Formation of regular black hole from baryonic matter

Ali \"Ovg\"un; Daniil Shatov; Vitalii Vertogradov

REVIEW 6 major objections 4 minor 3 cited by

Formation of regular black hole from baryonic matter

T0 review · 6 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Gravitational collapse of baryonic matter can end in a regular black hole if the center reaches a de Sitter-like equation of state, and the resulting shadow radius grows with the equation-of-state parameter.

desk verdict A useful pair of exact dynamical regular-black-hole interiors, but the printed junction conditions to the Husain exterior have algebraic errors that break the global construction as stated. read the letter →

arxiv 2502.00521 v2 pith:UMUVTADP submitted 2025-02-01 gr-qc

Vitalii Vertogradov , Ali Övgün , Daniil Shatov This is my paper · ORCID

classification gr-qc MSC 83C5783C7583C15 PACS 95.30.Sf04.70.-s97.60.Lf04.50.Kd

keywords blackholeregulargravitationalcollapsesingularityVaidyaspacetimedynamicalspacetimesshadowbaryonicmatter

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 ordinary baryonic collapse need not end in a spacetime singularity. Allowing the equation-of-state parameter $k(r,v)=P/\rho$ to vary in radius and time, the authors derive two exact interior solutions—one a dynamical analogue of a known regular black hole, the other a model with an exponential pressure profile—whose central energy density stays finite because a de Sitter-like core forms at $r=0$. They match these interiors to the Husain exterior, a generalized Vaidya spacetime with constant barotropic equation of state $P=\alpha\rho$, and claim the result is a globally well-defined dynamical regular black hole. If correct, this would answer a common objection to regular black holes: that singularity avoidance requires exotic matter, whereas here the collapsing fluid is baryonic. The paper also derives an observational signature: the shadow radius of the static exterior grows monotonically with $\alpha$ and saturates near a Schwarzschild-like value for large $\alpha$.

What carries the argument

The machinery is the generalized Vaidya metric $\mathrm{d}s^2 = -(1-2M(v,r)/r)\,\mathrm{d}v^2 + 2\epsilon\,\mathrm{d}v\,\mathrm{d}r + r^2\,\mathrm{d}\Omega^2$, with matter fixed by the generalized barotropic equation of state $P = k(r,v)\rho$. Expanding $k(v,r) = \sum_{i=0}^n k_i(v) r^i$ around $r=0$, the condition $\rho(0)$ finite forces $k_0 \le -1$; the paper chooses $k_0 = -1$, which gives a de Sitter core and keeps the Kretschmann scalar (17) finite at the center. This yields the two closed-form mass functions (21) and (28), each containing a free function of advanced time ($k_3(v)$ or $k_1(v)$) and an integration function $M_0(v)$ set to eliminate the central mass divergence. The matching step uses the Husain solution, a generalized Vaidya exterior with $P = \alpha\rho$, and enforces continuity of $M$, $\rho$, and $P$ at $r_b$, producing junction conditions (41)-(42) and (45)-(46). Shadows are computed from the static Husain metric using the photon-sphere and shadow relations (50)-(51).

What would settle it

Compute the Kretschmann scalar for the mass function (21) with $k_0 = -1 + \delta$ for any $\delta > 0$ in the expansion (10); Eq. (9) gives $\rho \sim r^{-2\delta}$, so the curvature diverges at $r=0$ and the regular-black-hole claim fails unless $k_0$ is exactly $-1$. Observationally, the model predicts a shadow radius that grows monotonically with $\alpha$, so a measured shadow below the photon-sphere value predicted for the allowed $\alpha$ range at fixed $M$ and $J$ would contradict this prediction.

Watch

Extended reading notes

Core claim

The paper's central claim is that Eqs. (21) and (28), with the integration function $M_0(v)$ fixed so the mass vanishes at the center, are exact singularity-free interior solutions describing collapse of baryonic matter. Regularity is achieved by requiring $k_0(v) = -1$ in the power-series expansion of $k(r,v)$, which makes the energy density finite at $r=0$ and the Kretschmann scalar finite there, producing a de Sitter core. These interiors are matched to the Husain metric (36) at a radius $r_b$ where the interior pressure-density ratio equals $\alpha$; the matching conditions (41)-(42) and (45)-(46) are asserted to guarantee continuity of the mass function, energy density, and pressure, giving a single dynamical regular black hole spacetime. The paper further claims that the matching interface implies a phase transition in the collapsing fluid that can postpone apparent-horizon formation, and that the static Husain exterior has a photon-sphere radius and shadow radius that increase monotonically with $\alpha$.

