Matter-induced global regularity in non-polynomial quasi-topological gravity with Born-Infeld electrodynamics

Hong-Lin Liu; Qing-Quan Jiang; Xia Zhou; Xue-Ling Mu; Zhong-Wen Feng

REVIEW

Matter-induced global regularity in non-polynomial quasi-topological gravity with Born-Infeld electrodynamics

Hong-Lin Liu, Zhong-Wen Feng, Qing-Quan Jiang, Xia Zhou, Xue-Ling Mu

General Relativity and Quantum Cosmology · Submitted 7 Sep 2026

Pith's verdict: Shows Born-Infeld charge can regularize a singular vacuum branch in NPQTG, but the horizon classification rests on an unproven unimodality assumption.

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arXiv ID2609.07576 (version 3)

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CitationHong-Lin Liu, Qing-Quan Jiang, Xia Zhou, Xue-Ling Mu, Zhong-Wen Feng. "Matter-induced global regularity in non-polynomial quasi-topological gravity with Born-Infeld electrodynamics." arXiv:2609.07576 (2026). https://pith.science/paper/GVVAJCGO

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Editor's note

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The paper does something genuinely interesting: for the characteristic function h(p)=p/(1+ℓ²p), the vacuum branch of NPQTG develops a finite-radius singularity, but coupling to Born-Infeld electrodynamics can remove it, yielding globally regular geometries with an AdS core. That is a clean converse to the usual story where adding charge spoils regularity. The analysis of the regularity condition (A < Areg(β)) is sound, and the parameter-space separation between Cext and Creg is proven analytically. The triple-degenerate inner-horizon branch is a nice addition, and the representative numerics check out.

The main soft spot is structural: the entire horizon classification and the existence of the RBH region assume that the curve Ah(ρ) has a unique local minimum and a single stationary inflection point. No proof of unimodality is given. The paper shows one slice and one numerical example, but if Ah(ρ) had two minima, the inequality Aext < Areg would not guarantee a contiguous RBH region, and the derivation of the triple-degenerate branch would be incomplete. A referee will push on this hard.

There are also smaller issues: Eq. (73a) has a clear typo (numerator vs denominator), the numerical endpoint ρ* is given without derivation, and Appendix A’s defense of the action-level singularity (individual coeffients diverge, only the reduced action is integrable) is a modeling premise that the authors should discuss more prominently, not bury in an appendix.

All that said, the central result—that Born-Infeld nonlinearities can regularize a vacuum-singular branch and that the regular sector has finite width in parameter space—rests on mathematics that is otherwise correct. I would recommend the paper for peer review: it deserves a careful referee, not a desk reject. But I expect the referee will ask for a proof or at least a numerical survey of the horizon curve’s shape over the full parameter space, and for a more transparent handling of the action-level question.

Referee report

Major objections (3)

[§IV.B, Eqs. (61)–(64)] The horizon classification into regular horizonless, degenerate, and two-horizon RBH configurations treats Cext as a single boundary defined by a local minimum of Ah(ρ,β,ξ). This is load-bearing for the statement that the RBH sector is a connected nonzero-width interval Aext<A<Areg. The paper does not prove that Ah has exactly one local minimum, or even exactly one critical point, for the full physical range β>0, ξ>0. The representative ξ=1 slice in Fig. 3 and the single numerical example do not rule out a second minimum or a minimum-maximum pair; in that case a horizontal line A=const could enter and leave the two-horizon phase more than once, and the trichotomy in Eq. (64) would not describe the horizon structure. Please add an analytical proof of uniqueness of the relevant minimum (or a complete classification of all extrema of Eq. (61)) before making the general claim.
[Appendix A and Eq. (9)] The covariance status of the singular core limit is not settled by Appendix A. Along the globally regular branch, p tends to -1/ℓ² at the center (Eq. 45), where the coefficient functions H2, H3, H4, and H4' in Eq. (9) all diverge. Appendix A verifies only that the spherically reduced on-shell Lagrangian (A5) is locally integrable. That legitimizes the reduced spherical variational problem, but it does not by itself show that the covariant action (2) is well defined on this configuration or that the solution follows from a well-posed covariant variational principle. Either exhibit the cancellations that make the covariant density finite on the solution, or state explicitly that the 'globally regular solution' is a solution of the spherically reduced NPQTG-Born-Infeld system. As written, the headline claim is stronger than what is verified.
[§IV.C, Eqs. (72)–(73)] The construction of the triple-degenerate branch Ctri relies on branch and interval claims that are asserted rather than derived. The text states that the physical solutions belong to the χ(-)- branch, that the admissible interval is 0<ρ_-<ρ*_- with ρ*_-≃0.0421631, and that throughout this interval one has Aext<Atri<Areg and r+>r-, but no calculation, table, or systematic scan is provided beyond the single marked point ρ_-=0.03. Since Ctri is one of the advertised results, please supply the missing derivation or a reproducible numerical verification covering the entire interval.

