An asymptotically conical regular black hole in four-dimensional nonpolynomial quasitopological gravity

Jia-Zhou Liu; Shao-Wen Wei; Yu-Xiao Liu

REVIEW

An asymptotically conical regular black hole in four-dimensional nonpolynomial quasitopological gravity

Jia-Zhou Liu, Shao-Wen Wei, Yu-Xiao Liu

General Relativity and Quantum Cosmology · Submitted 26 Oct 2026

Pith's verdict: A clean analytic example of an asymptotically conical regular black hole with consistent Wald thermodynamics, though the covariant-action stationarity at the center is argued only on shell.

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arXiv ID2610.35248 (version 2)

Pith IDILRFAYS5, Pith's permanent record of this paper and its review

CitationJia-Zhou Liu, Shao-Wen Wei, Yu-Xiao Liu. "An asymptotically conical regular black hole in four-dimensional nonpolynomial quasitopological gravity." arXiv:2610.35248 (2026). https://pith.science/paper/ILRFAYS5

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Claim paper

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Editor's note

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Short version: if you work on regular black holes in higher-curvature gravity, this is worth reading. The paper does exactly what it says: it takes the Borissova–Carballo-Rubio first-integral framework, keeps a constant G=γ instead of setting it to zero, and obtains exact conical asymptotics with a deficit 4π(1-c). Linear H gives a conical Schwarzschild-like geometry; rational H=6ψ/(1-L²ψ) gives a Hayward-type core with finite Ricci and Kretschmann invariants and vanishing Weyl at r=0. The γ→0 limit recovers Hayward, the large-r cone and 1/r mass term are independent of L, and the first L-dependent correction enters at order r^-2. That is a real extension of Ref. [8], not a repackaging: the earlier regular examples sit in G=0, and the conical solutions from global-monopole or Lorentz-violating models belong to different theories.

The thermodynamics is the strongest part. The Hamiltonian/Iyer–Wald charge gives dE=T dS_w for both branches, with Wald entropy containing rational and logarithmic terms in the regular case. I checked the horizon counting, extremal mass, and the reconstruction formulas; they are internally consistent. The paper is also honest about scope: no extended thermodynamics, fixed couplings, and a stated openness to other choices of the reduced functions. The citation pattern is appropriate.

The genuine soft spot is the one the stress test flags. The reconstructed H2, H3, H4 in Eq. (69) diverge at ψ=L^-2, which is exactly r=0 along the selected branch. The authors know this: they restrict to the real branch 0<ψ0. It is a completeness gap, not a demonstrated error: the on-shell cancellation of the δ^-2 and δ^-1 terms is suggestive, and the curvature data are regular. A referee should ask the authors to close it or to state the weaker claim explicitly.

Who gets value: people working on regular black holes in modified gravity and on Wald thermodynamics for nonpolynomial theories. It deserves a serious referee, and I would send it out with requests for revision. Bring it to a reading group if you want a concrete worked example of the Iyer–Wald machinery in a non-Einstein theory.

Referee report

Significance

Assuming the construction holds, this is a genuinely useful contribution to the regular-black-hole program in covariant higher-curvature gravity. To my knowledge it is the first four-dimensional nonpolynomial quasitopological vacuum sector in which a nonzero constant part of G fixes the asymptotic conical geometry while the nonlinear completion of H controls the core, so that the two are decoupled. The paper contains substantial checkable content and receives credit for it: exact algebraic metric functions; explicit large- and small-r expansions; an independent external benchmark in the gamma->0 Hayward limit; and a thermodynamic derivation from the Iyer-Wald charge rather than a heuristic prescription. I spot-checked the asymptotic matching (14)-(16), the parametrization (23), the discriminant (37), the horizon mass and extremality conditions (50)-(54), and the first-law identities (97)-(99) and (109)-(112), and found them internally consistent. The main unresolved point, the covariant stationarity of the regular solution at the center, is stated precisely below and is, in my view, fixable within the manuscript's scope.

