Gravitational Collapse and Formation of Regular Black Holes: Dymnikova, Hayward, and Beyond

Vitalii Vertogradov

REVIEW 3 major objections 6 minor 2 cited by

Gravitational Collapse and Formation of Regular Black Holes: Dymnikova, Hayward, and Beyond

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Baryonic collapse can end in any regular black hole solution, with a radiation signal fixed by the initial and final matter.

desk verdict Useful reverse-engineering formula, but the formation claim is an identity, not a mechanism. read the letter →

arxiv 2504.19292 v1 pith:DIGRXQ22 submitted 2025-04-27 gr-qc

Vitalii Vertogradov This is my paper · ORCID

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

keywords blackholegravitationalcollapseregulardeSittercorebaryonicmatterradiationemissionVaidyaspacetimeDymnikovametric

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 addresses a long-standing objection to regular black holes: ordinary baryonic matter cannot by itself become the exotic de Sitter core these solutions require. The author models stellar collapse as a two-fluid process in which baryon-like matter with equation of state $P=\alpha\rho$ converts into a new type of matter while emitting electromagnetic radiation. Solving the coupled continuity equations yields a formula for the radiation density in terms of the initial and final matter, and shows that the Dymnikova and Hayward regular black hole metrics are recovered as special cases. The general claim is that any solution of Einstein's equations can be represented as the endpoint of such a transition, provided the conversion efficiency is chosen appropriately. If the emitted radiation is observable, different regular black hole models would leave distinguishable collapse signatures.

What carries the argument

The load-bearing device is the outgoing Vaidya-type metric $ds^2=-f(v,r)dv^2+2dvdr+r^2d\Omega^2$ with mass function $M(v,r)$, in which the Einstein equations separate into flux, density, and pressure components. The collapse is modeled as a two-fluid system where baryon-like matter ($P=\alpha\rho$) converts at a rate $\beta(v,r)$ into radiation ($P=\rho/3$); the continuity equations are split so that total energy-momentum is conserved while the components exchange energy. The argument turns on the identity obtained by requiring the total density $\rho_r+\rho_b$ to equal a prescribed target density $\rho_{\rm new}$: this fixes $\beta$ and yields $\rho_r=\rho_{0r}(\alpha\rho_{\rm new}-P_{\rm new})$. That identity converts a dynamical collapse problem into an algebraic statement about any chosen Einstein solution.

What would settle it

Compute the predicted radiation density from Eq. (59) for a concrete target, such as the Hayward metric, at radii where $P_{\rm new}\ge\alpha\rho_{\rm new}$ using any $\alpha$ in $(1/3,1)$; a negative $\rho_r$ would show the transition cannot be realized globally without extra assumptions. More generally, detecting collapse radiation whose energy density disagrees with $\rho_{0r}(\alpha\rho_{\rm new}-P_{\rm new})$ for the claimed final metric would falsify the mechanism.

Watch

Extended reading notes

Core claim

The central discovery is that the transition from baryon-like matter to the matter supporting a regular black hole is not arbitrary: the two-fluid conversion rate $\beta(v,r)$ is fixed by the target solution's density and pressure through Eq. (56), and the radiation density released is $\rho_r=\rho_{0r}(\alpha\rho_{\rm new}-P_{\rm new})$ with $\rho_{0r}=1/(\alpha-\tfrac13)$ (Eqs. (59) and (66)). With this choice, the sum of the radiation density and the remaining baryonic density equals exactly the target density $\rho_{\rm new}$, so the end state is the desired Einstein solution. Applied to the Dymnikova and Hayward metrics, the formulas reproduce their known mass functions; applied to any other solution, they predict a definite radiation density once the baryonic equation-of-state parameter $\alpha>1/3$ is specified. The paper therefore claims that any black hole whose metric solves Einstein's equations can be formed by this collapse mechanism, and that the accompanying radiation carries information about which final state was produced.

Load-bearing premise

The entire construction assumes the conversion efficiency of baryonic matter into new matter can be freely chosen at every point and time, with no independent physics limiting it; if no real process can produce that rate, the result is a formal identity rather than a physical prediction.