Minor objections (3)

[Eq. (51)] The displayed expressions for R(r) and K(r) contain misaligned nested brackets that make the formulas difficult to verify; please typeset them cleanly and consider using the auxiliary functions D(r) and Σ(r) consistently throughout.
[Fig. 4] The notation χ(−)- in the caption and text is easy to misread as a doubled minus sign; define it explicitly as the lower branch χ^(-)_- from Eq. (72).
[Fig. 2] The curves labeled critical (A=Areg) are described as diverging at a finite radius, but the plotted black curves appear finite because of sampling; please indicate the divergence point explicitly or use a zoomed panel.

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The paper

The paper claims

Born-Infeld electrodynamics can remove the finite-radius singularity of a vacuum-singular gravitational branch in non-polynomial quasi-topological gravity, producing globally regular charged spacetimes including black holes with a…

The reading

This paper shows that in four-dimensional non-polynomial quasi-topological gravity, which has a singular vacuum branch, coupling to Born-Infeld electrodynamics can remove the finite-radius curvature singularity and produce globally regular spacetimes. The charged solutions exist in a finite region of parameter space and range from horizonless configurations to regular black holes with distinct inner and outer horizons. This provides a converse example to the usual case in which charge spoils regularity. The authors also identify a continuous branch of regular black holes with a triple-degenerate inner horizon and vanishing surface gravity there.

What carries the argument

The key mechanism is the combination of the non-polynomial characteristic relation $p = H_{\rm BI}/(1-\ell^{2}H_{\rm BI})$ and the Born-Infeld radial function $H_{\rm BI}(r)$, whose parameter-dependent maximum is kept below the inversion singularity by the condition $A < A_{\rm reg}(\beta)$. The triple-degenerate inner horizon arises from a stationary inflection point of the horizon curve $A_{h}(\rho,\beta,\xi)$, yielding $\kappa_{-}=0$.

Extended reading notes

Core claim

The paper constructs exact static spherically symmetric charged solutions in NPQTG with characteristic function h(p)=p/(1+\$ell^{{2}}$p) coupled to Born-Infeld electrodynamics. It finds that the Born-Infeld contribution modifies the radial profile of $H_{\rm BI}(r)$ so that its maximum stays below the critical value $1/\ell^{2}$ for $A < A_{\rm reg}(\beta)$, where $A = \ell^{2}\mu (b/Q)^{3/2}$ and $A_{\rm reg}(\beta)$ depends on $\beta=8\pi G b^{2}\ell^{2}/3$. This prevents the curvature variable $p(r)$ from diverging, yielding a globally regular geometry with an asymptotically flat exterior and an AdS-type core. Within this regular sector, the condition $A_{\rm ext}(\beta,\xi) < A < A_{\rm reg}(\beta)$ guarantees regular black holes with two distinct horizons. Moreover, a continuous branch $C_{\rm tri}$ is found where the inner horizon is triple-degenerate ($\kappa_{-}=0$) while the outer horizon is simple ($\kappa_{+}\neq 0$).

Load-bearing premise

The central assumption is that the spherical reduction of the covariant action, which remains locally integrable even though the covariant coefficient functions diverge at the AdS core, is a sufficient condition for the solution to be considered a legitimate solution of the full covariant theory.

Editorial extensions

If this is right

The model provides a proof of principle that nonlinear electrodynamics can cure finite-radius singularities in higher-curvature gravity, expanding the class of regular black hole solutions derivable from an action principle.
Regular black holes in this theory occupy a finite region of parameter space, not a fine-tuned line, making them potentially relevant for astrophysical modeling.
The triple-degenerate inner horizon with $\kappa_{-}=0$ suggests a mechanism to suppress the mass-inflation instability, which could affect the interior structure of astrophysical black holes.
The solution family can serve as a test case for studying the interplay between higher-curvature gravity and nonlinear matter in the strong-field regime.

Reading between the lines

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

The regularization mechanism may extend to other nonlinear electrodynamics theories with a maximum field strength, such as exponential or logarithmic models, potentially producing similar globally regular solutions.
If non-polynomial quasi-topological gravity represents an effective low-energy limit of a quantum gravity theory, these regular solutions could be relevant for describing the end state of gravitational collapse in a UV-complete setting.
The occurrence of a triple-degenerate inner horizon with vanishing surface gravity may be a generic feature of regular black holes with certain parameter tuning, and further perturbative studies could test whether it persists under dynamical perturbations.

