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Regular Magnetic Black Holes and Monopoles from 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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Kirill A. Bronnikov (VNIIMS, Moscow; PFUR, Moscow)

General Relativity and Quantum Cosmology · Submitted 3 Jun 2000 · 19 references

Pith's verdict: Two clean no-go theorems plus a duality argument that undercuts the Ayon-Beato–Garcia regular electric black holes; the constructive half is a standard quadrature, and the abstract's 'iff' is a bit stronger than the prose proves.

T0 · AI review

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arXiv IDgr-qc/0006014 (version 2)
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CitationKirill A. Bronnikov (VNIIMS, Moscow), Moscow; PFUR. "Regular Magnetic Black Holes and Monopoles from Nonlinear Electrodynamics." arXiv:gr-qc/0006014 (2000). https://pith.science/paper/YTXNTPPA
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The paper

The paper claims

Regular static, spherically symmetric black holes and monopoles in nonlinear electrodynamics exist only with magnetic charge and a Lagrangian that tends to a finite limit at strong field.

The reading

General relativity predicts singularities, and physicists have long tried to remove them by changing the matter source. One candidate is nonlinear electrodynamics, a generalization of Maxwell's theory in which the electromagnetic Lagrangian is an arbitrary function of the field invariant rather than the field invariant itself. The paper asks whether such a theory, coupled to gravity, can give a static, spherically symmetric spacetime with no singularity at the center. The answer is a sharp no for electric charge. The proof is short. At a regular center every curvature invariant is finite, so every component of the energy-momentum tensor is finite. For a purely electric field, finiteness of the energy density forces the Lagrangian derivative to diverge at the center, which contradicts the assumed Maxwell weak-field limit. Adding magnetic charge does not help, because the magnetic contribution blows up faster. A second theorem rules out the escape routes of a wormhole throat or an infinitely long horn, using only the fact that the electromagnetic energy-momentum tensor has equal time-time and radial-radial components. What survives is the purely magnetic case. If the Lagrangian tends to a finite constant as the field invariant goes to infinity, the metric near the center approaches de Sitter space with a finite effective cosmological constant, and the whole spacetime is regular. The mass is then fixed by the charge and the Lagrangian. The paper gives an explicit example and compares it with earlier claimed regular electric solutions, showing via a Legendre-transform duality that those solutions require different Lagrangians in different regions of space.
Extended reading notes

Core claim

The paper's load-bearing assertion is the 'if and only if' in the abstract: general relativity coupled to nonlinear electrodynamics with a Maxwell weak-field limit admits a static, spherically symmetric, globally regular metric only when the electric charge is zero and L(F) tends to a finite limit as F goes to infinity. The 'only if' half rests on Theorem 1 (no regular center with nonzero electric charge, including dyonic) and Theorem 2 (no wormhole throat or horn when T^0_0 = T^1_1). The 'if' half is the explicit quadrature construction of §3, which yields a de Sitter core and a mass fixed by regularity.

Load-bearing premise

The necessity direction of the second condition, that a regular center with magnetic charge forces L(F) to a finite limit as F goes to infinity, is argued in a single sentence in §3 ('It is easily seen that a space-time with a regular center is indeed obtained for any L(F) such that L tends to L_infinity'). The converse, that a diverging L(F) necessarily produces a singular metric, is not proved explicitly; it is plausible because T^0_0 = L/2 would then diverge and M(r) would grow faster than r^3, but the paper does not close this step. The 'if and only if' in the abstract is therefore slightly stronger than what §3 establishes.

Original abstract

It is shown that general relativity coupled to nonlinear electrodynamics (NED) with the Lagrangian $L(F)$, $F = F_mn F^mn$ having a correct weak field limit, leads to nontrivial static, spherically symmetric solutions with a globally regular metric if and only if the electric charge is zero and $L(F)$ tends to a finite limit as $F \to \infty$. Properties and examples of such solutions, which include magnetic black holes and soliton-like objects (monopoles), are discussed. Magnetic solutions are compared with their electric counterparts. A duality between solutions of different theories specified in two alternative formulations of NED (called $FP$ duality) is used as a tool for this comparison.

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, 3,689 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.68 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.57 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

It is shown that general relativity coupled to nonlinear electrodynamics (NED) with the Lagrangian $L(F)$, $F = F_mn F^mn$ having a correct weak field limit, leads to nontrivial static, spherically symmetric solutions with a globally regular metric if and only if the electric charge is zero and $L(F)$ tends to a finite limit as $F \to \infty$. Properties and examples of such solutions, which include magnetic black holes and soliton-like objects (monopoles), are discussed. Magnetic solutions are compared with their electric counterparts. A duality between solutions of different theories specified in two alternative formulations of NED (called $FP$ duality) is used as a tool for this comparison.

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 It is shown that general relativity coupled to nonlinear electrodynamics (NED) with the Lagrangian $L(F)$, $F = F_mn F^mn$ having a correct weak field limit, leads to nontrivial static, spherically symmetric solutions with a globally regular metric if and only if the electric charge is zero and $L(F)$ tends to a finite limit as $F \to \infty$. result 0.99 0.88 0.88 asserted 1.00
s1 Properties and examples of such solutions, which include magnetic black holes and soliton-like objects (monopoles), are discussed. contribution 0.96 0.86 0.08 asserted 0.83
s2 Magnetic solutions are compared with their electric counterparts. contribution 0.84 0.82 0.03 asserted 0.54
s3 A duality between solutions of different theories specified in two alternative formulations of NED (called $FP$ duality) is used as a tool for this comparison. method 0.99 0.72 0.04 asserted 0.99

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
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. An example of qualitative behavior of H(P), F(P), and L(F) in an electric solution qe is found by integration in (19) from 0 to ∞. This is how the regular solutions of [8–10] were obtained with the following choices of H(P): [8] : H(P) = P 1 − 3Π (1 + Π)3 + 6 q 2s  Π 1 + Π5/2 , (20) [9] : H(P) = P  cosh2 (s √ Π), (21) [10] : H(P) = P exp(−s √ Π) (1 + Π)5/2  1 + 3 s √ Π + Π ,(22) where Π = p −q 2P/2 and s = |q|/(2m), q = qe and m being free parameters identified with the charge and mass of the configuration, respectively. The functions (20)–(22) beha

Figures reproduced from arXiv:gr-qc/0006014 by the authors.

References
How later papers used it

238 later papers cite it

Evidence: substantial. 236 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 38 of the 354 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 paper
Uses its method or tool 2 of 3 papers with this label
How the literature uses it

200 arXiv papers cite this in 302 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 · 180 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 1 paper
Uses its method or tool 3 papers
Compares with it 3 papers
Cites it as support 1 paper
Cites it as background 102 papers

Discussion

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