Editorial extensions

If this is right

The Dymnikova and Hayward regular black hole metrics become realizable endpoints of baryonic collapse with radiation emission, so the exotic de Sitter core need not be assumed from the start.
The radiation density formula $\rho_r=\rho_{0r}(\alpha\rho_{\rm new}-P_{\rm new})$ gives a distinct predicted signal for each target metric, so observations of collapse could in principle rule out some regular black hole models.
The viability condition $\alpha>1/3$ restricts the baryonic equation of state; $\alpha=1/3$ is degenerate because it describes a radiation-to-radiation transition.
Because the method works for an arbitrary Einstein solution, it extends beyond de Sitter-core models and supplies a common formation mechanism for spherical black holes generally.
Generalizing the baryon equation of state to $\bar P=\omega\rho$ through $\alpha=(3\omega+1)/2$ connects the result to realistic baryonic matter and to dynamic Kiselev-type solutions.

Reading between the lines

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

The construction is a representation result rather than a dynamical prediction: $\beta$ is chosen from the target solution, so the paper does not establish which microphysical process realizes the conversion. Without such a process, the derived relation is a consistency condition on the endpoint, not a proof that collapse will reach it.
A natural next step would be to impose a realistic $\beta$ from particle-physics phase transitions and compute the resulting radiation luminosity; the formula would then predict which regular metric emerges and how bright its collapse signal is.
Positivity of radiation density restricts the class of target solutions: wherever $P_{\rm new}>\alpha\rho_{\rm new}$, Eq. (59) gives negative $\rho_r$ unless the transition is confined to the core, so the universality claim likely needs a locality cutoff.
The identity can also be run in reverse: from observed collapse radiation and a known baryonic equation of state, one could reconstruct the target density and pressure and identify the black hole metric responsible.

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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,

The short version: this paper contains one genuinely new algebraic result, but the headline claim—that this explains how baryonic collapse forms Dymnikova, Hayward, or any regular black hole—does not stand up. The construction is reverse engineered: β is defined from the target density and pressure in Eq (56), and ρ0r is normalized in Eq (66) so that the total density is the target. No independent physics fixes β. The "formation" is an identity, not a derivation.

What the paper does well: the two-fluid machinery is set up cleanly, and Eq (59), ρ_r = ρ0r(αρ_new − P_new), is a compact and correct relation once you accept the definition of β. That is new relative to the earlier dust and quark-matter papers, and it could be useful as a consistency check for people doing reverse engineering of regular black holes. The algebraic steps in the Hayward section check out aside from the typos noted below.

The soft spots are not minor. First, the physical interpretation fails at the component level: ρ_b goes negative at the center for the Dymnikova example (Eq 34 gives −(8/3)ρ0r), and ρ_r can go negative at large r in the Hayward case. Negative baryonic density is not a transition; it indicates that the split is fictitious. Second, there are concrete errors: Eq (36) has 6/r0^2 where Eq (13) and the metric-derived density give 3/r0^2, so ρ0r is off by a factor of two; the promised Bardeen example never appears; and the exponents in Eqs (45) and (48) are misprinted (they should be r^{2α+1}, not r^{1−2α}). None of these are fatal to Eq (59), but they undermine the paper's reliability.

The paper would be more honest if reframed as "any static, spherically symmetric solution can be written as the endpoint of a two-fluid transition with a carefully chosen efficiency." That is a kinematic statement, not a collapse mechanism. Read that way, it is a small but useful tool. Read as written, it overclaims. The heavy self-citation is not itself a problem—the cited papers are the direct predecessors.

Who gets value: specialists in regular black holes who want a compact way to compute what radiation density a given target metric would require. I would not cite it in its current form, and I would be skeptical of any observational claims until magnitudes and timescales appear.

For peer review: send it out if you want a referee to force the reframing, but I'd expect a recommendation of major revision at best. The algebraic core is checkable and not without interest; the interpretation needs to be cut down.

Referee Report

3 major / 6 minor

Summary. The paper studies a two-fluid model in which baryon-like matter (P = αρ) converts into radiation with a position-dependent transition rate β(v,r). The total density is the sum of the two components, and the authors choose β so that the total stress-energy matches known regular black hole solutions. Specific constructions are given for the Dymnikova and Hayward metrics in §III, and §IV generalizes the construction to arbitrary Einstein solutions. The paper concludes that gravitational collapse of baryonic matter can form Dymnikova, Hayward, and other regular black holes, and that the accompanying radiation density provides an observational discriminant between models.

