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Minimally Deformed Regular Bardeen Black Hole Solutions in Rastall Theory

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Two new Bardeen black hole solutions in Rastall gravity preserve regularity while gaining a stable thermodynamic window at the cost of asymptotic flatness and energy conditions.

desk verdict The MGD-Bardeen-Rastall construction is legitimate, but the omitted h(r) and unverified horizon coincidence undercut every thermodynamic claim. read the letter →

arxiv 2412.01158 v1 pith:HAPI4KVP submitted 2024-12-02 gr-qc

M. Sharif , and Malick Sallah This is my paper · ORCID
classification gr-qc PACS 04.50.Kd04.40.Dg04.40.-b
keywords RastallgravityregularBardeenblackholeminimalgeometricdeformationgravitationaldecouplingHawkingtemperaturethermodynamicstabilityenergyconditionsanisotropicfluid
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 aims to build new regular black hole solutions by applying the minimal geometric deformation scheme to the Bardeen black hole within Rastall gravity. The authors split the field equations into a Bardeen seed and an extra source, close the extra source with a linear equation of state, and obtain two deformed metrics. They argue the deformed spacetimes remain regular at the core, although neither is asymptotically flat and both require exotic matter. Thermodynamically, they report acceptable Hawking temperature behavior and stability (positive specific heat and non-negative Hessian trace) in the interval $2.65 < r_H \le 4$, where $r_H$ is the Bardeen horizon radius of Eq. (28). The central claim is that the minimally deformed metrics are regular black holes with a thermodynamically stable window.

What carries the argument

The minimal geometric deformation (MGD) scheme of gravitational decoupling is the engine: only the radial metric component is deformed, $e^{-\eta_2} = \eta_4(r)+\sigma h(r)$, while the temporal component is kept fixed. The deformation function $h(r)$ solves the decoupled extra-source system (23)-(25) under the linear equation of state $\chi^0_0+\lambda\chi^1_1+\tau\chi^2_2=0$, which becomes the traceless condition for model I and the barotropic condition for model II. This splits the two-source Rastall equations into two individually conserved subsystems, so the full metric is a linear combination of the Bardeen seed and the deformation. The paper's thermodynamic conclusions all pass through the Killing horizon radius $r_H$ of the undeformed Bardeen metric given by Eq. (28).

What would settle it

Evaluate $h(r)$ from Eq. (32) for model I and from Eq. (35) for model II at the Bardeen horizon $r_H$ of Eq. (28) for the plotted parameters. If $h(r_H) \neq 0$ for either model, the true horizon of the deformed metric is shifted, and the Hawking temperature, specific heat, and Hessian trace reported in Section 5 are evaluated at the wrong surface, invalidating the claimed stability window.

Watch

Extended reading notes

Core claim

The paper claims that minimally deforming the regular Bardeen metric (27) through $e^{-\eta_2} = \eta_4(r)+\sigma h(r)$, with $h$ fixed by a traceless equation of state (model I: $\chi^0_0+\chi^1_1+2\chi^2_2=0$) or a barotropic equation of state (model II: $\delta\chi^0_0-\chi^1_1=0$), produces two new regular black hole solutions in Rastall theory. The deformation functions are regular at the core, so the Bardeen regularity is preserved. The radial metric component grows without bound at large $r$, however, so both spacetimes fail to be asymptotically flat; and several energy conditions are violated, indicating exotic matter. At the Bardeen horizon $r_H$ from Eq. (28), the Hawking temperature increases as mass decreases, and both the specific heat and the Hessian trace of the Helmholtz free energy are non-negative on $2.65 < r_H \le 4$, which the authors read as thermodynamic stability. The authors explicitly state that the deformed metric is a proper black hole only if its horizon coincides with the Bardeen Killing horizon, and all thermodynamic quantities are evaluated at that surface.

Load-bearing premise

The argument assumes the deformation function vanishes at the Bardeen horizon, $h(r_H)=0$, so the deformed metric's event horizon coincides with the seed Killing horizon $r_H$ of Eq. (28); the paper states this coincidence is necessary for a proper black hole but never verifies it.

Editorial extensions

If this is right

  • If the horizon coincidence holds, both models describe regular black holes in Rastall theory, since the deformation functions are nonsingular at the core.
  • Neither model is asymptotically flat, so the solutions cannot represent isolated astrophysical black holes in the usual sense.
  • Both models violate some energy conditions, indicating that the effective source is exotic.
  • In both models the specific heat is positive for $2.5 \le r_H \le 4$ and the Hessian trace is non-negative for $2.65 < r_H \le 4$, so the authors conclude thermodynamic stability in that common interval.
  • Smaller black holes radiate more, with Hawking temperature increasing as mass decreases, and the decoupling parameter raises the temperature in both models.

