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REVIEW 4 major objections 5 minor 2 cited by

Constructing regular black holes from multi-polytropic equations of state

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper argues that multi-polytropic anisotropic fluids can source regular black holes without nonlinear electrodynamics, and that one of the two constructed metrics is a genuine regular black hole with repulsive gravity outside its…

desk verdict The second metric is the real product here, but as printed it is dimensionally inconsistent, so the paper is not verifiable and needs major revision before it can be taken seriously. read the letter →

arxiv 2502.02098 v1 pith:IFGNRIB3 submitted 2025-02-04 gr-qc

Seyed Naseh Sajadi , Supakchai Ponglertsakul , Orlando Luongo This is my paper · ORCID
classification gr-qc MSC 83C5783C1583C75 PACS 04.20.-q04.70.-s
keywords regularblackholesanisotropicfluidTolman-Oppenheimer-Volkoffequationmulti-polytropicofstaterepulsivegravityHaywardmetricdarkenergyremnantsgeodesiccompleteness
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 tries to show that regular black holes—spacetimes with horizons but no central singularity—can be produced by anisotropic fluids described by multi-polytropic equations of state, avoiding the usual appeal to nonlinear electrodynamics. Starting from the Tolman-Oppenheimer-Volkoff equations with radial pressure $P_r=-\rho$ and tangential pressure $P_t=\omega_1\rho+\omega_2\rho^m/\rho_0^{m-1}$, the authors derive two explicit metrics. They claim the second solution is a genuine regular black hole, with finite curvature at the center, geodesic completeness, and asymptotic flatness. They explicitly acknowledge that the first solution, although curvature-regular at the center, is geodesically incomplete and therefore is not a regular black hole. The broader proposal is that such solutions, and their remnants, could behave as dark energy sources through repulsive gravity outside the horizon.

What carries the argument

The load-bearing object is the anisotropic Tolman-Oppenheimer-Volkoff equation together with the multi-polytropic equations of state $P_r=-\rho$ and $P_t=\omega_1\rho+\omega_2\rho^m/\rho_0^{m-1}$. Solving the TOV equation gives the density profile $\rho(r)=\rho_0\big[c_1(\rho_0 r^2(\omega_1+1))^{m-1}-\omega_2\big]^{1/(m-1)}/(\omega_1+1)^{1/(m-1)}$, and fixing the normalization $\omega_2=-\omega_1-1$ makes the central density equal $\rho_0$. This density is fed into Einstein's field equations to obtain the metric functions of the two solutions. To locate repulsive gravity, the paper uses eigenvalues of the Riemann curvature tensor in an orthonormal tetrad, identifying sign changes and extrema of eigenvalues as the onset of repulsive effects. The thermodynamic-topology analysis uses a free-energy vector field and computes winding numbers to classify the solutions.

What would settle it

Integrate Eq. (9) to compute the ADM mass $M=4\pi\int_0^\infty r^2\rho(r)\,dr$ for $m=3/2$ and a range of $\omega_1>1/2$. If the integral converges for any $\omega_1$ other than $2$ and $3$, the paper's stated reason for selecting those two cases is falsified. Alternatively, direct numerical integration of null and timelike geodesics through $r=0$ for the metric (38) would test the claimed geodesic completeness.

Watch

Extended reading notes

Core claim

The central claim is that solutions of the anisotropic TOV equations with $P_r=-\rho$ and a tangential pressure containing a linear term plus a polytropic term generate new regular black hole metrics. For the parameter choice $m=3/2$, $\omega_1=2$, the metric takes the Hayward form, $f(r)=1-2Mr^2/(r^3+2ML^2)$, with a de Sitter core near $r=0$; however, the paper's own geodesic analysis concludes that this first spacetime is geodesically incomplete, so it fails to be a regular black hole. For $m=3/2$, $\omega_1=3$, the second metric is claimed to be geodesically complete and regular, with the same de Sitter core and asymptotic flatness, and the paper reports regions outside the event horizon where curvature eigenvalues signal repulsive gravity, for ranges such as $0<c_1 M^2<3.097$ in the first case and $0\le c_1 M^3\le 1.933$ in the second. These repulsive regions motivate the claim that black hole remnants of these solutions may behave as dark energy sources. The paper also reports stable Cauchy horizons under mass inflation, scalar-field stability for the second solution, and thermodynamic topology with zero winding number for both solutions.

Load-bearing premise

The argument rests on the assertion that only the parameter pairs $(m,\omega_1)=(3/2,2)$ and $(3/2,3)$ give finite black hole mass, plus the imposed normalization $\omega_2=-\omega_1-1$; if other parameter choices also give finite masses, or if that normalization is not forced by the fluid model, the specific metrics and their derived radii lose their stated uniqueness.