Major objections (2)

[III B, Eq. (69), Appendix A] The covariant action (4) with the reconstructed functions (69) is not defined on any open neighborhood of the regular solution, and Eq. (74) addresses a different object than the one that the exactness claim requires. Setting delta=1-L^2 psi, one has H3=4/delta^2-gamma/3, H2=O(delta^{-2}), H4=-1/delta+O(ln delta), and H4'=O(delta^{-2}); along the selected branch delta=r^3/(2 M L^2)+O(r^4) (Eq. (73)), so the metric attains psi=L^{-2} exactly at r=0. Consequently, off-shell metrics arbitrarily close to the solution have a Lagrangian density in (4) that diverges in a neighborhood of the center, and the action is not finite there. Eq. (74) proves only that the on-shell, angularly integrated reduced Lagrangian density is O(r^2 ln r) after the central expansion is substituted; that is a statement about the value of the action on the solution, not about the well-definedness of the action as a functional or about its first variation.
[III B and IV; Eqs. (70)-(74) and the symmetric-criticality statement in Sec. III A] The exactness claim for the regular solution as a stationary point of the covariant four-dimensional theory is therefore not closed. The principle of symmetric criticality is invoked only on the domain where the covariant representatives are defined, but the solution reaches the boundary of that domain at r=0; the branch condition 0<psi<L^{-2} in Eq. (70) has its endpoint attained at the center. Missing are (i) a demonstration that the covariant Euler-Lagrange expressions, evaluated on the solution, vanish in the limiting or distributional sense as r->0, which would involve H_i'(P) terms with delta^{-3} falloff; and (ii) a computation of the boundary terms at r=0 arising from integration by parts in the variational calculation. Absent these, the solution is established as a solution of the reduced equations for r>0 with a locally integrable on-shell action, which is weaker than the abstract's exact covariant claim. The thermodynamic derivation in Sec. IV uses only the asymptotic and bifurcation spheres, with r>r_h>L, so it is unaffected by this point. I recommend either supplying a direct computation of the covariant field equations near the center or a boundary-term argument, or explicitly scoping the exactness statement to the reduced theory on r>0 with the covariant extension understood by continuity.

Minor objections (5)

[III B, Eq. (74)] The cancellation of the delta^{-2} and delta^{-1} terms leading to Eq. (74) is stated without derivation; since this cancellation is the sole support for the regular-center action claim, please display the central expansion of the reduced Lagrangian in an appendix.
[II-IV] The symbol N is used both for the lapse normalization (e.g., Eq. (28)) and inside G_N for Newton's constant; please rename one of them to avoid confusion in Secs. III-IV.
[II B, Eq. (70)] An explicit sentence stating that psi=L^{-2} sits at the boundary of the reconstruction domain, and that the solution attains this boundary at r=0, would help readers connect Eq. (70) with Eqs. (72)-(74).
[IV B, Eq. (110)] The logarithmic term 2L^2 ln(psi_0 D_h) diverges as r_h->L^+ (the limit where M->infinity); a brief comment on this asymptotic limit would be useful.
[Abstract and conclusion] Once the covariant-extension issue is resolved, please align the wording exact black hole solution in the abstract with the precise sense in which the solution is exact, for example reduced equations on r>0 with covariant extension by continuity.

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The paper

The paper claims

An asymptotically conical regular black hole is constructed in four-dimensional nonpolynomial quasitopological gravity.

The reading

The paper builds two vacuum black hole solutions in a four-dimensional nonpolynomial quasitopological gravity. Both share the same asymptotic conical geometry, controlled by a constant part of the reduced function $G(\psi)=\gamma$. A linear choice of the second reduced function $H(\psi)=6\psi$ gives a singular Schwarzschild-like metric. A nonlinear rational deformation $H(\psi)=6\psi/(1-L^2\psi)$ produces an exact algebraic metric whose center approaches de Sitter space with finite curvature invariants, while preserving the leading asymptotic cone. The two solutions therefore demonstrate that an identical far-field conical deficit can disguise qualitatively different cores. Both models satisfy the first law of black hole thermodynamics, and the regular model's Wald entropy contains rational and logarithmic terms originating from the higher-curvature couplings.

What carries the argument

The central object is the first integral of the spherically reduced equations, Eq. (8): $r(1+2f)G(\psi)+r^3 H(\psi)=12M$, where $\psi=(1-f)/r^2$. Choosing $G(\psi)=\gamma$ (constant) and $H(\psi)=6\psi/(1-L^2\psi)$ makes this equation quadratic in $\psi$ and allows an algebraic solution for $f(r)$. The function $H_4(\psi)$ reconstructed from this data (Eq. (69)) enters the Wald entropy formula $S_w=\pi r_h^2 H_4(r_h^{-2})/G_N$ (Eq. (87)), giving the rational and logarithmic terms.