Reading between the lines

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

  • The horizon-coincidence condition is the load-bearing link between the seed Bardeen solution and the deformed one; if $h(r_H)$ is not zero, the thermodynamic analysis in Section 5 is evaluated at the wrong surface.
  • The loss of asymptotic flatness suggests these solutions describe localized cores embedded in a non-flat background; a natural next step is to match them to an exterior Rastall vacuum or cosmological region.
  • The same construction could be tested with other regular seeds, such as Hayward or Simpson-Visser geometries, to see whether the stability window depends on the seed or on the deformation scheme.
  • A direct numerical check of $h(r_H)$ for both equation-of-state choices would settle whether the reported stability interval is real or an artifact of evaluating at the Bardeen horizon.
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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 / 3 minor

Summary. The manuscript constructs two families of static, spherically symmetric metrics by applying the minimal geometric deformation (MGD) scheme to the regular Bardeen black hole within Rastall gravity. The seed Bardeen solution is used for the first decoupled subsystem, and two equations of state for the additional source close the second subsystem, giving two models (traceless source and barotropic source). The authors analyze the effective density and pressures, asymptotic flatness, energy conditions, Hawking temperature, specific heat, and a Hessian-matrix stability criterion. They conclude that both models preserve regularity, violate asymptotic flatness, involve exotic matter, and are thermodynamically stable in the interval 2.65 < r_H ≤ 4.

Significance. If the central construction were sound, the paper would provide a modest but useful addition to the literature on regular black holes in modified gravity, and the explicit thermodynamic stability analysis would be a concrete result for future comparisons. The MGD setup is clearly presented, the two subsystems are carefully separated, and the authors are transparent about the costs of their models (violation of asymptotic flatness and energy conditions). However, the claimed significance is not yet established because the paper does not verify the horizon-coincidence condition that it itself identifies as necessary for a proper black hole, and the deformation function h(r) is not given explicitly. These gaps affect the meaning of every subsequent thermodynamic quantity, so the central claim remains unverified rather than demonstrated.

major comments (3)
  1. [Sec. 4, Eq. (29)] The paper states that coincidence of the Killing horizon e^{η1}=0 and the causal horizon e^{-η2}=0 is necessary for Eq. (29) to describe a proper black hole, and all thermodynamic quantities in Section 5 are evaluated at the Bardeen Killing horizon r_H of Eq. (28). For g_tt=B=1-2Mr^2/(r^2+e^2)^{3/2} and g^{rr}=B+σh(r), the causal-horizon condition at r_H requires h(r_H)=0. This condition is never verified, and the explicit h(r) from Eqs. (32) and (35) is not given. The plotted deformation functions in Figures 1 and 5 are positive throughout the displayed range, so the condition h(r_H)=0 is not evident. Consequently, the surface r=r_H may not be a null horizon, and the Hawking-temperature formula (36), which contains |g_tt,r|/√(-g_tt g_rr), diverges at r_H unless h(r_H)=0; the finite T_H curves in Figures 9–10 are then not consequences of the stated metric. This gap affects all of Section 5, including the specific-heat and Hessian stability intervals, and the central claim that the solutions are proper black holes.
  2. [Sec. 4.1, Figure 1; Sec. 4.2, Figure 5] Regularity of the deformed metric is inferred from plots of h(r) that begin at r=0.5, not from the limiting behavior as r→0. A function that looks finite on [0.5,4] can diverge at the origin, so the statement that the extended model is regular at the core is not supported by the evidence shown. This is load-bearing because the word 'regular' in the title and abstract is the main claimed improvement over the singular Bardeen solution; the authors should provide the explicit h(r) and its small-r expansion, or state and prove the regularity condition.
  3. [Sec. 4.1, parameter choices; Sec. 5, Figs. 9–14] The text says that for all plots M=1 and e=1, but for these values the Bardeen factor 1-2Mr^2/(r^2+e^2)^{3/2} never vanishes (its minimum is about 0.23), so there is no horizon to evaluate. Figures 9–14 instead use M or r_H in the range 2.6–4, which corresponds to different M values. The parameter sets used for the thermodynamic claims are therefore not the same as those used for the matter and energy-condition plots, and the reader cannot reproduce the stability intervals without additional information. The authors should specify the exact (M,e) values used in each figure.
minor comments (3)
  1. [Sec. 1, first paragraph] The sentence 'Bardeen [14] determined that the radius of the photon sphere for a Schwarzschild black hole is 3M' is historically inaccurate; the photon-sphere radius 3M is due to Synge, whose work is already cited as reference [13] immediately before. Please correct the attribution.
  2. [Sec. 4, Eq. (28)] The expression for r_H in Eq. (28) contains multiple cube roots and is not obviously real for all values of e and M; the domain of validity of this expression should be stated explicitly.
  3. [Sec. 5.3, Eq. (39)] The Hessian matrix is written as a 2×2 matrix of derivatives of the free energy with respect to TH and V, but the paper does not define how E is computed for these solutions; a brief definition of E would make the stability criterion reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the MGD construction derives the deformed metric and its thermodynamics from the seed Bardeen solution and an EoS constraint; no fitted parameter is relabeled as a prediction, and self-citations are contextual.