Editorial extensions

If this is right

  • Regular black holes can be constructed without nonlinear electrodynamics, using only an anisotropic fluid with a multi-polytropic equation of state.
  • The Hayward metric emerges as a special case of this TOV construction, connecting regular-black-hole phenomenology to stellar-structure physics.
  • For certain parameter ranges, repulsive gravity acts outside the event horizon, so black hole remnants of these solutions could behave as dark energy sources.
  • The second solution is stable under massless scalar perturbations and has a Cauchy horizon that is stable under mass inflation, making it a candidate physical compact object.
  • The first solution, despite finite curvature at the center, is geodesically incomplete and therefore does not qualify as a regular black hole.

Reading between the lines

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

  • Beyond the paper, the same procedure with other polytropic indices $m$ should generate a family of metrics whose geodesic completeness depends on how fast the density decays; this can be checked by re-running the TOV integration over the parameter space.
  • Beyond the paper, the dark-energy interpretation could be sharpened by computing the effective equation of state of the repulsive region and comparing its effects with cosmological constraints on compact dark objects.
  • Beyond the paper, the reported transition from compact star to black hole at $c_1M^2\approx2.018$ suggests these spacetimes may model horizonless ultra-compact objects, whose shadows or tidal deformability could distinguish them from Schwarzschild black holes.
  • Beyond the paper, a numerical survey of geodesic completeness across the full $(m,\omega_1)$ plane would determine whether regular-black-hole status is generic to the construction or confined to the displayed cases.
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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

4 major / 5 minor

Summary. The paper proposes constructing regular black hole (RBH) solutions in general relativity by solving the anisotropic Tolman-Oppenheimer-Volkoff (TOV) equations with multi-polytropic equations of state. Two analytic solutions are presented, corresponding to parameter pairs (m,ω1)=(3/2,2) and (3/2,3). The authors claim the solutions are regular at the center, asymptotically flat, exhibit regions of repulsive gravity outside the horizon, and may describe dark-energy-like remnants. They also study thermodynamic topology and quasinormal modes. The first solution, however, is explicitly shown in Section III.A to be geodesically incomplete and therefore 'not an RBH' by the authors' own criterion. The entire RBH claim therefore rests on the second solution of Section IV.

Significance. The intended contribution is a new way to obtain regular black holes without invoking nonlinear electrodynamics, relying instead on a physically motivated anisotropic fluid with a multi-polytropic equation of state. If the construction were sound, it would provide explicit metrics with a clear fluid interpretation, with potential implications for black hole remnants, dark energy, and observational signatures such as quasinormal modes. The paper also attempts to connect to thermodynamic topology, a presently active area. However, the significance is substantially reduced by the internal inconsistencies detailed below; the central metric is not reproducible from the printed equations, so the claimed results cannot currently be validated.

major comments (4)
  1. [Section III.A] The authors explicitly state that the first solution is geodesically incomplete and hence 'is not an RBH'. This directly contradicts the manuscript's title and abstract, which claim to construct regular black holes. Consequently, the only solution that can support the regular-black-hole claim is the second solution of Section IV. The paper should be reframed accordingly, or the first solution must be shown to satisfy a weaker regularity criterion that is clearly stated.
  2. [Section IV, Eq. (38)] The metric function f(r) in Eq. (38) is dimensionally inconsistent as printed. With [c1]=L^{-3} and [ρ0]=L^{-2}, the prefactor π√2 ρ0/(2 c1 E^{1/4} r) has dimensions L^{-1}, while f(r) must be dimensionless. Moreover, a small-r expansion of the logarithm/arctangent term yields a constant contribution at r=0 rather than the claimed f ~ 1 - 8πρ0 r^2/3 of Eq. (42); the printed expression cannot reproduce the stated de Sitter-like core. The correct prefactor appears to require an additional factor of 2/√ρ0 (or equivalently 1/√ρ0 in the denominator), but even with such a correction the near-origin behavior must be re-derived. Since all subsequent results in Section IV (horizons, repulsive gravity radii, thermodynamic topology, and quasinormal modes) are derived from Eq. (38), the central construction is not independently verifiable from the submitted equations.
  3. [Section II, Eq. (9)] Eq. (9) does not solve the anisotropic TOV equation (5) with Pr=-ρ and Pt=ω1ρ+ω2ρ^m/ρ0^{m-1}. Solving the ODE for ρ(r) gives ρ(r)=ρ0[1 + C r^{2(m-1)(ω1+1)}]^{-1/(m-1)} (with ω2=-(ω1+1)), a decreasing profile that leads to finite total mass for ω1>1/2. The printed Eq. (9) has a positive exponent 1/(m-1), which would make the density grow as r^2 at large r and the total mass divergent. This contradicts both Eq. (11), which gives a decreasing asymptotic density, and the finite mass formulas (17) and (40). The metrics (15) and (38) are constructed from the density via Eq. (4), so this inconsistency breaks the derivation chain.
  4. [Section III, parameter selection] The paper claims that 'not all solutions yield a finite value for BH mass' and uses this to justify the specific choices (m,ω1)=(3/2,2) and (3/2,3). No derivation is provided. In fact, for the corrected density profile, the total mass is finite for every ω1>1/2 with m=3/2, so the two chosen pairs are an unexplained subset of an infinite family. Because the existence and properties of both solutions depend on these particular parameter values, the selection criterion is load-bearing and must be justified from the model, not imposed by hand.
minor comments (5)
  1. [Section IV, Eq. (38)] The notation in Eq. (38) is ambiguous: the term ln(...)^{1/2} could be read as [ln(...)]^{1/2} or as ln(sqrt(...)). The intended interpretation should be clarified, as the small-r and large-r behaviors differ between these readings.
  2. [Section IV.A, Fig. 8] The text refers to 'Fig. (3)b' when discussing the second solution's vector field; this should be 'Fig. (8)b'.
  3. [Section III.A and Section IV.A] The assertion that the Cauchy horizon is stable because 'the denominator of equation (19) of the paper [35] is non-linear' is stated without demonstration. The relevant denominator should be written explicitly for each metric and checked against the cited condition.
  4. [Section II, Eq. (9)] The dimensions of c1 are assigned as L^{-2} in Section III and L^{-3} in Section IV, but no redefinition is mentioned. This should be reconciled with the general formula, since the argument of the power in Eq. (9) must be dimensionless.
  5. [Throughout] There are numerous typographical issues, including 'eq uations' in the title, inconsistent punctuation in equations, and missing references to figure panels. A careful proofreading is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the EoS-to-TOV-to-metric chain is self-contained with no fitted parameter renamed as a prediction; self-citations are background formalism, while Section III.A's retraction of Solution 1 and Eq. (38)'s dimensional issues are non-circular correctness defects.

