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REVIEW 3 major objections 6 minor 62 references

Regular Power-Maxwell Black Holes

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper constructs a new family of singularity-free, asymptotically dynamical black holes sourced by power-law nonlinear electrodynamics and shows that stable branches with positive heat capacity exist.

desk verdict Solid magnetic regular black hole construction; electric counterpart claim needs scoping before the paper is citable. read the letter →

arxiv 2506.13676 v2 pith:GBYWITV3 submitted 2025-06-16 gr-qc

Yi-bo Liang , Hong-Rong Li This is my paper · ORCID
classification gr-qc MSC 83C5783C2283C15 PACS 04.70.-s04.20.Jb
keywords regularblackholesnonlinearelectrodynamicspower-MaxwellFPdualityenergyconditionsholethermodynamicseffectivemetricPenrosediagrams
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

This paper builds a new family of spherically symmetric spacetimes sourced by nonlinear electrodynamics with a power-law Maxwell Lagrangian, and claims they are regular at the center while behaving like power-Maxwell at infinity—asymptotically dynamical but not de Sitter. The authors show that, for suitable parameters, the same family contains black holes with three horizons, that the weak and dominant energy conditions can hold, and that electric solutions exist through electromagnetic duality under a single-valuedness condition. They also derive the first law and Smarr formula with the ADM mass entering the Lagrangian, and exhibit black-hole branches with positive heat capacity. If correct, this provides a concrete matter-sourced alternative to singular power-Maxwell black holes and to earlier regular black hole models, with thermodynamically stable configurations.

What carries the argument

The load-bearing mechanism is the reverse-engineering identity $L(f)=4m'(r)/r^2$, which converts a chosen regular metric function $m(r)$ in $A(r)=1-2m(r)/r$ into a nonlinear electrodynamics Lagrangian. The paper combines this with FP duality—the Legendre-type map between magnetic and electric formulations—and, where that duality fails, with an auxiliary-scalar representation $L(f,\phi)=h(\phi)f+j(\phi)$, to obtain electric counterparts and a generalized duality. These identities carry the derivation of the Lagrangian, the energy conditions, the effective photon metric, and the first law.

What would settle it

Choose $n=1/4$ with the parameters used in Fig. 3, compute the electric dual $H(p)$ from the full Lagrangian Eq. (7), and test whether $f(p)$ is monotonic over the full radial interval; if $f(p)$ has more than one extremum, or if $L_f$ changes sign inside the horizon, the electric solution is not a valid nonlinear electrodynamics solution and the central claim fails for that case.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the metric ansatz of Eq. (6)—built from a regular mass function with terms $r^{2}/(r^{\nu}+b^{\nu})^{4n/\nu}$ and $r^{2}/(r^{\sigma}+q^{\sigma})^{3/\sigma}$—is an exact solution of nonlinear electrodynamics with the Lagrangian in Eq. (7). The magnetic solution is constructed first and the electric one follows by FP duality, but the electric Lagrangian is single-valued only when $0<\nu\le\gamma$ (with $\gamma=3-4n$) in the $\alpha=0$ sector, and at $n=1/2$ the electric dual exists only in an auxiliary-scalar formulation. The spacetimes admit black holes with three horizons for restricted parameter ranges, satisfy weak and dominant energy conditions under stated inequalities, and the ADM mass enters the Lagrangian so the first law and Smarr formula take corrected forms; branches with positive heat capacity exist.

Load-bearing premise

The argument stands on the assumption that the Lagrangian obtained by reverse engineering from the chosen regular metric is a genuine single-valued nonlinear electrodynamics source at every radius; if $L(f)$ branches or $L_f$ changes sign in the interior, the spacetime is not a solution of the theory being proposed.

Editorial extensions

If this is right

  • For $0<n<1/2$, regular black holes exist only when the dimensionless charge $\tilde{\lambda}$ lies between two extremal values; at the boundaries the black hole is extremal, and outside them no black hole forms.
  • Electric counterparts are unique when $0<\nu\le\gamma$ in the $\alpha=0$ sector; outside that window the electric Lagrangian branches, and at $n=1/2$ an ordinary electric dual does not exist.
  • The effective photon metric is regular for magnetic solutions without black holes under the same parameter condition, but singular for electric solutions; photons can follow spacelike trajectories and suffer infinite redshift or blueshift at the regular center.
  • The ADM mass enters the Lagrangian, so the standard first law acquires a correction factor $\Delta$; the Smarr formula follows from homogeneity of degree $1/2$ in entropy and charges.
  • Heat capacity changes sign at two phase transitions, leaving an interval with positive heat capacity where the black hole is locally thermodynamically stable.