Load-bearing premise

The whole construction rests on the assumption that the collapsing fluid's equation-of-state parameter is exactly $-1$ at the center; without a microphysical mechanism that guarantees this value, the central density diverges and the singularity reappears.

Editorial extensions

If this is right

If the regularity condition $k_0 = -1$ is exactly realized during collapse, the Kretschmann scalar at $r=0$ stays finite, so no curvature singularity forms despite the horizon being present.
The matched interior-exterior spacetime is globally defined only up to the radius where the interior pressure grows with $r$ and the dominant energy condition is violated; beyond that radius the Husain exterior takes over, so the model describes a regular core surrounded by collapsing baryonic matter.
Because the shadow radius increases monotonically with $\alpha$ and saturates for large $\alpha$, the model predicts that a black hole with larger equation-of-state parameter casts a larger shadow at fixed mass and $J$.
When $k_3(v)$ decreases to zero, the two apparent horizons merge and disappear; the paper argues that continuation beyond the first zero of $k_3(v)$ is unphysical, so the regular black hole's apparent-horizon structure has a finite lifetime.
The phase transition at the matching interface can delay apparent-horizon formation, allowing electromagnetic radiation emitted during the transition to escape to a distant observer.

Reading between the lines

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

Beyond the paper's own claims, the regularity result is conditional on a phase transition whose microphysics is not specified; if the central equation-of-state parameter only approaches $-1$ asymptotically, the central density still diverges as $r^{-2(1+k_0)}$.
A testable extension would be to compute the shadow in the full dynamical Husain exterior using the slow-evolution method the authors cite, to see whether the monotonic dependence on $\alpha$ survives during collapse rather than only in the static limit.
The matching conditions enforce continuity of $M$, $\rho$, and $P$, but the paper does not demonstrate the full junction conditions; if a thin shell is required at $r_b$, the global solution would have a surface layer and the phase-transition picture would change.
The horizon merge-and-vanish behavior when $k_3(v)$ passes through zero implies a natural endpoint for the regular black hole; this could be compared with observational signatures of black-hole disappearance if the model is applied to a specific collapse scenario.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Quick take: this is a constructive paper with two new exact, time-dependent regular black hole interiors (generalized Dymnikova and a Hagedorn-fluid-like model), and the explicit attempt to match them to the Husain exterior is the most interesting part. The shadow scan is routine but clearly presented. The central idea is not wildly original — the authors are upfront that these generalize known solutions — but the specific mass functions and the proposed junction are new.

The main problem is that the matching section as printed doesn't work. Eq. (41) is missing a factor of k3(v); without it the dimensions don't balance. Eq. (46) should have numerator 2α^2+8α+9, not 9+6α+5α^2+2α^3. I checked the algebra independently; the stress-test note is correct. These aren't cosmetic typos — they are the explicit conditions claimed to produce a smooth match, so the global construction is not established as written. Also, the paper only matches M, ρ, and P, and asserts that is sufficient for smoothness, but a proper junction also requires checking the extrinsic curvature (or specifying a thin shell). That's not done.

There's also a conceptual circularity: the regular center is put in by hand via k0=-1, which forces a de Sitter core. So the title's claim that baryonic collapse leads to a regular black hole is really a statement about a model with an assumed phase transition. The authors do acknowledge the interior is only valid near the center, which helps, but the headline overstates the derivation.

The shadow section uses α=2..6 in Table I even though Eq. (49) restricts α to [-1,1] for energy conditions. That's an inconsistency, and the text in the conclusion says they used the dynamical Husain solution while Section IV explicitly says they used the static one. Both are fixable but need attention.

On the positive side, the interior density and pressure follow correctly from the field equations, the horizon plots are physically instructive, and the authors are honest about limitations. The errors are concrete and likely correctable. I'd send this to a competent referee, expecting major revision. It's not ready as is, but the underlying construction is worth refereeing rather than desk-rejecting.

Recommendation: engage with it, but require the matching algebra to be fixed and the energy-condition range to be respected in the shadow analysis.

