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Formation of Frozen Stars from collapsing matter by tunneling

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Collapsing matter tunnels into frozen stars with probability one, as a Euclidean-action calculation shows the transition rate is unity up to negligible corrections.

desk verdict A serious extension of the frozen-star program whose central claim—probability-unity tunneling—is unsupported: no bounce solution is constructed and the determinant is not computed. read the letter →

arxiv 2508.02100 v1 pith:U52VKQ75 submitted 2025-08-04 gr-qc

Ram Brustein , A.J.M. Medved , Tamar Simhon This is my paper
classification gr-qc
keywords frozenstarblackholemimickerquantumtunnelingEuclideanactionfalsevacuumdecayBorn-Infeldstringfluidtransitionprobabilityultracompactobject
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 argues that any collapsing shell of ordinary matter inevitably transitions into a frozen star, a horizonless ultracompact object that mimics a black hole from the outside, through a quantum tunneling process. The authors calculate the relevant Euclidean partition function, including the Born–Infeld matter action and quadratic fluctuations, and find that both the exponential factor and the prefactors equal one. Hence the transition probability is unity up to negligibly small corrections, so the formation of a frozen star requires no classical formation mechanism. If correct, this answers a central challenge for black-hole-mimicker models: how such exotic objects can form from standard matter. The paper extends an earlier claim that the transition was merely likely to a claim of inevitability.

What carries the argument

The central object is the Euclidean partition function for false-vacuum decay, adapted from the Gibbons–Hawking method, with the transition probability identified with $Z$. The paper evaluates $Z$ by splitting the Euclidean action into Einstein–Hilbert bulk, transitional layer, boundary-at-infinity, and Born–Infeld matter contributions, and by computing the determinant prefactor via the linearized equations of motion. The radially dependent inverse temperature $\beta(r)=4\pi r$, derived from the static heat equation, is what makes the bulk action cancel the transitional-layer action in the Einstein–Hilbert sector, while the relation between transverse pressure and the Born–Infeld source makes the matter action vanish as well.

What would settle it

The claim would collapse if a full quadratic fluctuation determinant around the frozen star solution were found to differ from one, or if the collapsing shell's free energy contributed a non-negligible Euclidean action. Concretely, one could attempt to construct the explicit instanton solution interpolating between the Minkowski interior and the frozen-star interior; if no such saddle point exists, the rate formula (22) would have nothing to mediate the transition.

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Extended reading notes

Core claim

The central claim is that a collapsing shell of mass $M$ with a Minkowski interior tunnels with probability one into a frozen star of the same mass. The argument treats the outer transitional layer of the frozen star as a Euclidean instanton that mediates the decay from the false vacuum of the empty interior to the true vacuum of the frozen star. For both the Einstein–Hilbert and Born–Infeld parts of the action, the Euclidean action of the frozen star interior vanishes: the bulk and transitional-layer contributions cancel each other, and the boundary term at infinity is shared with the exterior Schwarzschild geometry, so the action difference with the Minkowski interior is zero. The determinant prefactor is one because the frozen star is ultrastable, meaning that linearized metric and gauge-field perturbations vanish identically, while the false vacuum is empty Minkowski space. The final result is $\Gamma_{\text{matter shell}\to\text{frozen star}}=1$ up to negligibly small corrections, making the transition inevitable.

Load-bearing premise

The result assumes that the Euclidean version of the frozen star's outer transitional layer is a legitimate quantum-gravitational instanton for the shell-to-frozen-star transition, and that the standard false-vacuum formula applies even though no explicit bounce solution connecting the two interiors is constructed and the action difference is zero.

Editorial extensions

If this is right

  • Every collapsing shell in the relevant mass range ends up as a frozen star rather than a spacetime with a singularity and horizon, if the calculation holds.
  • The frozen star's bulk, whose entropy is the Bekenstein–Hawking value, effectively performs the sum over microstates, so no explicit enumeration of fuzzball states is required.
  • The transition is triggered when the shell reaches the outermost edge of the transitional layer, just outside its Schwarzschild radius, where local temperatures become exponentially large.
  • The frozen star's entropy satisfies the area law while its interior action cancels, preserving the generalized Gibbons–Hawking result for ultracompact objects.
  • Observational searches that cannot distinguish a frozen star from a black hole by exterior geometry are consistent with this formation channel.