Reading between the lines

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

  • The auxiliary-scalar formulation may extend the duality construction to other nonlinear electrodynamics models whose magnetic and electric sectors are not Legendre-dual; a direct test is whether square-root Maxwell electric solutions exist in that formulation for other regular metrics.
  • Because the photon effective metric differs from the spacetime metric and permits spacelike photon trajectories, the shadow and photon-ring observables of these black holes would not match a standard ray-tracing in the spacetime metric; computing those observables is a natural next step.
  • The reverse-engineering route could be applied to other regular mass functions, but the paper checks single-valuedness analytically only when one of the two sources vanishes; verifying monotonicity of $f(p)$ for general $\lambda$ and $\alpha$ is the key open technical condition.
  • If these solutions are dynamically stable, the positive-heat-capacity branch makes them plausible end states of gravitational collapse without a curvature singularity, though dynamical stability is not addressed here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper constructs a multi-parameter family of static, spherically symmetric, regular spacetimes, Eq. (6), and interprets them as solutions of general relativity coupled to nonlinear electrodynamics with the Lagrangian of Eq. (7). The construction is inverse: a regular mass function is chosen and L = 4m'(r)/r^2 is integrated, yielding a Lagrangian that reduces to power-Maxwell behavior L proportional to f^n at infinity and is regular as f goes to infinity. The paper analyzes: (i) the m = 0 regularized power-Maxwell spacetimes and their Penrose diagrams; (ii) black-hole solutions with up to three horizons, with existence windows characterized by Eq. (13) and numerics in Figs. 3-4; (iii) electric counterparts via FP duality, including a monotonicity/single-valuedness result (0 < nu <= gamma) for the alpha = 0 sector, the explicit failure of uniqueness for m not equal to 0 (Fig. 6), and an auxiliary-scalar formulation, Eqs. (23)-(24), that supplies a dual for the n = 1/2 square-root Maxwell case in which the conventional dual vanishes; (iv) photon propagation in the effective metric, which is regular (though non-Lorentzian in patches) for certain magnetic configurations and singular for electric configurations, with spacelike photon trajectories; and (v) black-hole thermodynamics: a first law with a correction factor Delta, a Smarr formula, Eq. (71), and parameter intervals with positive heat capacity.

Significance. If the central claims hold, the paper provides a new family of singularity-free black holes with power-Maxwell asymptotics, dynamical non-de Sitter asymptotics, and thermodynamically stable branches, extending the Bardeen/Fan-Wang regular-black-hole program to a new source class. The inverse-construction algebra leading to Eq. (7) is internally consistent: I checked that L = 4m'/r^2 reduces Eq. (6) to the stated Lagrangian and to the correct power-Maxwell limit, and that Eq. (13) follows algebraically from substituting rM2 into Eq. (12). The treatment of FP duality is unusually candid: the n = 1/2 obstruction (H(p) = 0), the m nonzero multivaluedness of f(p), and the auxiliary-scalar remedy are all stated explicitly rather than hidden. The thermodynamic identities are explicit and checkable, and the paper makes concrete parameter-specific predictions (three-horizon windows; positive-heat-capacity intervals) that are falsifiable through the numerical analysis in Figs. 3, 4, 7, and 8. The principal qualification is scope: several headline claims are established only for the magnetic sector or for m = 0, and the electric black-hole source is not a single-valued NED Lagrangian.

