pith. sign in

arxiv: 2606.23746 · v1 · pith:MUMXI4L2new · submitted 2026-06-21 · 🌀 gr-qc

Physical properties of charged black holes from the nonlinear electrodynamics model based on electric potential regularization

S. Habib Mazharimousavi This is my paper

Pith reviewed 2026-06-26 09:40 UTC · model grok-4.3

classification 🌀 gr-qc
keywords nonlinear electrodynamicscharged black holesblack hole thermodynamicsevent horizonsblack hole shadowsReissner-Nordstromcausal structurenull geodesics
0
0 comments X

The pith

Nonlinear electrodynamics from a regularized point-charge potential produces charged black holes with up to three horizons and modified thermodynamics and shadows.

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

The paper constructs static spherically symmetric black hole solutions in Einstein gravity minimally coupled to a nonlinear electrodynamics model built from a regularized electric potential. The resulting metrics depend on three parameters: ADM mass M, charge Q, and nonlinear scale r0, and they admit single, double, or triple horizons along with naked singularities whose type depends on the combination m minus two-thirds q squared. Thermodynamic analysis yields Hawking temperature and heat capacity that reveal two critical charge values separating thermally unstable regimes from stable phases bounded by Davies points, with extremal zero-temperature solutions appearing at large charge. Null geodesic and shadow calculations show deviations from Reissner-Nordström geometry, including possible disappearance of the photon sphere beyond a critical charge. A reader would care because these features could produce observable differences in black hole structure and light deflection if the nonlinear scale is physically realized.

Core claim

The nonlinear electrodynamics model constructed from a regularized electric potential yields static spherically symmetric black hole solutions characterized by ADM mass M, charge Q, and nonlinear scale r0; depending on the values of M and Q these solutions exhibit black holes with one, two or three horizons, naked singular geometries, and a central singularity whose character is fixed by m minus two-thirds q squared, while their thermodynamics are controlled by two critical charge parameters that determine the existence of stable phases and extremal configurations, and their null geodesics display noticeable deviations from the Reissner-Nordström case including the possible absence of a phot

What carries the argument

The regularized electric potential of a point charge that defines the NED Lagrangian and, after minimal coupling, integrates to the three-parameter family of metrics.

If this is right

  • Depending on M and Q the solutions exhibit single, double, and triple horizons as well as naked singular geometries.
  • Two critical charge parameters govern thermal behavior, with black holes unstable below the critical charge and a stable phase appearing above it within a finite horizon-radius range.
  • For sufficiently large charge, extremal configurations with vanishing temperature arise.
  • Nonlinear corrections produce deviations in null geodesics and black hole shadows, including possible absence of a photon sphere beyond a critical charge.
  • The central singularity is spacelike or timelike according to the sign of m minus two-thirds q squared.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Precise shadow observations could place upper bounds on the nonlinear scale r0 if the model applies to real black holes.
  • The triple-horizon cases may produce distinctive quasinormal-mode spectra or perturbation dynamics not present in Reissner-Nordström geometry.
  • The regularization procedure offers a minimal-coupling route to singularity resolution that could be compared with other regular black-hole constructions.
  • Thermodynamic stability windows might affect the endpoint of Hawking evaporation or the accretion behavior of these objects.

Load-bearing premise

Regularizing the electric potential of a point charge produces a specific NED Lagrangian whose associated stress-energy tensor integrates to the claimed three-parameter metric family.

What would settle it

An astrophysical measurement of a charged black hole shadow whose size or shape either matches the Reissner-Nordström prediction for all observed charges or deviates in a manner that cannot be reproduced by any value of the nonlinear scale r0.

Figures

Figures reproduced from arXiv: 2606.23746 by S. Habib Mazharimousavi.