full rationale

The paper's derivation chain is constructive rather than circular. Equation (27) supplies the seed Bardeen metric; the deformation h(r) is defined as the solution of the decoupled source equations (23)-(25) closed by an equation-of-state constraint, Eq. (31) for Model I and Eq. (34) for Model II. The minimally deformed line element (29) is then a direct linear combination of the seed metric and the solved deformation. The effective density and pressures (33), energy-condition plots, Hawking temperature (36), specific heat (37) with entropy (38), and Hessian trace (40) are all algebraic or graphical consequences of that constructed geometry; none is fitted to an external data set or to the stability claim being made. The paper does cite earlier MGD-in-Rastall works that include the authors' own papers (e.g., [47]), but the relevant field equations, decoupling, and horizon conditions are re-derived in the text, so those citations are contextual and not load-bearing. The main weakness is a correctness gap rather than a circular step: the paper states that coincidence of the Killing horizon e^{η1}=0 and causal horizon e^{-η2}=0 is necessary for a proper black hole, then evaluates all thermodynamic quantities at the seed Bardeen horizon r_H of Eq. (28), but it never verifies h(r_H)=0 for the explicit solutions, and the closed forms of h(r) are omitted. If h(r_H) were nonzero, the horizon and therefore the thermodynamic conclusions would be displaced. This is an unverified assumption in the black-hole interpretation, but it does not make any equation equivalent to its own input, so it should be scored as a correctness risk, not as circularity.

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

The central construction rests on Rastall's modified field equations, the MGD decomposition, an ad hoc equation of state, and unverified assumptions about the horizon and thermodynamic formulas. The paper introduces one invented entity, the extra matter source, without independent evidence. Several numerical parameters are chosen by hand, and one parameter, delta, is left unspecified.

free parameters (6)
  • Rastall parameter xi = 0.2, 0.6
    Nonminimal coupling parameter of Rastall theory; chosen by hand for all plots, not derived or measured.
  • Decoupling parameter sigma = 0.2, 0.4, 0.6, 0.8, 1.0
    Controls the strength of the additional source; chosen by hand for plots.
  • Magnetic charge e = 1.0
    Bardeen magnetic charge, fixed to 1 for all plots.
  • Mass M = 1 for geometry plots; 2.6-4 for thermodynamic plots
    Mass is set to 1 for the solution and energy condition plots, but the thermodynamic plots vary M without a clear restatement of the parameter set.
  • Barotropic parameter delta = not stated
    Nonnegative parameter in the barotropic equation of state (34); Model II and Figure 5 depend on it, but no value is given in the text.
  • Equation-of-state constants lambda, tau = lambda=1, tau=2 (Model I); lambda=-1/delta, tau=0 (Model II)
    Constants in the linear equation of state (30), chosen to close the system rather than derived from physics.
assumptions (6)
  • domain assumption Rastall field equations with nonminimal coupling (Eq. 2) are the correct gravitational theory.
    The entire paper is built inside Rastall gravity; this is assumed without independent experimental support.
  • domain assumption The MGD split into two individually conserved subsystems is valid, implying no energy exchange between the seed source and the extra source chi.
    The paper asserts conservation of each subsystem (Eqs. 22 and 26) as a necessary condition for applying MGD.
  • domain assumption The regular Bardeen metric (Eq. 27) is a valid seed solution of the first subsystem in Rastall theory.
    The first subsystem is solved by the Bardeen metric components; this is an input from the literature.
  • ad hoc to paper The linear equation of state (30) is a legitimate physical constraint that closes the second subsystem.
    The traceless and barotropic equations of state are chosen for convenience to obtain solutions, not derived from microphysics.
  • ad hoc to paper The deformed metric has its event horizon at the Bardeen Killing horizon r_H, requiring h(r_H)=0.
    The paper states the horizon coincidence is necessary but never verifies it; the thermodynamic analysis assumes it.
  • domain assumption The area-law entropy S=pi r_H^2 and the Hawking temperature formula (36) apply to the deformed metrics.
    Standard black hole thermodynamics are adopted without discussion of whether they hold for non-asymptotically flat Rastall spacetimes.
invented entities (1)
  • Additional matter source chi_{mu nu}
    purpose: Induces the geometric deformation and anisotropy needed to extend the Bardeen seed solution via MGD.
    The extra source is introduced as a mathematical fixture, constrained only by an ad hoc equation of state, with no microphysical origin or observable signature outside the constructed metric.