Reading between the lines

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

  • If the probability-unity result is literally true, standard stellar collapse would not produce even a transient trapping horizon, which sharpens the tension with classical collapse simulations that the paper does not address.
  • The mechanism may generalize to other horizonless ultracompact models whose interiors are ultrastable and whose Euclidean action vanishes; the key ingredients are zero action difference and unit determinant, not the specific Born–Infeld string fluid.
  • A concrete testable consequence is that the end state of collapse predicted here carries no curvature singularity and no event horizon, so horizon-scale images or ringdown observations of candidate black holes could in principle falsify the model.
  • The result depends on the collapsing shell's free energy being negligible; for realistic matter with significant entropy or temperature, the action difference might not vanish, and the transition rate would be suppressed.
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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 claims to compute the probability for a collapsing matter shell to quantum-mechanically tunnel into a frozen star of equal mass, finding it to be unity up to negligible corrections. The calculation uses the Euclidean partition function of Eq. (22), with the Einstein–Hilbert and Born–Infeld actions evaluated on the frozen-star interior and on the Minkowski interior of the collapsing shell. The authors argue that both actions vanish (Eqs. (30) and (39)), that the fluctuation determinants are unity (Section 5.3), and therefore that the transition is inevitable (Eq. (58)). The paper also reviews the frozen-star geometry and its Born–Infeld sourcing, and derives a radially dependent inverse temperature β=4πr from the heat equation.

Significance. If correct, the result would be significant: it would resolve a key formation problem for black-hole mimickers by making the transition from ordinary collapsing matter to a frozen star inevitable, without requiring a classical formation mechanism or a sum over microstates. The paper also provides a self-contained review of the frozen-star model and connects it to a Born–Infeld string-fluid Lagrangian. However, the central derivation rests on assumptions that are not established: no instanton solution connecting the two phases is constructed, the action difference is set to zero by cancellation, and the determinant prefactor is asserted rather than computed. The claimed probability-unity result is therefore not supported by the presented analysis.

major comments (4)
  1. [Section 5, Eq. (22)] The false-vacuum decay formula (22) is applied to two disconnected stationary configurations, the Minkowski interior and the frozen-star interior, without exhibiting the Euclidean solution that connects them. In the standard instanton formalism, the exponent is the action of a single bounce solution relative to the false vacuum, not the action difference between two separately evaluated backgrounds. Since no such bounce is constructed, the equality I_TV = I_FV = 0 yields e^0 only by assumption; the exponential factor in Eq. (58) is an input rather than a derived result.
  2. [Section 5.3.3, Eqs. (48)-(54)] The argument that the determinant prefactor is unity is invalid. The authors impose h(t,R−λ)=0 and ∂_r h(t,R−λ)=0, solve the linearized equations, find h=0, and conclude that the quadratic term in the action expansion vanishes and hence det=1. Vanishing of the linearized solution for a particular boundary-value problem does not imply vanishing of the second variation of the action; the determinant requires the full eigenvalue spectrum of the fluctuation operator, including modes that do not satisfy those boundary conditions. A free scalar field has a vanishing linearized solution at φ=0, yet its fluctuation determinant is not unity. No mode decomposition, eigenvalue computation, or regularization is presented, so the assertion det'(I''(φ_TV))=1 is not established.
  3. [Section 5.4, Eq. (58)] Even granting the action and determinant calculations, Eq. (58) does not follow from Eq. (22). The prefactor A in Eq. (22), the treatment of zero modes, and the normalization of the partition function are not computed; the identification of the partition function with a transition probability of exactly unity is not justified. More fundamentally, a transition probability of unity means there is no barrier, which contradicts the claim that a quantum tunneling process is being described. The cancellations in Eqs. (30) and (39) are built into the model through the specific choice of the temperature profile and the boundary conditions, so the conclusion is effectively an input rather than a consequence of the dynamics.
  4. [Sections 4 and 5.2] The derivation of the inverse temperature profile β=4πr in Section 4 uses the boundary condition (16), which presupposes that the temperature at the frozen-star surface is perturbatively close to the Hawking value. This same profile is then used in the action integrals of Section 5.2 to obtain the cancellations that set I_TV=0. The vanishing interior action is therefore not an independent result but is tied to prior assumptions about the frozen-star thermodynamics, making the probability-unity conclusion circular with the model's construction.
minor comments (5)
  1. [Abstract] There is a typo: 'which posses' should be 'which possess'.
  2. [Section 5.1, Eq. (27)] The notation in Eq. (27) is unclear: the integration limits are written as R(1+λ) and R(1−λ), and the first expression appears to omit the integration variable. Please rewrite with standard bounds R−λ and R+λ.
  3. [Section 5.1, Eq. (25)] The sign convention for the Euclidean action and the treatment of boundary terms should be stated explicitly, since the cancellation leading to Eq. (30) depends on the relative signs of the bulk and transitional-layer contributions.
  4. [References and Acknowledgments] Reference [51] lists 'P Paranjape' without a first initial, and the Acknowledgments contain the garbled string 'V ATAT (Israel planning and budgeting committee)', which should be corrected.
  5. [Figure 1] The caption uses 'FS' without defining the abbreviation in the figure; please define it or expand the label in the figure.

