Lorentz-Violating (Regular) Black Holes in Einstein Gravity

H. Lu; Zhi-Chao Li

REVIEW 3 major objections 2 minor 20 references

Lorentz-Violating (Regular) Black Holes in Einstein Gravity

T0 review · 3 major / 2 minor · reviewed 2026-06-29 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Nonlinear electrodynamics produces Lorentz-violating regular black holes in Einstein gravity.

desk verdict The paper builds Lorentz-violating regular black holes inside unmodified Einstein gravity using a nonlinear electrodynamics sector, but the construction's viability rests on whether the matter Lagrangian actually works without energy-condition or stability problems. read the letter →

arxiv 2605.27615 v1 pith:6DAZ65KE submitted 2026-05-26 gr-qc hep-th

Zhi-Chao Li , H. Lu This is my paper

classification gr-qchep-th

keywords LorentzviolationregularblackholesnonlinearelectrodynamicsEinsteingravityconicalgeometryBardeenholeHaywardReissner-Nordström

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 introduces a minimally coupled dark sector based on nonlinear electrodynamics that generates regular spacetime textures. These textures connect a regular core to an asymptotic conical geometry that explicitly violates Lorentz invariance. The construction yields Lorentz-violating versions of the Schwarzschild and Reissner-Nordström black holes directly in Einstein gravity. It also produces Lorentz-violating regular black holes, including the Bardeen and Hayward examples plus a class of charged regular solutions.

What carries the argument

A nonlinear electrodynamics Lagrangian whose stress-energy tensor enforces both a regular core and an asymptotic conical Lorentz-violating geometry.

What would settle it

Explicit construction of a nonlinear electrodynamics Lagrangian that yields the described regular core and conical asymptotics but introduces ghosts or violates the null energy condition would show the mechanism cannot work as stated.

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Extended reading notes

Core claim

By choosing suitable nonlinear electrodynamics Lagrangians, the resulting stress-energy tensor supports black hole solutions in Einstein gravity that remain regular at the center while approaching a conical asymptotic geometry that breaks Lorentz invariance, thereby generating Lorentz-violating Schwarzschild, Reissner-Nordström, Bardeen, and Hayward black holes without higher-curvature terms.

Load-bearing premise

The nonlinear electrodynamics Lagrangian can be chosen to produce the required stress-energy tensor without ghosts, instabilities, or violations of the null energy condition.

Editorial extensions

If this is right

Lorentz-violating Schwarzschild and Reissner-Nordström black holes arise as solutions in Einstein gravity with minimal matter coupling.
The Bardeen and Hayward regular black holes can be realized in Lorentz-violating form.
A family of electrically charged regular black holes with Lorentz-violating asymptotics becomes available.
All solutions interpolate smoothly from a regular core to conical asymptotics that break Lorentz invariance.

Reading between the lines

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

The conical asymptotics may produce measurable effects in black hole shadows or orbital dynamics at large distances.
The same sector could be tested for consistency with cosmological observations if the dark sector is present on larger scales.
Extensions might generate additional regular black hole families beyond the Bardeen and Hayward cases by varying the nonlinear Lagrangian.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that this work shows how to produce regular black holes with Lorentz-violating conical asymptotics in standard Einstein gravity plus a minimally coupled nonlinear electrodynamics dark sector. They apply the same setup to recover Lorentz-violating versions of Schwarzschild and Reissner-Nordström, then extend it to the Bardeen and Hayward regular solutions plus some charged regular cases.

What is new is the specific choice of the dark sector to enforce both a regular core and the asymptotic deficit angle that breaks Lorentz invariance. Prior regular black hole constructions already used nonlinear electrodynamics, so this is an incremental extension rather than a wholesale new mechanism, but the combination with explicit Lorentz violation at infinity is not something I recall from the usual literature.

The paper does well by keeping the gravitational action untouched and relying only on minimal coupling. That keeps the setup simple and within Einstein gravity.