Figure 1
Figure 1. Figure 1: FIG. 1: The metric function in Configuration 1, where [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The Penrose diagrams of the solutions in Configurations 1 and 2. Panel (a) corresponds to black hole solutions in Figs. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The metric function in Configuration 2, where [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The metric function in Configuration 3, where [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The Penrose diagram of the solutions in Configurations 3 and 4, corresponding to Fig. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The Penrose diagram of the black hole solutions presented in Fig. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The Penrose diagram of the black hole solutions presented in Fig. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The metric function in Configuration 4, where [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The Penrose diagrams of the black hole solutions presented in Fig. [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Generic plots of the Hawking temperature and heat capacity versus [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Plots of [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The radius of the shadow observed at infinity in terms of the dimensionless electric charge [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
read the original abstract

We investigate static, spherically symmetric black hole solutions arising from Einstein gravity minimally coupled to a nonlinear electrodynamics (NED) model constructed from a regularized electric potential of a point charge. The resulting spacetime is characterized by three parameters, namely, the ADM mass $% M $, the electric charge $Q$, and the nonlinear scale $r_{0}$. We show that, depending on the values of $M$ and $Q$, the solutions exhibit a rich causal structure comprising black holes with single, double, and triple horizons, as well as naked singular geometries. The nature of the central singularity is determined by the combination $m-\frac{2}{3}q^{2}$, allowing for both spacelike and timelike singularities. We perform a detailed thermodynamic analysis by deriving the Hawking temperature and heat capacity, revealing the existence of two critical charge parameters that govern the thermal behavior. Below a critical charge, the black holes are thermally unstable, whereas above it a stable phase emerges within a finite range of horizon radii bounded by Davies points. For sufficiently large charge, extremal configurations with vanishing temperature arise, further constraining the stability region. We also investigate observational signatures by analyzing null geodesics and black hole shadows, showing that nonlinear electrodynamics corrections lead to noticeable deviations from the Reissner-Nordstr\"{o}m geometry, including the possible absence of a photon sphere beyond a critical charge. Our results highlight that nonlinear electrodynamics significantly enriches the causal structure, thermodynamic phase space, and dynamical response of charged black holes, providing potentially observable deviations from the Reissner-Nordstr\"{o}m paradigm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript constructs a nonlinear electrodynamics model by regularizing the electric potential of a point charge, minimally coupled to Einstein gravity. This produces a three-parameter family of static spherically symmetric metrics (M, Q, r0). The paper analyzes the resulting causal structures (single/double/triple horizons, naked singularities with nature set by m−(2/3)q²), thermodynamics (Hawking temperature, heat capacity, two critical charges, stability windows bounded by Davies points, extremal configurations), and null geodesics (shadows with possible absence of photon spheres beyond a critical charge), claiming these features enrich the Reissner-Nordström paradigm.

Significance. If the model derivation is consistent and the metrics solve the coupled equations, the work supplies a concrete NED example with multiple horizons, thermodynamic critical points, and observable shadow deviations. The identification of charge-dependent stability regions and the potential disappearance of photon spheres would be of interest for strong-field tests and modified electrodynamics phenomenology.

major comments (3)
  1. [§2] §2 (model construction): the regularization procedure applied to the point-charge potential is presented, but neither the explicit Lagrangian L(F) nor the integration steps yielding a divergence-free, minimally coupled T_{\mu\nu} that produces the claimed three-parameter metric are shown. This is load-bearing, as all horizon counts, critical charges, and shadow results presuppose that the geometry solves the Einstein-NED system rather than being an ansatz.
  2. [§3] §3 (metric and horizons): without the explicit metric function f(r) or the field equations, the statements that solutions exhibit single/double/triple horizons depending on M and Q, and that the central singularity type is controlled by m−(2/3)q², cannot be verified independently.
  3. [§4] §4 (thermodynamics): the two critical charge parameters and the stability windows bounded by Davies points are reported, but their derivation from the temperature and heat capacity expressions relies on the unshown metric; the claim that sufficiently large charge produces extremal configurations with vanishing temperature therefore remains unconfirmed.
minor comments (2)
  1. [Abstract] The abstract contains a typographical artifact ("$% M $"); this should be cleaned in the final version.
  2. Notation for the nonlinear scale is introduced as r0 but later appears as r_{0}; consistent usage throughout would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript accordingly to ensure all derivations and expressions are explicitly presented for independent verification.

read point-by-point responses

  1. Referee: [§2] §2 (model construction): the regularization procedure applied to the point-charge potential is presented, but neither the explicit Lagrangian L(F) nor the integration steps yielding a divergence-free, minimally coupled T_{\mu u} that produces the claimed three-parameter metric are shown. This is load-bearing, as all horizon counts, critical charges, and shadow results presuppose that the geometry solves the Einstein-NED system rather than being an ansatz.