Circularity Check

3 steps flagged · score 8.0 of 10

The Γ=1 claim reduces to a zero Euclidean action difference and a unit determinant that are built into the frozen-star model and its authors' prior instanton identification, rather than derived from an explicit tunneling solution.

  1. self citation load bearing [Section 5, opening paragraphs (page 13-14)]
    "In [1], we proposed, following an idea first introduced by Mathur in the context of fuzzballs [35], that the correct description of this evolution must include a quantum-induced phase transition or, equivalently, a quantum tunneling event. We then argued that, from this perspective, it is natural to regard the Euclidean version of the outer transitional layer as a quantum-gravitational instanton which is mediating the transition from the empty interior of an infalling shell of conventional (standard model) matter to a same-sized sphere of exotic frozen star matter."

    The applicability of the false-vacuum formula (22) to this problem rests entirely on the identification of the Euclidean transitional layer as a quantum-gravitational instanton. That identification is explicitly attributed to the authors' own previous work [1], and no Euclidean solution of the Einstein-Born-Infeld equations interpolating between the Minkowski interior and the frozen-star interior is constructed here. The central claim therefore leans on a self-citation chain for its load-bearing premise.

  2. self definitional [Section 5.4, first paragraph after Eq. (22)]
    "Meanwhile, using Eq. (30) and Eq. (39), one can see that the action for the frozen star interior also vanishes, I_TV = 0 , up to perturbative-order corrections. Hence, the exponential is equal to unity up to perturbatively small corrections."

    The exponential factor in Eq. (22) becomes unity because both actions vanish: I_FV = 0 for empty Minkowski space, and I_TV = 0 from the cancellations in Eqs. (30) and (39). Those cancellations are properties of the static frozen-star model, engineered in prior work [1] so that the interior action contributes nothing to the area-entropy law; they are not actions of a tunneling bounce connecting the two configurations. With I_TV - I_FV = 0 as an input, exp[-(0)] = 1 is built in rather than derived, and the 'inevitable transition' is equivalent to the model's zero-action construction.

1 more flagged steps
  1. fitted input called prediction [Section 5.3.3, final paragraph]
    "as the linearized equations of motion vanish identically, so too does the quadratic term in the expansion of the frozen action. It then follows via det( A) = eTr lnA that the determinant of this quadratic term should be unity."

    The pre-exponential factor is set to unity by choosing boundary conditions that force the perturbations h_mu nu to vanish, and then inferring that a vanishing quadratic term implies unit determinant. A vanishing homogeneous solution does not determine the determinant of the quadratic fluctuation operator; the Gaussian approximation is degenerate rather than equal to unity. The boundary conditions are chosen ad hoc, so the 'prediction' det' = 1 is an imposed input, not a computed result. This makes the prefactor in Eq. (58) unity by construction.

full rationale

The central result, Eq. (58), is obtained by substituting I_TV = I_FV = 0 and both determinants equal to unity into Eq. (22). The equality I_TV = 0 is not a tunneling action; it is the static Euclidean action of the frozen-star interior, which vanishes because of cancellations already central to the authors' prior area-law derivation [1] and because the Born-Infeld action cancels between bulk and source. The equality I_FV = 0 is just the action of empty Minkowski space. The determinant equality is obtained by imposing trivial boundary conditions and by an invalid inference from vanishing linearized perturbations to a unit Gaussian determinant. Moreover, the entire applicability of Eq. (22) to two disconnected stationary points depends on a 'quantum-gravitational instanton' identification that the paper explicitly attributes to its own earlier work [1]. Thus the probability-unity claim is forced by the model's construction and by the self-citation chain rather than by an independent tunneling computation. While no data fitting is involved, the prediction reduces by definition to the input assumptions of zero action difference and unit prefactor, so the circularity score is 8.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new particle, force, or conserved quantity. Its central object, the frozen star and its string-fluid source, is carried over from the authors' earlier work, and the current contribution is a calculation within that model. The free parameters are the small scales ε, λ, and η that define the geometry and the transitional layer.