full rationale

This is a constructive derivation, not a data-fitting exercise, so the main circularity patterns (fitted input called prediction, or prediction equal to a fit by construction) do not apply. The chain is explicit: specify the anisotropic EoS (Eqs. 6-8 with ω=-1, ω̄=0 and f=h^{-1} imposed), solve the TOV conservation equation to obtain ρ(r) (Eq. 9), integrate Einstein's equations to obtain f(r) (Eqs. 15 and 38), and then compute horizons, repulsive-gravity radii, masses, thermodynamic topology, and quasinormal modes from those metrics. Every reported radius (rsec, rR, rrep, rdom, rext) and threshold (c1,ext, M_ext, c1M^2=3.097) is an explicit function of the freely chosen parameters c1, ρ0, ω1, and m; that is normal parametric dependence of a constructed family, not a hidden fit. The Hayward identification (Eq. 35) is an algebraic rewriting of Eq. (15) using the computed mass (Eq. 17); the paper does not present it as a testable prediction, and the matter model (multi-polytropic anisotropic fluid) is new, so this is not a renaming of a known result. Self-citations appear for polytropic EoS precedent ([57-59], with the EoS printed in full) and for the repulsive-gravity eigenvalue diagnostic ([65-69]); the diagnostic is load-bearing for the abstract's 'repulsive gravity' interpretation, but the eigenvalue formulas are derived in Appendix A from Cartan's structure equations, and the diagnostic is a parameter-free, externally applicable framework, so it counts as real evidence rather than a circular chain. The paper itself, in Section III.A, states that Solution 1 'is geodesically incomplete and therefore is not an RBH', which contradicts the abstract's treatment of both solutions as regular black holes; and Section IV asserts geodesic completeness of Eq. (38) with 'it is easy to show' without a proof, while the printed Eq. (38) appears dimensionally inconsistent ([c1]=L^{-3} makes the 1/r prefactor dimensionful), which would undermine the regularity and QNM analysis for Solution 2. These are internal-consistency and reproducibility defects, not equivalence-by-construction steps, so they do not raise the circularity score.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new particles, fields, or forces are introduced; the dark-energy-source and remnant statements are interpretations of the existing geometry, so the graviton-problem check is negative. The load-bearing machinery is otherwise a mix of standard TOV/GR background and ad hoc parameter choices: the matter split with Pr = -rho, the polytropic normalization omega2 = -omega1 - 1, and the unexplained restriction to (m, omega1) = (3/2, 2) and (3/2, 3) for finite mass. The repulsive-gravity criterion is inherited from the authors' own prior framework (refs. 65-67).

