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Regular Black Holes in Nonlocal Quasitopological Gravity

T0 review · 0 major / 4 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read Infinite-derivative completions of quasitopological gravity keep exact vacuum regular black holes while eliminating ghosts and strong-coupling instabilities.

desk verdict Clean nonlocal fix that keeps QT regular black holes exact and kills the strong-coupling degeneracy without adding ghosts. read the letter →

arxiv 2607.07790 v1 pith:ZRMS4NT2 submitted 2026-07-08 gr-qc hep-th

Pablo Bueno , Pablo A. Cano , Robie A. Hennigar , Ángel J. Murcia This is my paper · ORCID
classification gr-qchep-th PACS 04.50.Kd04.70.Bw04.60.-m
keywords regularblackholesquasitopologicalgravitynonlocalinfinitederivativesBirkhofftheoremghost-freestrongcouplinglimitingcurvature
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs nonlocal versions of quasitopological gravity by adding a quadratic term built from a form factor of the linearized quasitopological equations. These theories keep every exact spherically symmetric regular black-hole solution of the original local theories, including Hayward-like cores produced by infinite towers of curvature corrections. At the same time the nonlocal form factors raise the derivative order on every background, remove the strong-coupling pathology that appears when the local equations drop order on spheres, and leave only a healthy massless graviton around flat and de Sitter space. A linearized Birkhoff theorem follows: continuous spherical deformations of these regular solutions can only shift the mass. The construction therefore unites two previously separate singularity-resolution mechanisms—limiting curvature from pure gravity and nonlocal smearing of point sources—inside a single, ghost-free framework.

What carries the argument

The nonlocal operator F = (e^{Ω(ˆD)} − I)/ˆD, where ˆD is the second-order operator obtained by linearizing the truncated quasitopological equations Ê_ab. Because Ω is entire and ˆD becomes self-adjoint on the relevant subspaces, e^{Ω(ˆD)} has trivial kernel, so the only linearized solutions are those of the original local theory; higher derivatives never introduce new poles or order reduction.

What would settle it

Compute the full linearized spectrum of the nonlocal theory on a non-spherical, non-maximally-symmetric background (for example a Kerr or binary black-hole metric) and check whether any additional poles or strongly coupled modes appear that are absent from the original quasitopological theory.

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Extended reading notes

Core claim

Infinite-derivative completions of quasitopological gravity, defined by the action containing the quadratic term −½ Ê_ab F^{ab}_{cd} Ê^{cd} with the form factor F = (e^{Ω(ˆD)} − I)/ˆD, are ghost-free around maximally symmetric and spherically symmetric vacua, free of strong-coupling instabilities, and admit the exact regular black-hole solutions of the original quasitopological theories while satisfying a perturbative Birkhoff theorem.

Load-bearing premise

The linearized operator obtained from the truncated equations must become self-adjoint when restricted to spherical or maximally symmetric perturbations, so that the form factor forces every extra mode to vanish.

Editorial extensions

If this is right

  • Every exact vacuum regular black hole of local quasitopological gravity remains an exact solution of the nonlocal completion.
  • The Newtonian potential of a point mass is finite at the origin, combining nonlocal smearing with the limiting-curvature core.
  • Spherical gravitational waves cannot propagate on these backgrounds; continuous deformations only shift the mass.
  • Suitable entire functions Ω render the theory potentially super-renormalizable around flat space while preserving unitarity of the massless graviton.
  • Collapse of thin shells or dust can still form the same regular black holes, now inside a theory free of strong-coupling pathologies.

Reading between the lines

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

  • The same construction should extend immediately to the non-polynomial four-dimensional quasitopological densities already known to produce regular black holes, yielding the first ghost-free nonlocal regular black holes in four dimensions.
  • If the nonlocal scale is taken much smaller than the quasitopological scale, matter sources inside the de Sitter core are only weakly distorted, offering a controlled setting for studying mass inflation or Cauchy-horizon stability.
  • The form-factor technique may apply to other higher-curvature families whose equations drop order on special backgrounds, providing a general template for curing strong coupling without losing exact solutions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The manuscript constructs nonlocal completions of quasitopological (QT) gravity by adding a quadratic term −½ Ê_ab F^{ab}_{cd} Ê^{cd} to the QT Lagrangian, with the form factor F = (e^{Ω(ˆD)} − I)/ˆD built from the linearization of Ê_ab. The resulting theories inherit the exact spherically symmetric regular black-hole solutions of the original QT models (including the Hayward metric for the geometric series of couplings), remain free of Ostrogradski ghosts around maximally symmetric and spherically symmetric vacua, eliminate the strong-coupling pathology associated with order reduction of the principal symbol, and satisfy a perturbative Birkhoff theorem. Explicit propagator analysis around flat space shows that only the massless graviton propagates, while the Newtonian potentials of a point source are regularized by the nonlocal form factor and are given in closed form for Ω(x) = γ^{2}x^{2}.