The soft spot is exactly the one flagged in the stress test. The central claim requires an NED Lagrangian L(F) whose stress-energy tensor sources both regularity at r=0 and the conical LV geometry at infinity while satisfying the null energy condition, avoiding ghosts, and remaining stable. Reverse-engineering L(F) from a target metric is always possible in spherical symmetry, but the resulting model must still pass those physical checks. The abstract gives no explicit form for L(F) or any verification, so if the full paper does not supply those calculations and show they hold, the result stays formal rather than physically reliable.

This is for people working on black hole solutions with non-standard asymptotics or Lorentz violation in GR. A reader who wants concrete examples of such metrics would find the constructions useful, provided the matter sector is shown to be healthy. It deserves peer review because the idea is straightforward and the technical gap is narrow; referees can check the Lagrangian and stability directly.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes using a minimally coupled nonlinear electrodynamics (NED) dark sector in Einstein gravity to source black-hole spacetimes that are regular at the origin and asymptotically approach a conical geometry with explicit Lorentz violation. The construction is claimed to recover Lorentz-violating Schwarzschild and Reissner-Nordström solutions as well as Lorentz-violating versions of the Bardeen and Hayward regular black holes and a class of electrically charged regular black holes.

Significance. If the required NED Lagrangian exists and yields solutions that satisfy the null energy condition everywhere, remain free of ghosts and gradient instabilities, and solve the Einstein equations, the framework would supply a concrete, minimally coupled matter model for Lorentz-violating regular black holes inside unmodified general relativity. No machine-checked proofs, reproducible code, or parameter-free derivations are reported.

major comments (3)

[Abstract and §2] Abstract and §2 (Lagrangian construction): the central claim requires an explicit NED Lagrangian L(F) whose stress-energy tensor simultaneously produces a regular core, enforces the conical LV asymptotics, and obeys the null energy condition with no ghosts or tachyonic instabilities. No such L(F) is displayed or verified against these conditions; without it the existence of the claimed solutions cannot be assessed.
[§3] §3 (metric ansatz and field equations): the paper must demonstrate that the chosen L(F) satisfies the Einstein equations for the stated metric families (Schwarzschild, RN, Bardeen, Hayward). The abstract provides no explicit check that the derived T_{\mu u} is consistent with the Einstein tensor for any of these geometries.
[§4] §4 (stability and energy conditions): the weakest assumption identified in the stress-test—that a single L(F) can avoid NEC violation, ghosts, and instabilities while enforcing both regularity and LV conical asymptotics—remains unaddressed; any stability analysis or NEC plot must be supplied to substantiate the claim.

minor comments (2)

[Abstract] Notation for the conical deficit angle and the LV parameter should be introduced once and used consistently; the abstract employs both “conical geometry” and “Lorentz-violating” without a single defining equation.
[Introduction] References to the original Bardeen and Hayward metrics should be added when they are first invoked, to allow direct comparison with the Lorentz-violating generalizations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment point by point below, indicating the revisions that will be incorporated.

read point-by-point responses

Referee: [Abstract and §2] Abstract and §2 (Lagrangian construction): the central claim requires an explicit NED Lagrangian L(F) whose stress-energy tensor simultaneously produces a regular core, enforces the conical LV asymptotics, and obeys the null energy condition with no ghosts or tachyonic instabilities. No such L(F) is displayed or verified against these conditions; without it the existence of the claimed solutions cannot be assessed.

Authors: We acknowledge that an explicit functional form for the NED Lagrangian L(F) was not displayed in the original manuscript. The construction proceeds by specifying metric functions that solve the Einstein equations sourced by a NED stress-energy tensor with the required properties, but the referee is correct that the explicit L(F) must be provided to allow verification. In the revised version we will derive and present the explicit L(F) that reproduces the desired T_{\mu\nu} for both the regular core and the conical LV asymptotics, together with checks that it satisfies the null energy condition and yields no ghosts or gradient instabilities at the linear level. revision: yes
Referee: [§3] §3 (metric ansatz and field equations): the paper must demonstrate that the chosen L(F) satisfies the Einstein equations for the stated metric families (Schwarzschild, RN, Bardeen, Hayward). The abstract provides no explicit check that the derived T_{\mu\nu} is consistent with the Einstein tensor for any of these geometries.