major comments (3)
  1. [Sec. II A, after Eq. (6)] The dominant-energy-condition restriction is printed as '0 < nu <= [6-7n+2*sqrt(3(4n^2-7n+3))]/n > gamma', which is not a well-formed inequality as it stands, and no derivation is provided for it or for the accompanying assertion (after Eq. (4)) that the weak energy condition forces mu = 0. Since the abstract's claim that 'both the weak and dominant energy conditions are shown to be satisfiable' rests directly on these statements, the authors should state the bound in correct form (clarifying the role of gamma) and show the computation of the energy-momentum components and the resulting inequalities for rho, rho + p_r, and rho - p_theta.
  2. [Sec. II A, Eq. (13)] The three-horizon existence condition (13) is derived by substituting r-tilde = rM2, the location of the maximum of R2(r-tilde), into Eq. (12), justified only by the heuristic that R1(r-tilde) 'changes slowly'. The resulting inequality is therefore an approximation, although the text presents it as a necessary condition and the captions of Figs. 3 and 4 use lambda-M2 as exact boundary values. Please either supply a rigorous bound (for example by bracketing R1 between its limiting values) or explicitly label Eq. (13) as a heuristic criterion whose validity for the reported parameter windows is established by the numerical examples.
  3. [Secs. II B(iii), II C, V; Abstract] For the black-hole family (m not equal to 0, with both lambda and alpha nonzero), Fig. 6 shows that f(p) is non-monotonic, so the electric Lagrangian L(f) is multivalued and the metric (6) is not sourced by a single-valued electric NED; the conclusion itself concedes that 'uniqueness appears to fail' for m not equal to 0. The auxiliary-scalar representation that would restore a single-valued electric description is written out explicitly only for alpha = 0 (Eqs. (46)-(47)). The abstract's statements that 'electric counterparts [are] derived via FP duality' and that 'uniqueness conditions for the electric solutions are then established' therefore overreach for the black-hole spacetimes, and the electric-side analyses in Secs. III B and IV proceed in the multi-branched P-framework without stating that dependence. Please rescope the abstract and the electric-case claims to the m = 0 sector and the P-framework, or extend the auxiliary-scalar construction to m not equal to 0.
minor comments (6)
  1. [Passim] There are numerous typographical errors that should be corrected in a revision: 'gobally' (Fig. 1), 'condistions' (after Eq. (6)), 'approxinately' (Fig. 3), 'extermums' (Fig. 4), 'augular momentum' (below Eq. (53)), 'wrriten' (Sec. IV B), and '4n/nu' in Eq. (60).
  2. [Sec. IV A, Eqs. (59)-(60)] The symbol M is used both for the ADM mass (M = alpha q^3, a derived quantity) and for the horizon-determined mass function in Eq. (60), and the text alternates between these meanings; a distinct notation (e.g., M_ADM) would remove genuine ambiguity.
  3. [Sec. II C, below Eq. (32)] The sentence 'the regularized theory given by Eq. (32) is not well-defined in the limit q->0' appears to refer to the limit of the duality variable p (or r -> infinity), not to the charge parameter q; the notation should be made unambiguous.
  4. [Sec. II B, below Eq. (30)] The claim that the energy-momentum tensor remains well-defined in the limit lambda -> 0 'through careful consideration of the coupled dynamics' is asserted verbally; an explicit check using Eqs. (43)-(44) would be more convincing.
  5. [Sec. III A, near Eq. (54)] The statement that the Phi = 0 singularity is 'unnoticed by photons' should be reconciled with the fact that both dr/dtau and dphi/dtau vanish there; a brief comment on the (in)completeness of the effective-metric geodesics at these points would strengthen the discussion.
  6. [Sec. IV A, Eq. (65)] The step from Eq. (64) to Eq. (65) is terse; expanding the variation so that the reader can see the origin of the factor Delta and the role of the constraint M = alpha^{1/4} g^{3/2} would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

Inverse construction with acknowledged electric-sector limitations; no circularity found.

full rationale

The paper does not present a circular derivation. The metric family Eq. (6) is introduced as a regular ansatz ('To regularize the singularities ... directly assume a regular form for the metric'), and the NED Lagrangian Eq. (7) is obtained from the identity L = 4m'(r)/r^2, i.e., by solving the inverse problem for the source. The subsequent checks—energy conditions, horizon count, effective light metric, thermodynamics—are independent consistency conditions on that constructed solution, not fitted reproductions of the input metric. The electric side is constructed via the standard FP duality (citing Bronnikov and Moreno-Sarbach), and when the dual is not single-valued the paper says so explicitly: 'For cases m ≠ 0, and the presence of a black hole, uniqueness appears to fail' (Sec. V). The n = 1/2 square-root case is handled by an auxiliary-field formulation that the paper labels as extending, not concealing, the failure of ordinary FP duality. The appearance of the ADM mass in the Lagrangian is a substitution (M = α q^3), and the Smarr relation follows from the homogeneity of the resulting M(S,b,q); both are algebraic consequences rather than circular inputs. No load-bearing self-citation chain is present. The main caveat is that the electric black-hole Lagrangian branches (non-monotonic f(p), Fig. 6), which the paper acknowledges; that is a correctness/scope concern, not circularity.

Assumptions & free parameters 6 free parameters · 7 assumptions · 1 invented entities

The central construction rests on the standard NED-in-GR framework, the static spherically symmetric ansatz, the inverse method of deriving L(f) from a chosen mass function, the FP duality with a monotonicity condition, the effective-photon-metric formalism, and the area-law thermodynamics. The paper's model parameters (n, nu, sigma, lambda, alpha, g) are not fitted; they are free theory or solution parameters with ranges chosen to satisfy energy conditions and horizon existence. The only genuinely new postulated entity is the auxiliary scalar field used to restore duality at n=1/2; it has no independent falsifiable handle.