Best

Referee Report

6 major / 4 minor

Summary. The paper constructs two exact generalized-Vaidya interior solutions (Model 1 and Model 2) that are intended to describe the collapse of baryonic matter with a radius- and time-dependent barotropic equation-of-state parameter k(v,r). It then matches these interiors to a Husain exterior at a chosen radius r_b, studies the apparent-horizon structure, and computes photon-sphere and shadow radii for the static Husain metric. The central claims are that the matched construction gives a globally well-defined dynamical regular black hole formed from baryonic collapse, that the shadow radius grows monotonically with the exterior equation-of-state parameter α, and that the matching radius represents a physical phase transition that can delay apparent-horizon formation.

Significance. If the global matching were established, the paper would provide explicit analytic examples of dynamical regular black holes with an exterior described by a Husain-type collapsing fluid, which could be useful for studies of horizon formation, energy-condition violation, and shadow phenomenology. The authors are commendably transparent about important limitations: the interior pressure grows with radius, the dominant energy condition fails beyond a critical radius, and the power-series solutions are only valid near the center. However, the current version has algebraic errors in the matching conditions, an unjustified identification of continuity of matter variables with a smooth junction, and a shadow analysis that applies to a static family rather than to the constructed dynamical spacetime. These issues affect the paper's main claim of a self-consistent global regular black hole, so a substantial revision is required.

major comments (6)

[Section II, Eqs. (20)-(21)] Model 1 as derived does not follow from the stated expansion. With k0=-1, k1=k2=0, k3=k3(v), Eq. (12) gives ρ=ρ0 e^{-(2/3)k3 r^3} and hence M=M0-(ρ0/(4k3))e^{-(2/3)k3 r^3}, not the printed Eq. (21). The printed profile corresponds to an equation-of-state parameter with r^3 coefficient k3/6, i.e. to a different definition of k3. Since Eq. (21) is used in the energy density, pressure, matching conditions, and horizon analysis, this inconsistency must be corrected and the derivation re-examined.
[Section III, Eq. (41)] Eq. (41) is missing a factor of k3. Equating the interior density from Eq. (23) with the Husain density from Eq. (40) at r_b^3=6(α+1)/k3 gives ρ1 = M0 k3 r_b^{2α+2} e^{-2(α+1)/3} / [3(1-2α)], not the printed expression without k3. This dimensionally required factor changes the subsequent mass matching and must be restored.
[Section III, Eq. (46)] Eq. (46) is algebraically inconsistent with Eqs. (28) and (45). Using the matching radius r_b=(α+1)/k1, the interior mass at r_b is M0[1-e^{-2(α+1)}(2α^2+6α+5)] and the Husain mass contribution from Eq. (45) is 4M0(α+1)^3 e^{-2(α+1)}/(1-2α). The resulting M1 is M0[1-e^{-2(α+1)}(2α^2+8α+9)/(1-2α)], not the printed numerator 9+6α+5α^2+2α^3. Because Eq. (46) fixes the exterior mass function, the claimed smooth matching fails as printed.
[Section III, Eq. (37) and junction discussion] The statement that matching M, ρ, and P is sufficient for a smooth junction is not justified. For the metric (1), the extrinsic curvature of a surface r=const receives a contribution from Γ^r_{vv}, which depends on ∂_v M as well as on the radial function A=1-2M/r; the paper matches only M, ρ (proportional to M'), and P (proportional to M''), and does not enforce continuity of ∂_v M or explicitly compute the Darmois-Israel conditions [K_ab]=0. Unless this is shown, a surface layer or thin-shell contribution at r=r_b cannot be excluded, so the claim of a single globally well-defined spacetime is not established.
[Section II, Eqs. (13), (20); Section V] The central regularity is assumed rather than derived from baryonic collapse. The inequality k0≤-1 is necessary for finite central density, but the paper sets k0=-1 exactly, thereby imposing a de Sitter core by hand. If k0 > -1, the Kretschmann scalar diverges at the center. The phase-transition discussion in Section V is an interpretive outline, not a microphysical mechanism by which baryonic matter reaches k0=-1. The title claim 'from baryonic matter' is therefore stronger than what is demonstrated.
[Section IV, Table I and Fig. 5; Section VI] The shadow analysis is performed for the static Husain metric (48), not for the matched dynamical spacetime built from Eqs. (21)/(28) and (36), nor even for the time-dependent Husain exterior. The monotonic increase of r_ph and R_sh with α is thus a property of a different static family and does not directly test the collapse model. In addition, Table I varies α from 1 to 6, while Eq. (49) restricts the allowed energy-condition range to α∈[-1,1]; the monotonicity claim should either be restricted to the allowed interval or extended with justification. The last paragraph of Section VI also appears to contradict Section IV by saying the shadow was computed for the 'dynamical Husain solution rather than its static limit.'