Significance. If correct, the construction supplies the first concrete examples of nonlocal gravity theories that admit exact vacuum regular black holes while remaining ghost-free and free of strong-coupling instabilities. It unifies two previously separate singularity-resolution mechanisms—limiting curvature from infinite towers of higher-curvature terms and nonlocal smearing of sources—within a single, controllable framework. The explicit linearization, Lagrange-multiplier argument, and closed-form Newtonian potentials constitute reproducible technical results that can be checked independently and that open a well-defined route for studying dynamical collapse and matter coupling in these models.

minor comments (4)
  1. The self-adjointness of ˆD on the restricted subspace of perturbations that preserve ∇_c ∇_d P^{acbd}=0 is used crucially between Eqs. (18) and (19). A short explicit remark that this follows from the defining QT property (4) and the reduction of the linearized QT equations to Einstein’s on maximally symmetric backgrounds would make the argument fully self-contained.
  2. In the appendix the regularized hypergeometric functions that appear in the closed-form potentials (36)–(37) are written with a tilde; a one-line definition or standard reference would remove any ambiguity for readers less familiar with the notation.
  3. The claim of potential super-renormalizability is stated only for flat space. A brief sentence clarifying that the same UV fall-off is expected (but not proven) around other maximally symmetric backgrounds would prevent over-interpretation.
  4. A few typographical inconsistencies appear (e.g., “e.g.Section”, missing spaces after periods in the introduction). These are purely cosmetic and do not affect readability of the technical content.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: nonlocal completion and ghost-freedom argument are self-contained; QT RBHs and Birkhoff are imported from prior overlapping-author work but used as independent mathematical inputs.

  1. self citation load bearing [Introduction and Quasitopological Gravity section (eqs. (1)–(8) and surrounding text)]
    "Under mild assumptions on the couplings, the unique spherically symmetric solutions of these theories are regular black holes (RBHs).[26] … For each choice of couplings, the corresponding RBHs are the unique SS solutions of the QT theories—i.e., a Birkhoff theorem holds [25]."

    The existence, uniqueness and regularity of the vacuum SS solutions that the nonlocal theory is designed to preserve are imported wholesale from prior papers by the same author group ([14,25] and related works). While those results are mathematically independent and not re-derived or re-fitted here, the present paper's claim to 'admit exact … regular-black-hole solutions' rests entirely on that self-citation chain rather than on a new derivation internal to the nonlocal theory.

full rationale

The paper's central construction is the nonlocal action (10) with form factor F = (e^Ω(D̂) − I)/D̂. By design this action retains every solution of the original QT theory that satisfies Ê_ab = 0 (including the exact SS regular black holes), while the entire-function form factor forces the linearized spectrum around maximally-symmetric and SS backgrounds to coincide with that of QT gravity (only the massless graviton). The key technical step—self-adjointness of D̂ on the subspace of perturbations that still obey ∇_c ∇_d P^{acbd} = 0—is derived directly from the defining QT property (4) and the structure of the linearized operators; it is not smuggled in by definition or by an unverified self-citation. The existence of QT densities, their second-order spherical equations, the Birkhoff theorem, and the limiting-curvature RBHs are taken from earlier papers by overlapping authors. Those results are independent mathematical statements (explicit densities, second-order equations, uniqueness of SS solutions) and are not redefined or re-fitted here; they function as external inputs. Consequently the nonlocal completion and the ghost-freedom/strong-coupling claims constitute genuine new content rather than a circular re-packaging of the inputs. Score 2 reflects only the minor, non-load-bearing self-citation of the QT foundation.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The central claims rest on the existence of QT densities of arbitrary order (imported from prior work), the algebraic properties of entire form factors (standard in nonlocal gravity), and the technical assumption that ˆD is self-adjoint on the relevant perturbation subspaces. Free parameters are the QT couplings α_n and the scale γ that appears in the simplest entire function Ω. No new particles or forces are postulated; the only invented object is the specific nonlocal completion itself.

free parameters (3)
  • α_n (QT couplings)
    Dimensionful coefficients of the higher-curvature densities; their common scale α sets the size of the de-Sitter core. Chosen by hand to recover the Hayward metric or other regular profiles.
  • γ (nonlocality scale)
    Length scale appearing in the simplest entire function Ω(x)=γ² x²; controls the width of the Gaussian-like smearing of point sources. Free positive parameter.
  • Ω (entire function)
    Any entire function with Ω(0)=0 and Ω(x)→+∞ as |x|→∞; the concrete choice Ω(x)=γ² x² is made for analytic control of the Newtonian potentials.
assumptions (3)
  • domain assumption Quasitopological densities of arbitrary order exist in D≥5 and yield second-order equations on spherical symmetry with a Birkhoff theorem.
    Imported from the authors’ earlier series of papers; used as the starting point for the nonlocal completion.
  • standard math An entire function Ω with Ω(0)=0 has the property that e^{Ω(ˆD)} has trivial kernel on the space of metric perturbations.
    Standard fact about entire functions of operators; invoked to conclude that the only linearized solutions are those of the original QT theory.
  • ad hoc to paper The operator ˆD obtained by linearizing Ê_ab is self-adjoint when restricted to maximally symmetric backgrounds or to spherical perturbations of spherical solutions.
    Technical assumption needed to identify F with its self-adjoint part and to reduce the linearized equations to e^{Ω(ˆD)} e = 0.
invented entities (1)
  • Nonlocal Quasitopological (NLQT) action
    purpose: Quadratic nonlocal completion that preserves QT solutions while raising the derivative order uniformly and eliminating ghosts.
    Defined by eq. (10); the central new object of the paper. No independent experimental handle is provided beyond the theoretical consistency checks.

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Cite this review

Pith. "Pith review of Regular Black Holes in Nonlocal Quasitopological Gravity." pith.science (2026). https://pith.science/paper/ZRMS4NT2

@misc{pith2026260707790,
  author       = {Pith},
  title        = {Pith review of: Regular Black Holes in Nonlocal Quasitopological Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZRMS4NT2}},
  note         = {Machine review of arXiv:2607.07790}
}
read the original abstract

We present infinite-derivative completions of Quasitopological gravities that are ghost-free, avoid strong coupling instabilities and admit exact, spherically symmetric vacuum regular-black-hole solutions satisfying a perturbative Birkhoff theorem.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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