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Charging the Bardeen and Hayward regular black holes makes them singular; stricter regularity conditions are required.

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2026-07-10 17:08 UTC pith:X7KX2GTL

load-bearing objection Clean, explicit calculation showing that the classic charged Bardeen and Hayward metrics are singular, with concrete improved Ω functions that stay regular; solid incremental work inside the authors’ existing framework.

arxiv 2607.07831 v1 pith:X7KX2GTL submitted 2026-07-08 gr-qc

Charging up regular black holes

Ra\'ul Carballo-Rubio , Chiara Coviello , Vania Vellucci This is my paper
classification gr-qc PACS 04.70.Bw04.50.Kd04.20.Jb
keywords regular black holescharged black holesBardeen metricHayward metricEinstein–Maxwellspherical symmetryidentically conserved tensorselectrovacuum
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows that a broad class of regular black-hole geometries, built by deforming the Einstein tensor while keeping it identically conserved, does not automatically stay regular once electric charge is turned on. The authors integrate the deformed Einstein–Maxwell equations in spherical symmetry for the integrable subfamily of theories and obtain the charged analogues of the Bardeen and Hayward metrics; both become singular, one at the origin and one at a positive radius. They prove that the conditions needed for regularity are strictly stronger than those that already guarantee regularity in vacuum, and they supply two improved potentials that remain regular when charged. The result matters because it demonstrates that simply regularizing vacuum solutions is not enough for realistic situations that include charge, rotation, or matter; new ingredients are required.

Core claim

When the Einstein tensor is replaced by an identically conserved tensor built from second derivatives of the spherical metric functions, the charged solutions of the resulting field equations satisfy an algebraic relation that generalizes the vacuum mass function. For the standard Bardeen and Hayward choices this relation produces metrics that are singular, either because the radial derivative of the metric function fails to vanish at the origin or because a positive-radius root appears in the denominator. Regularity of electrovacuum solutions therefore imposes conditions stricter than those required for vacuum regularity alone.

What carries the argument

The deformed Einstein tensor G_μν(q,r) defined by two free functions α(r,χ) and β(r,χ) (with χ = ∇r · ∇r), together with the integrability condition γ = ∂_χ α − ∂_r β = 0 that reduces the charged problem to the single algebraic relation Ω(r,χ)|_χ=f = 4M − 2Q²/r.

Load-bearing premise

The analysis is restricted to the integrable subfamily of theories in which the two metric functions remain equal; outside that subfamily the charged equations stay coupled and the explicit singularity diagnoses no longer apply.

What would settle it

Construct or numerically evolve a charged solution belonging to a non-integrable theory (γ ≠ 0) that is regular at the origin and free of positive-radius singularities, or show that every regular vacuum deformation becomes singular once charge is introduced even without the integrability condition.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper constructs charged regular black holes within a general spherically symmetric deformation of the Einstein tensor that remains identically conserved and involves at most second derivatives of the metric functions. After reviewing the vacuum case (integrable subfamily γ = ∂_χ α − ∂_r β = 0), the authors couple the same master equations to Maxwell electrodynamics, obtain the algebraic relation Ω(r, χ)|_{χ=f} = 4M − 2Q^{2}/r, and thereby produce the explicit charged Bardeen and Hayward metrics (Eqs. (28)–(29)). Direct curvature diagnostics show that both become singular (non-vanishing ∂_r f at r = 0 for the former; a positive-radius root of the denominator that drives a divergent Kretschmann scalar for the latter). Improved potential functions Ω_{B2.0} and Ω_{H2.0} are proposed that remain regular when charged, and the broader implication for rotating or matter-filled regular black holes is discussed.

Significance. If correct, the result supplies a clean, model-independent demonstration that vacuum regularity does not automatically survive the introduction of charge, and it furnishes concrete existence proofs of regular charged solutions inside the same Ziprick–Kunstatter and Kunstatter–Maeda–Taves classes. The derivation is fully explicit, the singularity diagnostics rest on standard curvature criteria, and the improved Ω functions are new, falsifiable proposals. This is a useful intermediate step toward realistic (rotating, matter-supported) regular black holes and clarifies the relative stringency of regularity conditions across different matter sectors.

minor comments (4)
  1. The restriction to the integrable subfamily γ = 0 is clearly stated, yet a short remark on whether non-integrable theories could evade the singularity conclusions (or at least alter the form of the charged solutions) would strengthen the discussion in Sec. V.
  2. Fig. 1 is schematic; labeling the locations of the singularities (r = 0 versus r = r_{+}) more explicitly would help readers unfamiliar with the curvature criteria used in the text.
  3. The forthcoming companion paper [37] is cited for a more comprehensive regularity analysis; a one-sentence preview of the key additional conditions would make the present manuscript more self-contained.
  4. Typographical consistency: “Reissner–Nordström” appears with and without the umlaut; standardize throughout.