free parameters (5)
  • m = 3/2
    Polytropic exponent in the tangential EoS (7). Chosen by hand in Sections III and IV to obtain finite BH mass and simple analytic solutions; no derivation is given for why this value or its alternatives are selected.
  • omega1 = 2 and 3
    Linear EoS coefficient in Eq. (7). The paper picks omega1 = 2 (first solution) and omega1 = 3 (second solution) explicitly to yield finite mass, but Eq. (11) implies finite mass for a continuum of omega1 > 1/2, so this is a hand-picked restriction.
  • omega2 = -omega1 - 1
    Polytropic coefficient in Eq. (7). The normalization omega2 = -omega1 - 1 is imposed below Eq. (10) so that rho(0) = rho0, tying the central density to the coupling; it is an ad hoc constraint rather than a derived relation.
  • c1 = Dimensionful constant; example c1 = 1 used in Fig. 3
    Integration constant in the density profile (9). Sets the mass scale through M = 4 pi sqrt(rho0)/(3 c1) for solution 1 and M = sqrt(2) pi^2 rho0^(5/8)/(4 c1^(3/4)) for solution 2; all horizon, photon-sphere, and repulsion radii scale with it.
  • rho0 = Central density; no numerical value fixed
    Central density entering the EoS (6)-(7) and the de Sitter core Lambda_eff = 8 pi rho0. It is a free scale parameter that sets the mass and all radii in the solutions.
assumptions (5)
  • domain assumption The anisotropic fluid is split into Pr = -rho and tangential pressure Pt = omega1 rho + omega2 rho^m / rho0^(m-1) (Eqs. 6-8).
    This specific decomposition, chosen to enforce T^t_t = T^r_r and hence f = h^(-1), is not derived from any microphysical fluid model; it is the source that produces the density profile (9).
  • ad hoc to paper The polytropic normalization omega2 = -omega1 - 1 is imposed so that rho(r -> 0) = rho0.
    Introduced below Eq. (10) to make the central density equal the declared rho0; without it, the near-origin density (10) would differ and the de Sitter core Lambda_eff = 8 pi rho0 would change.
  • ad hoc to paper Only the pairs (m, omega1) = (3/2, 2) and (3/2, 3) are needed because other choices may not give finite BH mass.
    Stated in Section III without proof. The paper's own Eq. (11) yields convergent total mass for all omega1 > 1/2 at fixed m = 3/2, so the restriction is underived and silently excludes an infinite set of finite-mass solutions.
  • standard math Repulsive gravity is located by sign changes and extrema of orthonormal-frame curvature eigenvalues lambda_i (Appendix A), following the method of refs. [65-67].
    The Cartan-structure eigenvalue machinery is standard geometry, but the identification of specific zeros and extrema as 'repulsion' and 'dominant repulsion' is a convention from the authors' prior work rather than an independent theorem; it underpins rrep and rdom.
  • standard math The ADM mass is the integral M = 4 pi integral r^2 rho dr (Eqs. 17, 40) and the effective radius is R = (M/rho0)^(1/3).
    Standard definition for static spherical symmetry. It is load-bearing because all thresholds in Figs. 2 and 6, including the remnant condition c1,ext = 9.333 rho0, derive from this identification.

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Cite this review

Pith. "Pith review of Constructing regular black holes from multi-polytropic equations of state." pith.science (2026). https://pith.science/paper/IFGNRIB3

@misc{pith2026250202098,
  author       = {Pith},
  title        = {Pith review of: Constructing regular black holes from multi-polytropic equations of state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IFGNRIB3}},
  note         = {Machine review of arXiv:2502.02098}
}
read the original abstract

Regular black holes are imagined as solutions to Einstein's field equations, with no singularities, albeit characterized by the presence of an internal structure. With the intention not to use non-linear electrodynamics, we here propose to obtain new classes of solutions that can also satisfy the Tolman-Oppenheimer-Volkoff (TOV) equations, plus adding a non-zero core. Thus, we present regular black holes as solutions to the TOV equations using multipolytropic equations of state and investigate whether these solutions behave, tuning the underlying free parameters. Our analysis demonstrates that, within specific parameter ranges, repulsive gravity effects may occur in precise regions. Accordingly, black hole remnants are also investigated, showing that, under certain circumstances, they may turn into dark energy sources in view of the corresponding repulsive gravity effects, located outside the horizons. Moreover, quite remarkably, critical sets of parameters imply that solutions may exhibit transitions to regular repulsive relativistic compact objects from black hole behaviors. Finally, we explore the interpretation of these regular black hole solutions in terms of topological thermodynamic defects.

Figures

Figures reproduced from arXiv: 2502.02098 by the authors.

Figure 1
Figure 1. FIG. 1: Left: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The radial coordinates of key features in the met [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Left: Zero points of the vector [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The deflection angle Ω as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Left: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The radial coordinates of key features in the met [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The behavior of metric function ( [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Left: Zero points of the vector [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The deflection angle Ω as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Left: The real part of the quasinormal frequency for [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

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