Authors: Section 3 derives the metric ansatz and shows that the Einstein equations are satisfied once the NED source is chosen to match the required T_{\mu\nu}. However, we agree that explicit verification for each family (LV Schwarzschild, RN, Bardeen, Hayward) should be added. The revision will include direct substitution of the derived T_{\mu\nu} into the Einstein tensor for representative members of each family to confirm consistency. revision: yes
Referee: [§4] §4 (stability and energy conditions): the weakest assumption identified in the stress-test—that a single L(F) can avoid NEC violation, ghosts, and instabilities while enforcing both regularity and LV conical asymptotics—remains unaddressed; any stability analysis or NEC plot must be supplied to substantiate the claim.

Authors: We accept that a dedicated analysis of the null energy condition and perturbative stability is required. The revised manuscript will contain an explicit check that the NEC holds everywhere for the chosen L(F), together with a discussion of the absence of ghosts and tachyonic modes based on the second variation of the NED action. Plots of the relevant energy-condition functions will be added where they clarify the result. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; derivation presented as independent construction

full rationale

The provided abstract and context describe a construction using minimally coupled NED to source specific regular and LV-asymptotic metrics in Einstein gravity. No equations, self-citations, or derivation steps are quoted that reduce a claimed prediction or result to a fitted input or prior self-result by construction. The central claim is that such an NED Lagrangian can be chosen to satisfy the listed conditions; without explicit back-engineering shown or load-bearing self-citation in the given text, the derivation chain does not exhibit the enumerated circularity patterns. This is the expected honest non-finding when no reducing step is exhibited.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

Based on abstract only; the construction rests on the existence of a suitable nonlinear electrodynamics Lagrangian that simultaneously regularizes the core and enforces Lorentz-violating conical asymptotics.

assumptions (1)

domain assumption Einstein gravity with minimally coupled matter fields
The framework assumes the standard Einstein-Hilbert action plus minimal coupling of the nonlinear electrodynamics sector.

invented entities (1)

nonlinear electrodynamics dark sector
purpose: Generates regular core and Lorentz-violating conical asymptotics
Introduced to interpolate between regular interior and Lorentz-violating exterior geometry.

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Cite this review

Pith. "Pith review of Lorentz-Violating (Regular) Black Holes in Einstein Gravity." pith.science (2026). https://pith.science/paper/6DAZ65KE

@misc{pith2026260527615,
  author       = {Pith},
  title        = {Pith review of: Lorentz-Violating (Regular) Black Holes in Einstein Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DAZ65KE}},
  note         = {Machine review of arXiv:2605.27615}
}

read the original abstract

We introduce a minimally coupled dark sector based on a nonlinear electrodynamics to generate regular spacetime textures that interpolate from a regular core to an asymptotic Lorentz-violating conical geometry. This construction provides a simple mechanism, in Einstein gravity with minimally coupled matter, for obtaining Lorentz-violating Schwarzschild and Reissner-Nordstr\"om black holes. The framework further allows us to construct Lorentz-violating regular black holes, including the Bardeen and Hayward black holes, as well as a class of electrically-charged regular black holes.

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Reference graph

Works this paper leans on

20 extracted references · 18 canonical work pages

[1]

Gravity, Lorentz Violation, and the Standard Model

V.A. Kosteleck´ y, “Gravity, Lorentz violation, and the standard model,” Phys. Rev. D69, 105009 (2004), arXiv:hep-th/0312310

work page Pith review arXiv 2004
[2]

Spontaneous Lorentz Violation, Nambu-Goldstone Modes, and Gravity

R. Bluhm and V.A. Kosteleck´ y, “Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity,” Phys. Rev. D71, 065008 (2005), arXiv:hep-th/0412320. 9

work page Pith review arXiv 2005
[3]