Circularity Check

1 steps flagged

No significant circularity: charged singularity diagnoses follow by direct integration of the master equations, not by tautology or fitted input.

specific steps
  1. self citation load bearing [Sec. II–III, Eqs. (2)–(5), (11)–(14); citations [18], [16], [17]]
    "The most general field equations for gµν above, in which the spherically symmetric Einstein tensor is deformed into an identically conserved tensor constructed from up second derivatives of qab(x) and r(x), take the form [18]: Gµν(q,r)=8πTµν … These were studied in [18] in the notation used here, and in [16] for an equivalent vacuum theory …"

    The master equations and the vacuum potentials ΩB, ΩH that define the Bardeen and Hayward metrics are imported from the authors’ own prior works. This is ordinary framework reuse rather than a circular reduction of the new charged results; the charged integration and singularity diagnostics are performed independently inside the paper. The step is therefore only a minor self-citation and does not force the central claim.

full rationale

The paper starts from the master field equations (2)–(5) already developed for vacuum/Vaidya cases and integrates them for electrovacuum under the explicitly flagged integrability condition γ = 0. This yields the algebraic relation Ω(r,χ)|χ=f = 4M − 2Q²/r (Eq. 27), from which the charged Bardeen and Hayward metrics (28)–(29) are obtained by simple substitution of the known vacuum potentials. The singularity statements then follow from elementary local diagnostics (∂rf|r=0 ≠ 0; positive root of the denominator producing a divergent Kretschmann scalar). These steps are independent calculations, not re-labelings of prior results or fits to data. Self-citations to the authors’ vacuum and Vaidya papers supply the framework but are not load-bearing for the new charged conclusions; the improved regularizations (31) and (33) are new choices offered as existence proofs. The derivation is therefore self-contained against its own stated assumptions, with only the ordinary (non-circular) reliance on prior framework papers.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 1 invented entities

The central claim rests on a phenomenological deformation of the Einstein tensor (already introduced in earlier papers by the same group) together with the standard Maxwell stress-energy and the technical restriction to integrable theories. The free parameter ℓ sets the core scale; the improved Ω functions are new choices of that deformation. No new particles or forces are postulated.

free parameters (1)
  • ℓ (regularization length)
    Appears in every explicit metric (Bardeen, Hayward and the two improved versions). It is a free theory parameter that controls the size of the regular core; its value is not fixed by the paper.
axioms (4)
  • domain assumption The most general spherically symmetric field equations are those in which the Einstein tensor is replaced by an identically conserved tensor built from q_ab, r and at most second derivatives (Eqs. (2)–(5)).
    Taken as the starting point of the whole construction; justified by reference to earlier work but not re-derived here.
  • ad hoc to paper Integrability condition γ = ∂_χ α − ∂_r β = 0 (Eq. (12)).
    Imposed to obtain h = f and the algebraic relation for Ω; without it the charged equations remain coupled and the explicit solutions disappear.
  • domain assumption Electromagnetic field obeys the standard Maxwell equations with point-source charge Q and the standard Maxwell stress-energy tensor (Eqs. (7)–(9)).
    Assumed throughout; non-linear electrodynamics is mentioned only as a possible extension.
  • domain assumption Asymptotic recovery of general relativity: α, β → GR values as r → ∞.
    Required for the ADM mass and charge to be well-defined.
invented entities (1)
  • Improved potential functions Ω_{B2.0} and Ω_{H2.0} no independent evidence
    purpose: To produce charged metrics that remain regular at the origin and free of positive-radius singularities.
    New choices of the free functions α, β inside the existing Ziprick–Kunstatter and Kunstatter–Maeda–Taves classes; no independent observational handle is given.

pith-pipeline@v1.1.0-grok45 · 14227 in / 2503 out tokens · 31459 ms · 2026-07-10T17:08:22.686556+00:00 · methodology

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read the original abstract

We present a general construction of charged regular black holes as solutions of a generalization of the Einstein--Maxwell field equations in spherical symmetry in which the Einstein tensor is deformed into an identically conserved tensor containing up to second derivatives of the gravitational field. The generality of the construction allows us to define the field equations satisfied by generic regular black holes when becoming charged. The conditions that guarantee regularity of charged solutions are evaluated and shown to be more stringent than the regularity conditions for uncharged solutions. This implies, in particular, that the charged versions of the Bardeen and Hayward black holes become singular. Improved versions of the Bardeen and Hayward metrics that remain regular when charged are proposed. Our results indicate that regularizing the vacuum solutions of general relativity is, in general, not enough to yield regular solutions in other situations of physical interest. The implications that follow for the construction of realistic regular black holes, in which aspects such as rotation and the presence of matter fields are taken into account, are discussed.

Figures

Figures reproduced from arXiv: 2607.07831 by Chiara Coviello, Ra\'ul Carballo-Rubio, Vania Vellucci.

Figure 1
Figure 1. Figure 1: FIG. 1: The Schwarzschild black hole can be deformed [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

discussion (0)

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Reference graph

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