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Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Eloy Ayón-Beato, Alberto García (CINVESTAV-IPN, Mexico)

General Relativity and Quantum Cosmology · Submitted 11 Nov 1999 · 21 references

Pith's verdict: A genuine exact regular black hole, but the NLED source in Eq. (7) is built around the solution — H(P) carries the black hole's own mass through s = |q|/2m — and the paper never flags it.

T0 · AI review

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arXiv IDgr-qc/9911046 (version 1)
Pith IDXSK5WGBC, Pith's permanent record of this paper and its review
CitationAlberto Garc\'ia (CINVESTAV-IPN, Eloy Ay\'on-Beato, Mexico). "Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics." arXiv:gr-qc/9911046 (1999). https://pith.science/paper/XSK5WGBC
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The paper

The paper claims

An exact black hole solution in ordinary general relativity whose curvature stays finite at the center, sourced by a nonlinear electromagnetic field.

The reading

The paper asks whether Einstein's equations, with no extra dimensions and no quantum corrections, can produce a black hole that has no singularity at its center. The authors answer yes: they write down a static, spherically symmetric metric whose curvature scalars stay finite everywhere, including at $r=0$, and whose source is a nonlinear electrodynamic field that reduces to Maxwell's theory when the field is weak and satisfies the weak energy condition. At large distances the metric matches the Reissner–Nordström charged black hole, so the two parameters read as mass $m$ and electric charge $q$, and horizons exist only while $|q| < 2 s_c m$, with $s_c$ solving a pair of critical-point equations to give $s_c$ of about $0.317$, i.e. $|q|$ below roughly $0.6$ of the mass. Above that charge the geometry is still regular but has no horizon at all. If the construction holds, the familiar statement that classical black holes must conceal a curvature singularity is not a theorem about general relativity itself but a statement about which kinds of matter one is willing to allow.

What carries the argument

The load-bearing device is a Legendre-dual formulation of nonlinear electrodynamics. Starting from a Lagrangian $L(F)$ with $F = F_{mu nu}F^{mu nu}/4$, the authors define $H = 2F L_F - L$ and $P_{mu nu} = L_F F_{mu nu}$, so that $H$ becomes a function of $P = (L_F)^2 F$ and the action can be rewritten as $L = 2P H_P - H$. This dual description is what makes the integration tractable: for a static, spherically symmetric ansatz the field equation $nabla_mu P^{alpha mu} = 0$ integrates immediately to $P = -q^2/2r^4$ with $q$ the integration constant, and the $tt$ Einstein equation collapses to the single ordinary equation $r Q' - Q = 2 r^4 H(P)$ for the metric function $1 - 2m/r + Q(r)/r^2$. Feeding in the chosen $H(P)$ of Eq. (7) makes the integrand a total derivative, and the resulting $Q(r)$ reproduces the metric. The whole construction therefore rests on picking $H$ so that this one integral closes in elementary form.

Extended reading notes

Core claim

The authors present the metric $$-g_{tt} = 1 - 2m $r^{2}$/($r^{2}$+$q^{2}$)^{3/2} + $q^{2}$ $r^{2}$/($r^{2}$+$q^{2}$)^2,$$ with the electric field $E = q r^4 (r^2-5q^2)/(r^2+q^2)^4 + 15 q r^4 m /(2 (r^2+q^2)^{7/2})$, and claim it is the first exact, everywhere-regular black hole solution of general relativity coupled to a physically admissible source. The source is a nonlinear electrodynamics defined through a function $H(P)$ of the field invariant $P$, chosen so that $H$ reduces to Maxwell theory for weak fields and satisfies the weak energy condition ($H<0$, $H_P>0$). The solution has the same asymptotic form as Reissner–Nordström, with $m$ and $q$ playing the roles of mass and charge. The authors compute the Ricci scalar, the Ricci square, and the Riemann square and report that all three remain finite for every $r$, so the zeros of $-g_{tt}$ are coordinate singularities marking horizons rather than curvature singularities. Solving the two conditions for a degenerate horizon gives $s_c$ near $0.317$ and a critical radius $x_c$ near $1.58$, so black holes exist for $|q| < 2 s_c m$, become extreme at equality, and give way to a horizonless regular spacetime above it. The global structure is stated to be that of Reissner–Nordström with the singularity at $r=0$ smoothed out into an ordinary origin of spherical coordinates, and the authors note that the spacetime admits no Cauchy surface, so the Penrose singularity theorem is not contradicted.

Load-bearing premise

The load-bearing premise is that the nonlinear electrodynamics of Eq. (7) counts as a legitimate matter theory, even though its defining function $H(P)$ is written in terms of the solution's own charge $q$ and ratio $s = |q|/2m$ rather than being fixed independently of the metric it produces.

Editorial extensions

If this is right

  • Classical general relativity admits black hole solutions with bounded curvature everywhere, so a singularity at the center is a property of particular matter sources rather than of the theory itself.
  • For charges below about $0.6$ of the mass the solution has inner and event horizons and can be extended to a maximal analytic completion with the same Penrose diagram as Reissner–Nordström, minus the singular boundary.
  • At the critical charge $|q| = 2 s_c m$ the two horizons merge into a single degenerate horizon, and the extreme case remains smooth in Eddington–Finkelstein coordinates.
  • For $|q| > 2 s_c m$ the solution describes a globally regular, horizonless spacetime, giving a charged, non-singular alternative to a naked Reissner–Nordström singularity.
  • Because the spacetime has no Cauchy surface, it evades the hypotheses of the Penrose singularity theorem, showing that singularity theorems constrain global hyperbolicity as much as they constrain curvature.