Original abstract

We construct exact static, spherically symmetric charged solutions in four-dimensional non-polynomial quasi-topological gravity coupled to Born-Infeld electrodynamics. We focus on the model $h(p)=p/(1+\ell^{2}p)$, whose vacuum branch develops a curvature singularity at a finite radius. We show that Born-Infeld nonlinearities can remove this singularity within a finite region of parameter space, yielding globally regular geometries with an asymptotically flat exterior and a finite-curvature AdS-type core. The regular sector contains both horizonless configurations and RBHs, separated by a degenerate-horizon boundary. We further identify a continuous branch of regular black holes with a triple-degenerate inner horizon and a simple outer event horizon, satisfying $\kappa_-=0$ and $\kappa_+\neq0$. These results provide a converse example to cases in which introducing charge spoils the regularity of a black hole that is regular in vacuum. In the present model, Born-Infeld electrodynamics instead removes the finite-radius singularity of a gravitational branch that is singular in vacuum and supports globally regular charged geometries, including regular black holes with nontrivial inner-horizon structure.

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, 2026-09-23: 1 request, 5,768 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.18 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.87 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We construct exact static, spherically symmetric charged solutions in four-dimensional non-polynomial quasi-topological gravity coupled to Born-Infeld electrodynamics. We focus on the model $h(p)=p/(1+\ell^{2}p)$, whose vacuum branch develops a curvature singularity at a finite radius. We show that Born-Infeld nonlinearities can remove this singularity within a finite region of parameter space, yielding globally regular geometries with an asymptotically flat exterior and a finite-curvature AdS-type core. The regular sector contains both horizonless configurations and RBHs, separated by a degenerate-horizon boundary. We further identify a continuous branch of regular black holes with a triple-degenerate inner horizon and a simple outer event horizon, satisfying $\kappa_-=0$ and $\kappa_+\neq0$. These results provide a converse example to cases in which introducing charge spoils the regularity of a black hole that is regular in vacuum. In the present model, Born-Infeld electrodynamics instead removes the finite-radius singularity of a gravitational branch that is singular in vacuum and supports globally regular charged geometries, including regular black holes with nontrivial inner-horizon structure.

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 construct exact static, spherically symmetric charged solutions in four-dimensional non-polynomial quasi-topological gravity coupled to Born-Infeld electrodynamics.	contribution 1.00	0.96	0.09	asserted 1.00
s1	We focus on the model $h(p)=p/(1+\ell^{2}p)$, whose vacuum branch develops a curvature singularity at a finite radius.	method 0.43	0.67	0.04	asserted 1.00
s2	We show that Born-Infeld nonlinearities can remove this singularity within a finite region of parameter space, yielding globally regular geometries with an asymptotically flat exterior and a finite-curvature AdS-type core.	result 0.98	0.94	0.69	asserted 1.00
s3	The regular sector contains both horizonless configurations and RBHs, separated by a degenerate-horizon boundary.	result 0.99	0.92	0.12	asserted 1.00
s4	We further identify a continuous branch of regular black holes with a triple-degenerate inner horizon and a simple outer event horizon, satisfying $\kappa_-=0$ and $\kappa_+\neq0$.	result 0.97	0.95	0.15	asserted 1.00
s5	These results provide a converse example to cases in which introducing charge spoils the regularity of a black hole that is regular in vacuum.	interpretation 0.98	0.68	0.19	asserted 1.00
s6	In the present model, Born-Infeld electrodynamics instead removes the finite-radius singularity of a gravitational branch that is singular in vacuum and supports globally regular charged geometries, including regular black holes with nontrivial inner-horizon structure.	interpretation 0.56	0.92	0.70	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 v2

not examined

full text
figures and tables
references and citing sentences
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 1.** For A < Areg(β), the maximum of ℓ 2HBI remains below unity and the finite-radius singularity is avoided. At A = Areg, the maximum reaches the critical value, whereas for A > Areg it exceeds this value and the solution becomes singular. The regularity condition in Eq. (41) therefore identifies a finite parameter region in which the vacuum-singular branch is replaced by globally regular charged geometries. C. Central geometry and regularity Within the globally regular region, Eq. (20) shows that Z(r) approaches a finite limit as r → 0. Define ∆ = µ − 4πG√ bQ3/2J (0) 4 . (43) Th

**Figure 2.** FIG. 2: Dimensionless curvature invariants for representative regular, critical, and singular solutions. (a)

**Figure 3.** FIG. 3: Horizon structure of the globally regular solutions and its dependence on the model parameters. (a)

**Figure 4.** FIG. 4: Parameter-space structure of the triple-degenerate branch

**Figure 5.** FIG. 5: Representative RBH with a triple-degenerate inner horizon. (a) Full radial profile of the metric function.

**Figure 6.** FIG. 6: Dimensionless Hawking temperature

Figures reproduced from arXiv:2609.07576 by the authors.

References

Discussion

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