Significance. The algebraic core of the paper is explicit and self-contained: for any target (ρ_new, P_new) satisfying the continuity equation, the two-fluid decomposition with ρ0r = 1/(α - 1/3) reproduces the target's density and pressure. This is a clean inverse-problem observation. However, because β is defined by Eq. (56) in terms of the target solution and ρ0r is fixed by Eq. (66), the construction is a reverse-engineering identity rather than a physical formation mechanism. The examples also produce negative component densities, a factor-of-two normalization error appears in the Dymnikova section, and the Hayward section contains several notation slips. The claimed collapse mechanism and observable radiation signature are therefore not established by the presented analysis.

major comments (3)

[§IV, Eqs. (55)-(66)] The central construction is a definitional identity rather than a derivation of formation. Eq. (56) defines β(v,r) in terms of the target solution's ρ_new and P_new, and Eq. (66) fixes ρ0r so that Eq. (65) reduces exactly to ρ_new. Since β and ρ0r are free functions with no dynamical or microphysical input, the method represents any Einstein solution — including singular ones — as the endpoint of the two-fluid transition. The conclusion in §V that the process 'may lead to any other black hole' therefore does not establish a formation mechanism; it only exhibits an algebraic decomposition. To support the paper's physical claim one would need an independent equation for β (for example, from a phase-transition model) and an evolution from initial data, rather than a target-driven definition.
[§IV, Eq. (64); §III.A, Eq. (34); §III.B, Eq. (43)] The component densities are not physically viable. Eq. (64) gives ρ_b = -ρ0r(ρ_new/3 - P_new), which is negative in any de Sitter-like core where P_new ≈ -ρ_new. For the Dymnikova example Eq. (34) yields ρ_b(0) = -(8/3)ρ0r < 0, and for the Hayward example with α = 1/2 Eq. (43) gives ρ_r = ρ0r(αξ - η) → -9ξ < 0 as r → ∞. Thus the 'baryonic' and 'radiation' components violate the weak energy condition at the component level, so they cannot be interpreted as ordinary baryonic matter and electromagnetic radiation. The claimed detectable radiation signature is consequently not established.
[§III.A, Eqs. (13)-(14), (36)-(37)] The normalization of the Dymnikova density is inconsistent by a factor of two. Equations (13)-(14) define ε0 = 3/r0², but Eq. (36) states ρ_Dymnikova = 6/r0² e^{-r³/(2Mr0²)}. Direct computation from Eq. (16) gives ρ = 3/r0² e^{-r³/(2Mr0²)}. As a result, Eq. (37) sets ρ0r a factor of two too large. Although the metric in Eq. (39) is unaffected, the expression for the radiation density in Eq. (28) and its normalization inherit this error, so the quantitative predictions of the Dymnikova example are not reliable.

minor comments (6)

[§I and §III.A] The notation in Eq. (13) is inconsistent: the density is written with ε, while the de Sitter condition in Eq. (14) uses ε0; this should be unified.
[§III.B, Eq. (49)] Equation (49) states 'α = 1/3, b = -1', but the coefficient comparison actually fixes a = 1/3 and b = -1, leaving the barotropic parameter α free. As printed, the equation contradicts the later requirement α > 1/3 used in the same section.
[§III.B, Eq. (52)] Equation (52) reads 'ρ = ρ0r(α - 1/3 η)', which is dimensionally inconsistent; the intended expression is ρ = ρ0r(α - 1/3)ξ, consistent with Eqs. (43), (51), and (53).
[§IV, Eq. (24)] The statement that C(v) is set to zero because of the interaction between matter and radiation is not justified; the homogeneous solution is not removed by interaction. A cleaner justification is that C(v) must vanish for the total density to be finite at r → 0 when α > 0.
[§III, Eqs. (26) and (40)] The condition β' < 0, introduced as the requirement that denser matter transitions faster, is asserted but never verified for the explicit choices of β in the Dymnikova and Hayward examples; a check (or a restriction on the parameter ranges) should be provided.
[§III.B, Eq. (48)] The exponent in Eq. (48) is written as r^{1-2α}, but the substitution and the subsequent comparison with Eq. (47) require r^{1+2α}; this appears to be a typographical slip.

Circularity Check

3 steps flagged · score 8.0 of 10

Central result is an identity: β and ρ0r are chosen to reproduce the target metric, so 'formation' is reverse-engineering.

self definitional [Section IV, Eqs. (54)-(59), (65)-(66)]
"Our goal is to describe the transition of baryonic matter into a new type of matter ρnew ... To determine β, we solve the integral equation: ρnew = ρr + ρb. (55) ... The key result is: ρr = ρ0r (αρnew − Pnew). (59) ... The total energy density of the new matter is: ρr + ρb = ρ0r (α − 1/3)ρnew. (65) ... ρ0r = 1/(α − 1/3). (66)"