free parameters (3)
  • ε = ε^2 << 1
    Exponentially small constant defining the near-null interior metric in Eq. (1); chosen by hand and all results are at leading order in ε.
  • λ = λ << R, λ > ε^2 R
    Half-width of the outer transitional layer; chosen small and used throughout the action integrals and in the determinant boundary conditions.
  • η = η << R
    Radius of the regularized central core; chosen small so integrated quantities deviate only at order η/R, and used in the temperature discussion.
assumptions (7)
  • standard math Euclidean path-integral and semiclassical instanton formula (Eq. 22) from Gibbons-Hawking and Coleman, assuming a well-defined gravitational path integral and a stationary-phase saddle.
    Used in Section 5 to identify the partition function with the tunneling probability.
  • domain assumption Frozen star geometry (Eq. 1) with saturated null-energy condition and zero transverse pressure in the bulk.
    Background model carried from prior work [6,7] and reviewed in Section 2.
  • domain assumption The matter source is a Born-Infeld string fluid originating from Sen tachyon condensation.
    Matter model from [12,21], used in Section 3 to compute the Born-Infeld action.
  • domain assumption The frozen star is static and in thermal equilibrium, so the heat equation with boundary temperature close to the Hawking value determines β=4πr.
    Section 4 derives the temperature profile used in all later action integrals.
  • ad hoc to paper The outer transitional layer can be interpreted as a Euclidean instanton mediating the transition from a collapsing shell to a frozen star.
    Proposed by the authors in Section 5; no explicit Euclidean solution is constructed.
  • ad hoc to paper Boundary conditions h=0 and ∂_r h=0 at the inner edge of the transitional layer force all metric perturbations to vanish.
    Used in Section 5.3.3 to conclude the determinant prefactor is unity.
  • domain assumption A pressureless dust shell has vanishing Helmholtz free energy and hence vanishing Euclidean action.
    Used in Section 5.4 to set the false-vacuum action to zero.

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Cite this review

Pith. "Pith review of Formation of Frozen Stars from collapsing matter by tunneling." pith.science (2026). https://pith.science/paper/U52VKQ75

@misc{pith2026250802100,
  author       = {Pith},
  title        = {Pith review of: Formation of Frozen Stars from collapsing matter by tunneling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U52VKQ75}},
  note         = {Machine review of arXiv:2508.02100}
}
read the original abstract

The frozen star is a type of black hole mimicker: An ultracompact object whose exterior geometry resembles that of a general-relativistic black hole but differs in its matter composition and in the regularity of its interior geometry. It is sourced by a spherically symmetric collection of open-string flux tubes, which posses an extremely anisotropic energy-momentum-stress tensor with maximally negative radial pressure. The frozen star represents an effective classical description of the highly quantum, closed-string polymer model. A key challenge for any model of a black hole mimicker is to explain how such objects can form from a collapsing body of matter. We started to address this important problem in a previous article by adapting the Euclidean-action method of Gibbons and Hawking to show that the transition into a frozen star is likely. Here, we improve on our previous results by showing that the transition probability for a collapsing shell of matter to tunnel quantum mechanically into a frozen star is unity, up to negligible corrections. Our conclusion is that such a transition is therefore inevitable.

Figures

Figures reproduced from arXiv: 2508.02100 by the authors.

Figure 1
Figure 1. The collapsing shell tunneling to the true vacuum of the frozen star [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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  1. Quantum nucleation of black hole mimickers via chaos dominated tunneling

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    Chaos-dominated multichannel tunneling can eliminate exponential barrier suppression, enabling quantum nucleation of black shells and other black-hole mimickers.

  2. Defrosting the Born-Infeld dyonic frozen star with tachyon matter: spectrum of oscillations

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    A Born-Infeld Lagrangian with electric, magnetic, and tachyon charges reproduces the slow, long-lived oscillation spectrum of the defrosted frozen star.

Reference graph

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