free parameters (6)
  • n (power-Maxwell exponent)
    Exponent in L(f)=2lambda(f/2lambda)^n; restricted to 0<n<1/2 or n=1/2 to make A(r)<0 at infinity. It is a coupling choice, not fitted.
  • nu (magnetic regularization exponent)
    Introduced in Eq. (6); constrained by energy conditions and by electric-frame monotonicity (0<nu<=gamma); chosen by hand.
  • sigma (Bardeen/Fan-Wang exponent)
    Introduced in Eq. (6); constrained sigma>1 for energy conditions; chosen by hand.
  • lambda (NED coupling)
    Positive constant with dimension L^-2 in Eq. (1); sets the scale of the power-Maxwell term. It is a model coupling, not fitted to data.
  • alpha (Bardeen-type coupling)
    Positive constant with dimension L^-2 in Eq. (6); sets the scale of the mass term. It is a model coupling, not fitted to data.
  • g (magnetic charge)
    Integration constant from F=g sin(theta)dtheta wedge dphi; labels the solution family and enters b and q. It is an integration constant, not fitted.
assumptions (7)
  • standard math Einstein equations with NED stress-energy tensor are the governing equations; the action is S=(1/16pi) integral epsilon (R-L(f)).
    Standard general relativity coupled to nonlinear electrodynamics; this is the background framework, not proved.
  • domain assumption The spacetime is static and spherically symmetric with metric g=diag(-A(r),1/A(r),r^2,...).
    Symmetry ansatz in Eq. (2); restricts the class of solutions considered.
  • domain assumption For a chosen mass function m(r), the NED Lagrangian is L=4m'(r)/r^2.
    Standard inverse construction used in regular black hole literature; the paper relies on it to derive Eq. (5) and Eq. (7).
  • domain assumption FP duality maps magnetic Lagrangians to electric ones, provided f(p) is monotonic and L(f) is single-valued.
    Used in Sec. II B; the paper itself notes equivalence fails when f(p) is non-monotonic.
  • ad hoc to paper The auxiliary scalar representation L(f,phi)=h(phi)f+j(phi) is on-shell equivalent to L(f) and has a strictly larger solution space.
    Introduced in Sec. II B/C specifically to restore duality at n=1/2; no external evidence is provided.
  • domain assumption Photon propagation is governed by the effective metric h_ab=L_f g_ab - 4L_ff F_ac F^c_b.
    Standard NED result, used in Sec. III for frequency shifts and null geodesics.
  • domain assumption Black hole entropy satisfies S=pi r_2^2, and b and q are independent thermodynamic variables in the first law.
    Area law plus the extended first-law treatment of Sec. IV A; this is an interpretive assumption for regular black holes.
invented entities (1)
  • Auxiliary scalar field phi in the L(f,phi) formulation
    purpose: Restores electric-magnetic duality for square-root Maxwell (n=1/2) and enables a generalized duality transformation between magnetic and electric solution sectors.
    The scalar is a mathematical device introduced through a Legendre-type representation in Eqs. (23)-(24); no observational signature or independent falsifiable prediction is given. Its physical status is unclear, especially since the n=1/2 electric solution has vanishing stress-energy for g=0 despite nonzero F.

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Pith. "Pith review of Regular Power-Maxwell Black Holes." pith.science (2026). https://pith.science/paper/GBYWITV3

@misc{pith2026250613676,
  author       = {Pith},
  title        = {Pith review of: Regular Power-Maxwell Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBYWITV3}},
  note         = {Machine review of arXiv:2506.13676}
}
read the original abstract

We present a new class of regular, spherically symmetric spacetimes in nonlinear electrodynamics that are asymptotically dynamical but not de Sitter, exhibiting power-law Maxwell behavior at infinity. Generalizing to black holes, we derive their existence conditions and construct corresponding Penrose diagrams. Both the weak and dominant energy conditions are shown to be satisfiable. Magnetic solutions are first obtained, with electric counterparts derived via FP duality. Uniqueness conditions for the electric solutions are then established. Although electric duals are absent in square-root Maxwell theory, our auxiliary scalar formulation restores duality and enables a generalized duality transformation. The effective light propagation metric remains regular for particular magnetic configurations (without black holes) but becomes singular for electric cases. Additionally, spacelike photon trajectories are admitted in this spacetime. Finally, the ADM mass is shown to enter the Lagrangian, with the first law and Smarr formula derived, establishing the existence of thermodynamically stable black holes with positive heat capacity.

Figures

Figures reproduced from arXiv: 2506.13676 by the authors.

Figure 1
Figure 1. FIG. 1: Penrose diagram for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (which is determined by the propagation of light and will be examined in the next section), except for that a point on the line with r = 0 in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: When [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Penrose diagram of the regularized Black hole sourced by a power-law Maxwell field, in the case where [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The behaviour of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The behavior of the heat capacity [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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