The efficiency β is not an independently derived physical quantity: Eq. (56) is obtained by inverting the requirement ρnew = ρr + ρb, i.e. beta is solved from the target density and pressure. Equation (66) then fixes ρ0r so that Eq. (65) becomes ρr + ρb = ρnew identically. Consequently the 'formation' of an arbitrary Einstein solution is a tautology: the two-fluid system is built to have the target matter content. The only nontrivial output, Eq. (59), is a repackaging of ρnew and Pnew into a radiation part, not a prediction from initial conditions or microphysics.
fitted input called prediction [Section III A, Eqs. (26)-(39)]
"To achieve this, consider the function β in the form β(v,r)=8/3 − 3r^3/(2M(v)r0(v)^2)(1 + 3/(2+2α) − 3r^3/(2M(v)r0^2(v))). (26) ... we deduce the definition of ρ0r(v) as ρ0r ≡ 6/((2α−2/3)r0^2). (37) ... With this definition, solving Einstein's equations yields the dynamic Dymnikova black hole metric"

β is chosen ad hoc, Eq. (26), precisely so that the integral (27)-(33) evaluates to the exponential factors of the Dymnikova profile, and ρ0r is then defined, Eq. (37), to reproduce the Dymnikova central density 6/r0^2. Thus the Dymnikova metric (39) is not a consequence of a collapse mechanism; it was inserted as the target and the free functions were adjusted to match it. Calling this formation is a fitted-input prediction.

1 more flagged steps

fitted input called prediction [Section III B, Eqs. (40)-(53)]
"we choose the function β in the form: ... (40) ... Using the definition of η from (41) and setting ρ0r = 1/(α−1/3), we find: ρ = 12M(v)^2L(v)^2/(r^3+2M(v)L^2(v))^2, which corresponds precisely to the energy density associated with the Hayward solution."

The same pattern: β is engineered so that (42) gives ln|αξ−η|, and ρ0r is set to the value 1/(α−1/3) that makes the total density equal to the Hayward density (18). The Hayward solution is recovered only because β and ρ0r were fit to it. The accompanying baryonic density (51) is then negative in the de Sitter-like core, underscoring that these components are mathematical placeholders rather than physically motivated fluids.

full rationale

Section IV constructs a two-fluid split of an arbitrary target solution: after setting C(v)=0, Eq. (56) determines β from (ρ_new,P_new), and Eq. (66) fixes ρ0r. Equations (59), (64), and (65) then guarantee ρr+ρb=ρ_new identically. The Dymnikova and Hayward sections are special cases: the chosen β and ρ0r are fitted to those metrics. Thus the central claim—collapse of baryonic matter can form any Einstein solution—is true by construction and is not an independent prediction. The paper does not provide a microscopic or dynamical derivation of β; it only solves the algebraic inverse problem. There is no need to invoke self-citation circularity: the reduction is internal. The physical issues (negative ρb in the core, negative ρr in the Hayward tail) reinforce but are not the basis of this verdict.

Assumptions & free parameters 6 free parameters · 6 assumptions · 2 invented entities

The central construction rests on free functions alpha, rho0r, M, r0, L, and beta. beta is reverse-engineered from the target solution, and the normalization is set to match it, so the ledger is dominated by ad hoc choices.

free parameters (6)

alpha = unspecified, alpha > 1/3
Equation-of-state parameter for baryon-like matter P = alpha rho; restricted to alpha > 1/3 to make rho0r positive and satisfy the weak energy condition.
rho0r(v) = 1/(alpha - 1/3) or 6/((2 alpha - 2/3) r0^2)
Integration and normalization function chosen to match the target solution's density, Eq (37) and Eq (66).
M(v) = free function
Mass function in the Vaidya-like metric; no evolution equation is supplied.
r0(v) = free function
De Sitter scale in the Dymnikova case; chosen to reproduce that solution.
L(v) = free function
Hayward length scale; chosen to reproduce the Hayward solution.
beta(v,r) = defined by Eq (56)
Conversion efficiency is reverse-engineered from the target rho_new and P_new; no microphysical derivation is provided.

assumptions (6)

domain assumption Baryonic matter obeys P = alpha rho with alpha > 1/3.
Used throughout Sections III and IV; physical baryonic matter is more complex, and the paper maps to omega only afterwards.
domain assumption Total energy-momentum is conserved, but baryonic and radiation components exchange energy with source terms +/- beta rho_r.
The continuity system (22) defines the model.
ad hoc to paper C(v) is set to zero, removing the non-interacting baryonic solution.
Eq (24); this discards a free integration function without physical justification.
domain assumption beta' < 0 so that denser matter transitions faster.
Imposed after Eq (22); never verified for the specific Dymnikova or Hayward beta functions.
domain assumption The target solutions (Dymnikova, Hayward) are valid solutions of Einstein's equations.
Taken from [5,12]; the construction is designed to reproduce them.
standard math New matter satisfies the usual continuity equation (58).
Used to evaluate integrals in Section IV.