minor comments (4)

[Throughout] The exterior metric is referred to as 'Husain' in the abstract and references but 'Hussain' in the Section III heading; the spelling should be unified.
[Eq. (15) and Eq. (14)] Eq. (15) is presented as a way to eliminate M0(v) via the regular-center condition, but M0(v) is later reintroduced as a free time-dependent mass. The logical status of M0(v) should be clarified.
[Figures 1-8] The figures would benefit from axis labels and from explicit mention in the captions of which curves correspond to apparent horizons, NEC horizons, and the matching radius; several captions contain typographical errors such as 'conside' and 'accetion disk.'
[Section V, Eq. (53)] The expression for ̇M appears to use the same symbol for the time derivative of the exponential factor; please define all derivatives with respect to v and state the assumptions on k3(v) and M0(v) needed for the inequality.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: regularity is an imposed boundary condition, and the shadow is a direct computation from the exterior ansatz.

full rationale

The paper's derivation chain is self-contained and does not reduce any prediction to its inputs by construction. The regularity condition is an explicit assumption rather than a hidden fit: Eq. (13) imposes k0 <= -1 to keep rho(0) finite, and the models then set k0 = -1 in Eqs. (20) and (27), with M0 chosen via Eq. (19) to enforce M(v, 0) = 0. The paper itself acknowledges in Sec. V that the interior solutions are only near-center expansions and that ordinary baryonic matter must undergo a phase transition to a different high-energy phase, so the de Sitter core is a stated input, not a derived output. The shadow monotonicity is computed directly from the static Husain metric (48) using Eqs. (50)-(51) with fixed M = 1 and J = 0.5 in Table I; it is a mathematical property of that exterior ansatz, not a fitted prediction from the collapse model. Self-citations such as [94], [95], and [97] are used for comparison and horizon interpretation, but the central algebra (mass functions, matching conditions, shadow equations) is carried out in the paper itself, so these citations are not load-bearing. The matching section's claim that continuity of M, rho, and P suffices for a smooth junction is not demonstrated, and Eq. (46) is algebraically inconsistent with Eqs. (28) and (45); these are correctness defects rather than circular reductions. No equation is defined in terms of the result it purports to predict, and no fitted parameter is renamed as a prediction.

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

The ledger shows the result is a construction: regularity is imposed via k0=-1, the exterior is selected for convenience, and the matching conditions are asserted. The only new predictive content is the shadow scan, which depends on arbitrary J and alpha values, several outside the allowed energy-condition range.

free parameters (7)

k0(v) = -1
Central coefficient in the series expansion of the barotropic EoS parameter; set to -1 in Eq. (20) to enforce a de Sitter core and finite central density. This is the key input that guarantees regularity.
k3(v) (Model 1) = (mu-v)^2 in plots
Arbitrary time-dependent coefficient of the r^3 term; determines the horizon dynamics and collapse timescale. Chosen as (mu-v)^2 with mu=4 in Figures 1-2.
k1(v) (Model 2) = (mu-v)^2 in plots
Arbitrary time-dependent coefficient of the r term; chosen as (mu-v)^2 with mu=4 in Figures 3-4.
rho0(v) = 2/3 lambda k3(v) v (Model 1); 4 lambda k1^3(v) v (Model 2)
Central energy density, chosen to produce convenient plots with lambda=1; no physical derivation from baryonic matter.
alpha (exterior EoS parameter) = alpha=1..6 in Table I
Barotropic parameter of the Husain exterior; the shadow calculation scans it from 1 to 6, despite Eq. (49) restricting it to [-1,1] for energy conditions.
J and M in the static Husain shadow metric = J=0.5, M=1
Parameters of Eq. (48) chosen for the numerical shadow table; no tie to the collapse parameters is established.
M0(v) = 3rho0/(2k3) or rho0/(8k1^3) in the regular cases
Integration function of advanced time; in the regular cases it is tuned to cancel the central mass contribution, enforcing M(r=0)=0.