    Authors: We agree that the explicit Lagrangian L(F) and the detailed integration steps confirming a divergence-free T_{\mu u} were not presented with sufficient clarity. In the revised manuscript we will add the explicit form of L(F) derived from the regularized potential together with the integration steps showing that the resulting stress-energy tensor is divergence-free and yields the three-parameter metric through the Einstein equations. revision: yes

  2. Referee: [§3] §3 (metric and horizons): without the explicit metric function f(r) or the field equations, the statements that solutions exhibit single/double/triple horizons depending on M and Q, and that the central singularity type is controlled by m−(2/3)q², cannot be verified independently.

    Authors: We will include the explicit metric function f(r) and the relevant Einstein-NED field equations in the revised §3. This will enable direct verification of the single-, double-, and triple-horizon configurations as well as the classification of the central singularity according to the sign of m−(2/3)q². revision: yes

  3. Referee: [§4] §4 (thermodynamics): the two critical charge parameters and the stability windows bounded by Davies points are reported, but their derivation from the temperature and heat capacity expressions relies on the unshown metric; the claim that sufficiently large charge produces extremal configurations with vanishing temperature therefore remains unconfirmed.

    Authors: The thermodynamic quantities are obtained directly from the metric function. In the revised manuscript we will explicitly derive and display the Hawking temperature and heat-capacity expressions from f(r), thereby confirming the two critical charges, the stability windows delimited by Davies points, and the existence of extremal configurations with vanishing temperature for sufficiently large charge. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent construction

full rationale

The abstract describes a NED Lagrangian obtained via regularization of the point-charge potential, yielding a three-parameter (M, Q, r0) metric family whose causal structure, thermodynamics, and shadows are then analyzed. No equations, self-citations, or explicit reductions are supplied in the provided text that would make any reported feature (horizon multiplicity, critical charges, shadow size) equivalent by construction to the input regularization scale. The introduction of r0 as a model parameter and the subsequent study of its effects constitute standard parametric exploration rather than a fitted-input-called-prediction or self-definitional loop. Absent load-bearing self-citations or ansatz smuggling, the chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The model rests on the choice of regularization for the electric potential (an ad-hoc modeling step) and the standard minimal-coupling assumption; r0 is the only explicit free parameter identified in the abstract.

free parameters (1)
  • r0
    Nonlinear scale that sets the strength of the regularization in the NED Lagrangian; all reported horizon structures and critical charges are stated to depend on its value.
axioms (2)
  • domain assumption Einstein gravity is minimally coupled to the nonlinear electrodynamics stress-energy tensor
    Standard assumption invoked to obtain the field equations whose solutions are analyzed.
  • standard math The spacetime is static and spherically symmetric
    Metric ansatz used to reduce the Einstein-NED equations to ordinary differential equations.
invented entities (1)
  • Regularized electric potential of a point charge no independent evidence
    purpose: To define a finite NED Lagrangian that removes the central singularity of the Maxwell field while preserving asymptotic flatness.
    The regularization procedure is postulated as the starting point of the model; no independent observational or theoretical justification is supplied in the abstract.

pith-pipeline@v0.9.1-grok · 5826 in / 1741 out tokens · 30885 ms · 2026-06-26T09:40:55.450404+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 1 linked inside Pith

  1. [1]

    Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory, Sitzungsber

    K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) , 189, 196 (1916)

    1916

  2. [2]

    Reissner, Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie, Ann

    H. Reissner, Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie, Ann. Phys. 355, 106 (1916)

    1916

  3. [3]

    Nordström, On the energy of the gravitational field in Einstein’s theory, Proc

    G. Nordström, On the energy of the gravitational field in Einstein’s theory, Proc. Kon. Ned. Akad. Wet. 20, 1238 (1918)

    1918

  4. [4]

    S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, 1973)

    1973

  5. [5]

    Born and L

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

    1934

  6. [6]

    J. M. Bardeen, Non-singular general-relativistic gravitational collapse, Proc. Int. Conf. GR5, 174 (1968)

    1968

  7. [7]

    Ayón-Beato and A

    E. Ayón-Beato and A. García, The Bardeen model as a nonlinear magnetic monopole, Phys. Lett. B 493, 149 (2000)

    2000

  8. [8]

    Ayón-Beato and A

    E. Ayón-Beato and A. García, Regular black hole in general relativity coupled to nonlinear electrodynamics, Phys. Rev. Lett. 80, 505 (1998)

    1998

  9. [9]