invented entities (2)

Baryon-to-exotic transition efficiency beta(v,r)
purpose: Controls the conversion of baryonic matter into radiation and new exotic matter in Eq (22).
No microphysical mechanism is given; beta is chosen to reproduce each target solution, so it has no independent falsifiable handle.
New matter supporting de Sitter core (unspecified)
purpose: Forms the regular core after transition; its density and pressure rho_new and P_new are inputs.
The paper states it does not determine the nature of this matter; it is an input from existing regular black hole solutions.

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Cite this review

Pith. "Pith review of Gravitational Collapse and Formation of Regular Black Holes: Dymnikova, Hayward, and Beyond." pith.science (2026). https://pith.science/paper/DIGRXQ22

@misc{pith2026250419292,
  author       = {Pith},
  title        = {Pith review of: Gravitational Collapse and Formation of Regular Black Holes: Dymnikova, Hayward, and Beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DIGRXQ22}},
  note         = {Machine review of arXiv:2504.19292}
}

read the original abstract

The gravitational collapse of a star can lead to the formation of a regular black hole. However, a key factor in this process is the transition of ordinary baryonic matter into a substance that forms the de Sitter core. However, the formation of de Sitter core during gravitational collapse remains an open question, particularly since ordinary baryonic matter does not naturally transition into the exotic matter required to form a de Sitter core. In this paper, we investigate the gravitational collapse of baryonic matter and its potential to form well-known regular black hole solutions, such as those proposed by Dymnikova and Hayward. We model the collapse process as a transition of baryonic matter into a new type of matter, accompanied by the release of energy in the form of electromagnetic radiation. Using a generalized dynamical framework, we derive the energy density of the emitted radiation as a function of both the properties of the initial baryonic matter and the resulting exotic matter. Our findings demonstrate that the gravitational collapse can lead to the formation of various types of regular black holes, providing insights into the physical mechanisms underlying their creation. The detectable radiation signature offers a potential observational test for distinguishing between different black hole models.

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Forward citations

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Reference graph

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(61) Substituting β into the integrand of ρb, we obtain: r1+2α βe ∫ β − 8 3 r dr = r1+2α [ 2α 3 ρnew − ( 8 3 + 2α ) Pnew − P ′ newr ]

for ρ′ newr, we ﬁnd: g′(r) = r1+2α [2αaρ new + ((2α + 2)b − 2a) Pnew + bP ′ newr] . (61) Substituting β into the integrand of ρb, we obtain: r1+2α βe ∫ β − 8 3 r dr = r1+2α [ 2α 3 ρnew − ( 8 3 + 2α ) Pnew − P ′ newr ] . (62) Comparing coeﬃcients with ( 61), we ﬁnd: a = 1 3 , b...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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Speciﬁcally, the f ollowing condition must hold: lim r→0 ρ = ρ0

The energy density ρ must remain ﬁnite throughout spacetime. Speciﬁcally, the f ollowing condition must hold: lim r→0 ρ = ρ0. (9)

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The matter pressure must be ﬁnite throughout spacetime, s atisfying: lim r→0 P = P0. (10)

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The article is organized as follows

the properties of black hole shadow formation in dynamical case has been investigated [ 31]. The article is organized as follows. In Sec. 2, we discuss the general characteristics of regular black holes. In Sec. 3, we describe a model of gravitational c ollapse in which baryonic matter transitions into radiation and forms a new type of matter. In Sec. 4, ...

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The mass function must vanish at the center, i.e., lim r→0 M(v, r ) = 0 . (11) 5 A few remarks are in order regarding these three conditions. First, we say that a regular black hole possesses a de Sitter core if the following condition is satisﬁed: lim r→0 P = − lim r→0 ρ. (12) Second, there is no need to impose additional constraints on the ﬁrst and seco...

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From the energy density corresponding to Dymnikova’s solution, ρDymnikova = 6 r2 0 e − r3 2M r2 0 , (36) we deduce the deﬁnition of ρ0r(v) as ρ0r ≡ 6( 2α − 2 3 ) r2 0 . (37) With this deﬁnition, solving Einstein’s equations yields th e dynamic Dymnikova black hole metric: ds2 = −f (v, r )dv2 + 2dvdr + r2dΩ 2, (38) where f (v, r ) = 1 − 2M r ( 1 − e − r3 2...

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arXiv 2023