assumptions (6)

domain assumption The spacetime is of generalized Vaidya form (Eq. 1) with energy-momentum tensor (Eq. 2) throughout collapse.
The entire analysis is restricted to this ansatz; no derivation from a more fundamental action or microphysical matter model is given.
domain assumption The barotropic EoS parameter k(v,r) has a convergent power-series expansion around r=0 (Eq. 10).
Used to integrate Eq. (9) and to build the two explicit models; validity near the center is assumed, and the paper later limits the interior solutions to this region.
ad hoc to paper k0(v) = -1 exactly (Eq. 20), representing a de Sitter core.
This condition is what makes the Kretschmann scalar finite; it is imposed rather than derived from baryonic microphysics.
domain assumption The exterior Husain solution (Eq. 36) with constant barotropic P=alpha rho describes the matter surrounding the regular core.
The paper chooses this exterior because Vaidya matching is impossible; no independent evidence is given that collapsing baryonic matter obeys P=alpha rho outside the core.
domain assumption Matching at r=r_b requires only continuity of mass, energy density, and pressure (Section III).
Full junction conditions, such as continuity of the extrinsic curvature, are not derived; the paper asserts equality of these three functions is sufficient for smooth matching.
ad hoc to paper Shadows can be computed from the static limit (Eq. 48) of the Husain solution even though the collapse is dynamical.
The paper states the dynamical method is invalid for rapid collapse and then uses the static metric without a quantitative justification that the static limit applies to the fully formed black hole.

invented entities (1)

Novel non-singular phase of baryonic matter (de Sitter-like core)
purpose: Supports the regular center and avoids singularity formation during collapse.
Introduced in Section VI as a phase transition of baryonic matter at high compression; no microphysical model, particle content, or independent observable is provided beyond a qualitative electromagnetic transient.

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Cite this review

Pith. "Pith review of Formation of regular black hole from baryonic matter." pith.science (2026). https://pith.science/paper/UMUVTADP

@misc{pith2026250200521,
  author       = {Pith},
  title        = {Pith review of: Formation of regular black hole from baryonic matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMUVTADP}},
  note         = {Machine review of arXiv:2502.00521}
}

abstract

We present a family of exact, singularity-free solutions describing the collapse of baryonic matter characterized by a barotropic equation of state whose coefficient $\alpha(r,v)$ varies in both radius and time. By matching these interior solutions to the Husain exterior metric, we obtain a self-consistent, dynamical spacetime representing a regular black hole. Although the pressure profile of our models grows with radius and eventually violates the dominant energy condition beyond a critical surface-necessitating an external junction to ensure a globally well-defined spacetime-the interior solution remains non-singular throughout the collapse. We further analyze the optical properties of these regular black holes and find that both the photon sphere radius and the corresponding shadow radius increase monotonically as the local equation of state parameter $\alpha$ is raised. Moreover, the matching interface between the interior and exterior metrics naturally suggests a phase transition in the collapsing fluid, which can postpone the formation of an apparent horizon. Taken together, our results not only highlight novel physical features of horizon formation in regular collapse models but also identify characteristic shadow signatures that could be tested by future observations.

Figures

Figures reproduced from arXiv: 2502.00521 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: , the evolution of NEC horizons is depicted. FIG. 3. F(v, r) = 0 for µ = 4, λ = 1. The behavior of apparent horizons conside with apparent horizons in previous model (see 1 for detailed discussion). FIG. 4. Same graph as [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Figure shows the shadow radius and photon sphere radius [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: for the regime in which the energy conditions remain FIG. 7. The plot demonstrates the behavior of apparent horizons under the validity of energy conditions. Here, M0(v) = K3(v) = v. In this case, the energy conditions are satisfied throughout the entire spacetime: the…

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Forward citations

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Reviewed August 9, 2026 · model on record in the stance chip above.

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Also, we adopt the signature−+ ++. II. THEORETICAL FRAMEWORK In this section, we briefly examine the conditions required for the formation of a regular black hole. We derive a general formula and identify the criteria under which a regular black hole can emerge from gravitational collapse, considering an arbitrary EoS. To begin, we analyze a metric that r...

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