    K. A. Bronnikov, Regular magnetic black holes and monopoles from nonlinear electrodynamics, Phys. Rev. D 63, 044005 (2001)

    2001

  10. [10]

    Dymnikova, Regular electrically charged structures in nonlinear electrodynamics coupled to general relativity, Class

    I. Dymnikova, Regular electrically charged structures in nonlinear electrodynamics coupled to general relativity, Class. Quantum Grav. 21, 4417 (2004)

    2004

  11. [11]

    Z. Y. Fan and X. Wang, Construction of regular black holes in general relativity, Phys. Rev. D 94, 124027 (2016)

    2016

  12. [12]

    Balart and E

    L. Balart and E. C. Vagenas, Regular black holes with a nonlinear electrodynamics source, Phys. Rev. D 90, 124045 (2014)

    2014

  13. [13]

    S. I. Kruglov, Nonlinear electrodynamics and black holes, Ann. Phys. 378, 59 (2017)

    2017

  14. [14]

    S. H. Hendi, Black hole solutions in nonlinear electrodynamics, Eur. Phys. J. C 77, 296 (2017)

    2017

  15. [15]

    S. A. Hayward, Formation and evaporation of regular black holes, Phys. Rev. Lett. 96, 031103 (2006)

    2006

  16. [16]

    Bambi and L

    C. Bambi and L. Modesto, Rotating regular black holes, Phys. Lett. B 721, 329 (2013)

    2013

  17. [17]

    S. H. Mazharimousavi, Regular electric potential and nonlinear electrodynamics, Fortschritte der Physik (2026), submitted

    2026

  18. [18]

    J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973) . 14

    1973

  19. [19]

    S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975)

    1975

  20. [20]

    P. C. W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41, 1313 (1977)

    1977

  21. [21]

    Chamblin et al., Charged AdS black holes and phase transitions, Phys

    A. Chamblin et al., Charged AdS black holes and phase transitions, Phys. Rev. D 60, 064018 (1999)

    1999

  22. [22]

    Perlick et al., Influence of a plasma on the shadow of a spherically symmetric black hole, Phys

    V. Perlick et al., Influence of a plasma on the shadow of a spherically symmetric black hole, Phys. Rev. D 92, 104031 (2015)

    2015

  23. [23]

    I., Astrophys

    Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results. I., Astrophys. J. Lett. 875, L1 (2019)

    2019

  24. [24]

    VI., Astrophys

    Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results. VI., Astrophys. J. Lett. 875, L6 (2019)

    2019

  25. [25]

    Regge and J

    T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063 (1957)

    1957

  26. [26]

    F. J. Zerilli, Gravitational field of a particle falling in a Schwarzschild geometry, Phys. Rev. D 2, 2141 (1970)

    1970

  27. [27]

    K. D. Kokkotas and B. G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Relativity 2, 2 (1999)

    1999

  28. [28]

    Berti, V

    E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Class. Quantum Grav. 26, 163001 (2009)

    2009

  29. [29]

    Cardoso et al., Geodesic stability, Lyapunov exponents and quasinormal modes, Phys

    V. Cardoso et al., Geodesic stability, Lyapunov exponents and quasinormal modes, Phys. Rev. D 79, 064016 (2009)

    2009

  30. [30]

    R. M. Wald, Black Hole Entropy is the Noether Charge, Phys. Rev. D 48, R3427 (1993)

    1993

  31. [31]

    Iyer, and R

    V. Iyer, and R. M. Wald, Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy, Phys. Rev. D 50, 846 (1994)

    1994

  32. [32]

    D. A. Rasheed, Non-linear Electrodynamics: Zeroth and First Laws of Black Hole Mechanics, arXiv:hep-th/9702087v2

    Pith/arXiv arXiv

  33. [33]

    Breton, Smarr’s Formula for Black Holes with Nonlinear Electrodynamics, Gen

    N. Breton, Smarr’s Formula for Black Holes with Nonlinear Electrodynamics, Gen. Relativ. Gravit. 37, 643 (2005)

    2005

  34. [34]

    T. K. Dey, Born-Infeld Black Holes in the Presence of a Cosmological Constant, Phys. Lett. B 595, 484 (2004)

    2004

  35. [35]

    S. H. Hendi, Asymptotic Reissner-Nordström black holes, Ann. Phys. 333, 282 (2013)

    2013