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Pith. "Pith review of Minimally Deformed Regular Bardeen Black Hole Solutions in Rastall Theory." pith.science (2026). https://pith.science/paper/HAPI4KVP

@misc{pith2026241201158,
  author       = {Pith},
  title        = {Pith review of: Minimally Deformed Regular Bardeen Black Hole Solutions in Rastall Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HAPI4KVP}},
  note         = {Machine review of arXiv:2412.01158}
}
read the original abstract

In this study, we utilize the minimal geometric deformation technique of gravitational decoupling to extend the regular Bardeen black hole, leading to the derivation of new black hole solutions within the framework of Rastall theory. By decoupling the field equations associated with an extended matter source into two subsystems, we address the first subsystem using the metric components of the regular Bardeen black hole. The second subsystem, incorporating the effects of the additional source, is solved through a constraint imposed by a linear equation of state. By linearly combining the solutions of these subsystems, we obtain two extended models. We then explore the distinct physical properties of these models for specific values of the Rastall and decoupling parameters. Our investigations encompass effective thermodynamic variables such as density and anisotropic pressure, asymptotic flatness, energy conditions, and thermodynamic properties including Hawking temperature, entropy, and specific heat. The results reveal that both models violate asymptotic flatness of the resulting spacetimes. The violation of energy conditions indicate the presence of exotic matter, for both models. Nonetheless, the energy density, radial pressure, as well as the Hawking temperature exhibit acceptable behavior, while the specific heat and Hessian matrix suggest thermodynamic stability.

Figures

Figures reproduced from arXiv: 2412.01158 by the authors.

Figure 1
Figure 1. Graphs of deformation function h(r) against r with ξ = 0.2 (left), 0.6 (right) for model I. It is fundamental to investigate whether the extended model preserves the regularity of the Bardeen black hole. It is observed that the inclusion of the deformation function h(r) creates the divergence of the extended mod￾els (29), from the standard Bardeen black hole (27). This implies that the extended model is regular if t… view at source ↗
Figure 2
Figure 2. Graph of e −η2 against r with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model I. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.00 0.05 0.10 0.15 r Ž Ρ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.20 -0.15 -0.10 -0.05 0.00 r Ž Pr 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.15 -0.10 -0.05 0.00 r Ž Pt [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Graphs of effective parameters against r with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model I. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Graphs of energy conditions against r with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model I. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Graphs of deformation function h(r) against r with ξ = 0.2 (red), 0.6 (blue) for model II. where δ denotes a nonnegative parameter which incorporates information about the temperature. It can be observed that this equation is a special case of the linear EoS (30) with …
Figure 6
Figure 6. Figure 6: Graph of e −η2 against r with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model II. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.00 0.05 0.10 0.15 0.20 r Ž Ρ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.15 -0.10 -0.05 0.00 r Ž Pr…
Figure 7
Figure 7. Figure 7: Graphs of effective parameters against r with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model II. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Graphs of effective parameters against r with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model II. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Graph of TH against M with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model I. 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.008 0.010 0.012 0.014 0.016 0.018 0.020 M TH [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Graph of TH against M with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model II. expression for the Hawking temperature is TH = k 2π = 1 4π [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Graph of C against rH with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model I. 5.2 Specific Heat The specific heat is a critical thermodynamic parameter for evaluating the thermal stability of black holes, …
Figure 12
Figure 12. Figure 12: Graph of C against rH with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model II. observe a direct variation of the specific heat to the Rastall parameter. The decoupling parameter varies inversely to the spe…
Figure 13
Figure 13. Figure 13: Graph of T r(H) against rH with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model I. We therefore determine the stability by using the trace of the Hessian matrix given by T r(H) = H11 + H22. (40) Here, the …
Figure 14
Figure 14. Figure 14: Graph of T r(H) against rH with ξ = 0.2 (solid), 0.6 (dashed), σ = 0.2 (orange), 0.4 (gray), 0.6 (cyan), 0.8 (magenta) and 1 (black) for model II. 6 Conclusions In this study, we have identified minimally decoupled regular Bardeen black hole solutions within the frame…

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Reviewed August 12, 2026 · model on record in the stance chip above.