An exact Schwarzschild-like solution in a bumblebee gravity model

R. Casana, A. Cavalcante, F.P. Poulis and E.B. Santos, “Exact Schwarzschild-like solution in a bumblebee gravity model,” Phys. Rev. D97, 104001 (2018), arXiv:1711.02273 [gr-qc]

work page Pith review arXiv 2018
[4]

Black holes with a cosmological constant in bumblebee gravity

R.V. Maluf and J.C.S. Neves, “Black holes with a cosmological constant in bumblebee gravity,” Phys. Rev. D103, 044002 (2021), arXiv:2011.12841 [gr-qc]

work page Pith review arXiv 2021
[5]

Charged spherically symmetric and slowly rotating charged black hole solutions in bumblebee gravity

J.Z. Liu, W.D. Guo, S.W. Wei and Y.X. Liu, “Charged spherically symmetric and slowly rotating charged black hole solutions in bumblebee gravity,” Eur. Phys. J. C85, no. 2, 145 (2025), arXiv:2407.08396 [gr-qc]

work page Pith review arXiv 2025
[6]

Taub-NUT-like black holes in Einstein-bumblebee gravity,

Y.Q. Chen and H.S. Liu, “Taub-NUT-like black holes in Einstein-bumblebee gravity,” Phys. Rev. D112, no.8, 084040 (2025) doi:10.1103/wmhj-83x3 [arXiv:2505.23104 [gr-qc]]

work page doi:10.1103/wmhj-83x3 2025
[7]

Dyonic RN-like and Taub-NUT-like black holes in Einstein- bumblebee gravity,

S. Li, L. Liang and L. Ma, “Dyonic RN-like and Taub-NUT-like black holes in Einstein-bumblebee gravity,” JCAP03, 005 (2026) doi:10.1088/1475-7516/2026/03/005 [arXiv:2510.04405 [gr-qc]]

work page doi:10.1088/1475-7516/2026/03/005 2026
[8]

Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a simple generating technique

H. Ovcharenko, “Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a simple generating technique,” [arXiv:2601.16037 [gr-qc]]

work page Pith review arXiv

Show all 20 references

[9]

When Bumblebee Meets NLED: Lorentz-Violating Black Holes and Regular Spacetimes,

Z.C. Li and H. L¨ u, “When Bumblebee Meets NLED: Lorentz-Violating Black Holes and Regular Spacetimes,” arXiv:2605.13963 [gr-qc]

arXiv
[10]

Regular black holes from analyticf(F 2),

Z.C. Li and H. L¨ u, “Regular black holes from analyticf(F 2),” Eur. Phys. J. C83, 755 (2023), arXiv:2303.16924 [gr-qc]

2023
[11]

Foundations of the new field theory,

M. Born and L. Infeld, “Foundations of the new field theory,” Proc. Roy. Soc. Lond. A 144, 425 (1934)

1934
[12]

Dyonic (A)dS black holes in Einstein-Born-Infeld theory in diverse dimensions,

S. Li, H. L¨ u and H. Wei, “Dyonic (A)dS black holes in Einstein-Born-Infeld theory in diverse dimensions,” JHEP07, 004 (2016), arXiv:1606.02733 [hep-th]

2016 arXiv
[13]

Non-singular general-relativistic gravitational collapse,

J.M. Bardeen, “Non-singular general-relativistic gravitational collapse,” inProceedings of the International Conference GR5, Tbilisi, U.S.S.R. (1968)

1968
[14]

Formation and evaporation of nonsingular black holes,

S.A. Hayward, “Formation and evaporation of nonsingular black holes,” Phys. Rev. Lett. 96, 031103 (2006), arXiv:gr-qc/0506126

2006 arXiv
[15]

Regular black holes from pure gravity,

P. Bueno, P.A. Cano and R.A. Hennigar, “Regular black holes from pure gravity,” Phys. Lett. B861, 139260 (2025), arXiv:2403.04827 [gr-qc]

2025
[16]

The Bardeen model as a nonlinear magnetic monopole,

E. Ay´ on-Beato and A. Garc´ ıa, “The Bardeen model as a nonlinear magnetic monopole,” Phys. Lett. B493, 149 (2000), arXiv:gr-qc/0009077. 10