Reading between the lines

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

  • The function $H(P)$ in Eq. (7) is written explicitly in terms of the solution's own charge $q$ and the ratio $s = |q|/2m$, so the source is not a single Lagrangian fixed in advance and then solved; a reader should ask whether a charge-independent $H(P)$ can generate the same family of metrics, since that is what would make this a solution of one definite theory rather than a metric with a matching
  • Because the metric differs from Reissner–Nordström only at order $1/r^3$ in $-g_{tt}$ and $1/r^3$ in $E$, any test performed far from the horizon will look like ordinary charged-black-hole physics; distinguishing the two requires probes at or inside the horizon scale, which is where the regular core actually differs.
  • The absence of a Cauchy surface is a strong global property, not a technicality: it bears on whether the spacetime can be treated as an initial-value problem at all, and therefore on how one would count its degrees of freedom or quantize the field on it.
  • The same integral-closing trick, in which a chosen $H(P)$ makes $r Q' - Q = 2 r^4 H(P)$ integrable in closed form, is a general recipe; it suggests a small family of regular charged solutions could be catalogued by which functions $H$ produce elementary $Q(r)$.
Original abstract

The first regular exact black hole solution in General Relativity is presented. The source is a nonlinear electrodynamic field satisfying the weak energy condition, which in the limit of weak field becomes the Maxwell field. The solution corresponds to a charged black hole with |q| \leq 2 s_c m \approx 0.6 m, having the metric, the curvature invariants, and the electric field regular everywhere.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,925 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.96 · states limitation 0.56 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.21 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The first regular exact black hole solution in General Relativity is presented. The source is a nonlinear electrodynamic field satisfying the weak energy condition, which in the limit of weak field becomes the Maxwell field. The solution corresponds to a charged black hole with |q| \leq 2 s_c m \approx 0.6 m, having the metric, the curvature invariants, and the electric field regular everywhere.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The first regular exact black hole solution in General Relativity is presented. contribution 0.93 0.93 0.78 asserted 1.00
s1 The source is a nonlinear electrodynamic field satisfying the weak energy condition, which in the limit of weak field becomes the Maxwell field. contribution 0.60 0.66 0.09 asserted 1.00
s2 The solution corresponds to a charged black hole with |q| \leq 2 s_c m \approx 0.6 m, having the metric, the curvature invariants, and the electric field regular everywhere. result 0.95 0.90 0.47 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
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • 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. FIG. 1. Behavior of
Figure 2. FIG. 2. Regular behavior of the Ricci,
Figure 3. FIG. 3. Behavior of

Figures reproduced from arXiv:gr-qc/9911046 by the authors.

References
How later papers used it

326 later papers cite it

Evidence: substantial. 325 of the citing papers share no author with it, and 20 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 430 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 of 3 papers with this label
Compares with it 1 of 10 papers with this label

  • Compared its own result with this paper's metric, recovering the Ayon-Beato and Garcia metric in the limit l_0 to 0 and H to 1, whose approximated form is the well-known RN-dS metric.

    Charged Black Holes with Yukawa Potential (2024) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

275 arXiv papers cite this in 361 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 182 sentences without a confident label

Contests it 1 paper
  • This has implications for the many claims in the literature (e.g. [ref] ) that certain NLED-Einstein theories have regular black hole solutions[Typically, the regular “core” replacing a Reissner-Nordstrom-type singularity supports a de Sitter metric, and hence violates the SEC.].

    Causality and Energy Conditions in Nonlinear Electrodynamics · Discussion · contest 0.84

Builds on it 2 papers
Uses its method or tool 3 papers
  • Thus, the field equations [ref] now reduce to the following form [equation] Solving the last equation by following the procedure introduced in Ref. [ref] leads to the line element of the regular MOG static spherically symmetric dark compact object as follows [equation] where $dΩ^2=dθ^2+sin^2θdφ^2$ is the line element of the unit 2-sphere, and also we have defined [equation] which satisfies the weak energy condition [ref] .

    Accretion onto a static spherically symmetric regular MOG dark compact object · Regular MOG static spherically symmetric dark compact object · 0.81

  • In order to specify this nonlinear electrodynamics, we identify [equation] where [equation] In the static spherically symmetric case, we have [equation] Following a standard procedure, we then obtain the electric field [equation] We can describe the underlying nonlinear electrodynamics within the $P$-dual formalism [ref] , which yields the Lagrangian [equation] with [equation] where [equation]

    Hairy black holes by gravitational decoupling · Case 3 · 0.95

  • In order to find the Lagrangian $L$ of the underlying theory, we will use the $P$-dual formalism [ref] , which is based on the Legendre transformation [equation] where $H$ represents the Hamiltonian in the dual formulation.

    Regular hairy black holes through Minkowski deformation · Additional Lagrangian · 1.00

Compares with it 9 papers
  • Interestingly, the new solution is regular and resembles the Ayón-Beato–García spacetime [ref] , provided the T-duality scale is redefined in terms of the electric charge, $l_0→Q$.

    Charged black holes from T-duality · Final remarks · 0.99

  • Also, in the limit $ℓ_0 →0$ and $ℋ→1$ we get the Ayon-Beato and Garcia metric [ref] , whose approximated form reads, [equation] which is the well-known RN-dS metric.

    Charged Black Holes with Yukawa Potential · Nonsingular BH solution: BH solution in the spacetime region: $ℓ_0≤r≪λ$ · 0.95

  • Unlike models requiring ad hoc exotic matter [ref] or speculative quantum gravity [ref] , our mechanism emerges naturally from the UV-completion of a scalar field theory, with a direct analogue in the shear-thickening behavior of non-Newtonian fluids.

    Exact, non-singular black holes from a phantom DBI Field as primordial dark matter · Introduction · 0.85

Cites it as background 152 papers

Discussion

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