Extended reading notes

Core claim

The paper discovers that in four-dimensional nonpolynomial quasitopological gravity, a constant component $G(\psi)=\gamma$ and a nonlinear deformation $H(\psi)=6\psi/(1-L^2\psi)$ combine in the first integral (Eq. (8)) to yield an exact static metric $f(r)$ (Eq. (41)) that is algebraically determined. At large radius $f(r)\to c-2m/r$ with $c=1/(1+\ell)$, giving a solid-angle deficit $4\pi(1-c)$. Near $r=0$ the metric behaves as $f(r)=1-r^2/L^2+\cdots$, so the core is de Sitter with Ricci scalar $12/L^2$, Kretschmann invariant $24/L^4$, and vanishing Weyl curvature. The solution can have two horizons, one degenerate extremal horizon, or no horizon, depending on the mass. Its Wald entropy takes the closed form $S_w=\frac{\pi}{G_N}\big[(1-\gamma/3)r_h^2 - L^2 r_h^2/(r_h^2-L^2) + 2L^2\ln(\psi_0 D_h)\big]$, and together with the Hamiltonian energy $E=N M/G_N$ it satisfies $dE=T dS_w$.

Load-bearing premise

The reconstructed functions $H_i(\psi)$ diverge at the branch endpoint $\psi=L^{-2}$ reached at $r=0$; although the on-shell reduced Lagrangian is locally integrable, it remains unproven that the full four-dimensional covariant action and its variation are well-defined in a neighborhood of the center, rather than only through the spherical limiting procedure.

Editorial extensions

If this is right

An asymptotic conical deficit can coexist with a regular, de Sitter core in a vacuum higher-curvature theory, not requiring matter fields.
The regular black hole satisfies the standard first law of black hole thermodynamics, confirming its consistency as an equilibrium thermodynamic object.
The Wald entropy contains a logarithmic term proportional to $L^2$, which may encode ultraviolet information about the central region.
The same asymptotic data ($c$, $m$) are shared by the singular and regular branches, so asymptotic measurements cannot distinguish the core geometry.
The existence of an extremal configuration with zero temperature and finite mass provides a clear endpoint for Hawking evaporation in this model.

Reading between the lines

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

The construction suggests that nonpolynomial quasitopological gravity can host regular black holes without introducing matter or diverging curvature, offering a purely geometric resolution of the central singularity.
The logarithmic entropy term resembles corrections found in other quantum-gravity approaches, raising the possibility that the scale $L$ has a microscopic origin (e.g., a fundamental length).
The extension to rotating or charged regular conical solutions is an open problem that would test the robustness of the method.
Connecting the solid-angle deficit $\ell$ to a global monopole charge or a Lorentz-violating condensate could link this vacuum solution to effective field theory models.

Original abstract

In this work, we construct an asymptotically conical regular black hole in four-dimensional nonpolynomial quasitopological gravity. Starting from a linear choice of the reduced function, which yields an asymptotically conical Schwarzschild-like solution, we introduce a nonlinear deformation that preserves the leading asymptotic cone while producing an exact black hole solution with a de Sitter core and finite curvature invariants. We further establish the corresponding black hole thermodynamics for both solutions within the Hamiltonian and Wald frameworks. The entropy contains rational and logarithmic terms in the regular model, and both solutions satisfy the first law of black hole thermodynamics.

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, 2026-10-27: 1 request, 3,676 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.47 · 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.

In this work, we construct an asymptotically conical regular black hole in four-dimensional nonpolynomial quasitopological gravity. Starting from a linear choice of the reduced function, which yields an asymptotically conical Schwarzschild-like solution, we introduce a nonlinear deformation that preserves the leading asymptotic cone while producing an exact black hole solution with a de Sitter core and finite curvature invariants. We further establish the corresponding black hole thermodynamics for both solutions within the Hamiltonian and Wald frameworks. The entropy contains rational and logarithmic terms in the regular model, and both solutions satisfy the first law of black hole thermodynamics.

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	In this work, we construct an asymptotically conical regular black hole in four-dimensional nonpolynomial quasitopological gravity.	contribution 0.99	0.96	0.63	asserted 1.00
s1	Starting from a linear choice of the reduced function, which yields an asymptotically conical Schwarzschild-like solution, we introduce a nonlinear deformation that preserves the leading asymptotic cone while producing an exact black hole solution with a de Sitter core and finite curvature invariants.	method 0.78	0.94	0.34	asserted 1.00
s2	We further establish the corresponding black hole thermodynamics for both solutions within the Hamiltonian and Wald frameworks.	contribution 0.83	0.95	0.09	asserted 1.00
s3	The entropy contains rational and logarithmic terms in the regular model, and both solutions satisfy the first law of black hole thermodynamics.	result 0.99	0.91	0.15	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

sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
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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

Forward citations

Cited by 2 Pith papers

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Exact Regular Planar Black Holes solutions in Bumblebee gravity
gr-qc 2026-11 conditional novelty 5.0 of 10

Exact planar black hole solutions in bumblebee gravity with a temporal/radial bumblebee field, coupling xi=1/4, and curvature invariants finite at r=0.

Discussion

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