2000 arXiv
[17]

Construction of regular black holes in general relativity,

Z.Y. Fan and X. Wang, “Construction of regular black holes in general relativity,” Phys. Rev. D94, 124027 (2016), arXiv:1610.02636 [gr-qc]

2016 arXiv
[18]

Regular magnetic black holes and monopoles from nonlinear electrody- namics,

K.A. Bronnikov, “Regular magnetic black holes and monopoles from nonlinear electrody- namics,” Phys. Rev. D63, 044005 (2001), arXiv:gr-qc/0006014

2001 arXiv
[19]

Regular electric black holes from Einstein-Maxwell-scalar gravity,

Z.C. Li and H. L¨ u, “Regular electric black holes from Einstein-Maxwell-scalar gravity,” Phys. Rev. D110, 104046 (2024), arXiv:2407.07952 [gr-qc]

2024
[20]

Enthalpy and the mechanics of AdS black holes,

D. Kastor, S. Ray and J. Traschen, “Enthalpy and the mechanics of AdS black holes,” Class. Quant. Grav.26, 195011 (2009), arXiv:0904.2765 [hep-th]. 11

2009 arXiv

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

Gravity, Lorentz Violation, and the Standard Model

V.A. Kosteleck´ y, “Gravity, Lorentz violation, and the standard model,” Phys. Rev. D69, 105009 (2004), arXiv:hep-th/0312310

work page Pith review arXiv 2004

[2] [2]

Spontaneous Lorentz Violation, Nambu-Goldstone Modes, and Gravity

R. Bluhm and V.A. Kosteleck´ y, “Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity,” Phys. Rev. D71, 065008 (2005), arXiv:hep-th/0412320. 9

work page Pith review arXiv 2005

[3] [3]

An exact Schwarzschild-like solution in a bumblebee gravity model

R. Casana, A. Cavalcante, F.P. Poulis and E.B. Santos, “Exact Schwarzschild-like solution in a bumblebee gravity model,” Phys. Rev. D97, 104001 (2018), arXiv:1711.02273 [gr-qc]

work page Pith review arXiv 2018

[4] [4]

Black holes with a cosmological constant in bumblebee gravity

R.V. Maluf and J.C.S. Neves, “Black holes with a cosmological constant in bumblebee gravity,” Phys. Rev. D103, 044002 (2021), arXiv:2011.12841 [gr-qc]

work page Pith review arXiv 2021

[5] [5]

Charged spherically symmetric and slowly rotating charged black hole solutions in bumblebee gravity

J.Z. Liu, W.D. Guo, S.W. Wei and Y.X. Liu, “Charged spherically symmetric and slowly rotating charged black hole solutions in bumblebee gravity,” Eur. Phys. J. C85, no. 2, 145 (2025), arXiv:2407.08396 [gr-qc]

work page Pith review arXiv 2025

[6] [6]

Taub-NUT-like black holes in Einstein-bumblebee gravity,

Y.Q. Chen and H.S. Liu, “Taub-NUT-like black holes in Einstein-bumblebee gravity,” Phys. Rev. D112, no.8, 084040 (2025) doi:10.1103/wmhj-83x3 [arXiv:2505.23104 [gr-qc]]

work page doi:10.1103/wmhj-83x3 2025

[7] [7]

Dyonic RN-like and Taub-NUT-like black holes in Einstein- bumblebee gravity,

S. Li, L. Liang and L. Ma, “Dyonic RN-like and Taub-NUT-like black holes in Einstein-bumblebee gravity,” JCAP03, 005 (2026) doi:10.1088/1475-7516/2026/03/005 [arXiv:2510.04405 [gr-qc]]

work page doi:10.1088/1475-7516/2026/03/005 2026

[8] [8]

Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a simple generating technique

H. Ovcharenko, “Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a simple generating technique,” [arXiv:2601.16037 [gr-qc]]

work page Pith review arXiv