continued Euler densities available in five dimensions. The ansatz is the maximally symmetric static metric (3.1) coupled to the most general symmetry-compatible spin connections (3.6), (3.11), (3.16), whose unknown radial functions $A_i(r)$ are fixed by the field equations. Two identities carry the argument: the sector condition (3.8), $\alpha_2 = \alpha_1^2/(12\alpha_0)$, which makes the momentum $\rho^{01}$ vanish and thereby zeroes every conserved charge in Nester's formula (4.3); and the common metric function, whose double zero at extremality leads, through the scaling $r = r_h + \epsilon\rho$, $t = \tau/\epsilon$, to the regular $\mathrm{AdS}_2$ factor shared by all three near-horizon geometries.

Extended reading notes

Core claim

On the paper's own terms, the central discovery is an explicit two-constant family of vacuum solutions to the five-dimensional Lovelock field equations (2.2)–(2.3), valid in the coupling sector $\alpha_2 = \alpha_1^2/(12\alpha_0)$ with $\alpha_0/\alpha_1 < 0$ enforced for reality. The metric takes $N = B = \sqrt{C_1/r^6 - (2C_0/7)r - (3\alpha_0/(2\alpha_1))r^2}$ for every horizon class $k = 1, 0, -1$, while the spin connections differ by topology: the component $A_0 = (6\alpha_0/\alpha_1)r + C_0$ provides the torsion and the components $A_4$ (spherical and hyperbolic) or $A_2$ (planar) carry the horizon curvature. The torsion 2-forms are nonzero in all three sectors, and the authors stress that torsion is essential to the solutions, not imposed on a Riemannian background. Conserved charges computed with Nester's boundary formula all vanish, (4.6), because the sector condition makes the relevant covariant momentum $\rho^{01}$ identically zero; the authors add that the on-shell action also vanishes, giving a second route to zero charge and entropy. Finally, the extremal near-horizon limits are regular and take the forms $\mathrm{AdS}_2 \times S^3$, $\mathrm{AdS}_2 \times \mathbb{R}^3$, and $\mathrm{AdS}_2 \times H^3$, with torsion surviving in both factors; the authors acknowledge, citing the known failure of such limits in the presence of torsion, that regularity had to be checked case by case, and for the hyperbolic case require $\ell^2 C_0^2 \geq 3$ so the spin connection stays real.

Load-bearing premise

The load-bearing premise is the software-checked identity that the displayed metric and spin connections — $N = B$ as given together with (3.7), (3.12), (3.17) — satisfy the full nonlinear field equations (2.2)–(2.3), asserted only in the passage 'Solving the field equations with the help of software packages [21] gives' with no substitution shown, and the identity holds only inside the restricted sector $\alpha_2 = \alpha_1^2/(12\alpha_0)$.

Editorial extensions

If this is right

5D Lovelock gravity now has an exact torsion black hole in every maximally symmetric horizon class, so torsion solutions are not confined to one topology.
Because all seven charges vanish identically (4.6), a black hole with genuine torsion degrees of freedom can exist without mass or entropy, extending the known massless Lovelock black holes into the torsionful sector.
The extremal limits give three regular near-horizon geometries, $\mathrm{AdS}_2 \times (S^3, \mathbb{R}^3, H^3)$, that preserve torsion in both factors, providing concrete seed geometries for a future classification of extremal solutions with torsion.
Since the metric function is the same for all three topologies, any physical distinction between the spherical, planar, and hyperbolic holes is carried entirely by the spin connection — a controlled setting for how torsion encodes horizon geometry.

Reading between the lines

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

If the sector condition is read as a fine-tuning, the paper's results do not establish torsion black holes for generic Lovelock couplings; the natural next test is whether a one- or two-parameter family survives away from $\alpha_2 = \alpha_1^2/(12\alpha_0)$, for example as a perturbative expansion in $\delta = \alpha_2 - \alpha_1^2/(12\alpha_0)$.
The vanishing of all charges follows from a single momentum component being identically zero in the sector, so the masslessness is structural rather than a boundary-condition accident; one could conjecture that any solution in this sector, including rotating or boosted ones, is forced to have zero charges.
The paper leaves the field-equation check to computer algebra; an independent hand-verification of the torsion equation (2.3), which is first order in the torsion, would be a cheap and decisive check of the central identity, and a published version with the substitution written out would make the result easier to build on.
If the near-horizon $\mathrm{AdS}_2$ factor is universal across horizon topologies, applying the same limit to non-maximally-symmetric horizons would test whether the classification of extremal torsion solutions in this sector factorizes into an $\mathrm{AdS}_2$ piece times a torsionful spatial base.

Original abstract

We analyze the existence of static black hole solutions with maximally symmetric horizons and nontrivial torsion in five-dimensional Lovelock gravity. For each symmetry class, we obtain a black hole solution characterized by two independent constants. We also determine the conserved charges associated with these solutions.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-08: 1 request, 2,840 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.60 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We analyze the existence of static black hole solutions with maximally symmetric horizons and nontrivial torsion in five-dimensional Lovelock gravity. For each symmetry class, we obtain a black hole solution characterized by two independent constants. We also determine the conserved charges associated with these solutions.

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 analyze the existence of static black hole solutions with maximally symmetric horizons and nontrivial torsion in five-dimensional Lovelock gravity.	contribution 0.99	0.92	0.13	asserted 0.69
s1	For each symmetry class, we obtain a black hole solution characterized by two independent constants.	result 0.99	0.95	0.32	asserted 1.00
s2	We also determine the conserved charges associated with these solutions.	result 0.59	0.95	0.14	asserted 1.00

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

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arXiv listing abstract v1

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full text
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P abstract

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S sections
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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

Discussion

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