Vacuum decays around spinning black holes

Kazushige Ueda; Masahide Yamaguchi; Naritaka Oshita

REVIEW 3 major objections 4 minor 1 cited by

Vacuum decays around spinning black holes

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A spinning black hole suppresses vacuum decay around it, if the nucleated bubble is an ellipsoid set by the Kerr geometry.

desk verdict First real attempt at vacuum decay around a Kerr hole, with honest caveats, but the suppression claim hangs on an unproven ellipsoidal bubble shape. read the letter →

arxiv 1909.01378 v3 pith:LKOKK6NO submitted 2019-09-03 hep-th gr-qc

Naritaka Oshita , Kazushige Ueda , Masahide Yamaguchi This is my paper · ORCID

classification hep-thgr-qc MSC 83C5781T20

keywords vacuumdecayfalseKerrblackholethin-wallapproximationIsraeljunctionconditionEuclideanactionColeman-deLucciabubbleprimordialholes

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

The paper asks whether a black hole's spin changes how strongly the hole catalyzes the decay of a metastable vacuum. Working in the thin-wall approximation, it treats the nucleated bubble as an ellipsoid of constant Boyer-Lindquist radius, with the shape fixed by the Kerr metric, and joins the interior and exterior spacetimes with the Israel junction conditions. It finds that spin raises the Euclidean action of the bounce, so a spinning black hole decays more slowly than a non-spinning one of the same mass; in the near-extremal example shown the action is about 85 percent larger, and in some cases it exceeds the Coleman-de Luccia value, meaning the false vacuum is stabilized relative to decay in empty space. The critical seed mass below which a growing bubble exists also increases with spin, by up to roughly 30 percent, so spinning holes catalyze decay over a wider mass range but more weakly. A denser bubble wall reduces the spin dependence.

What carries the argument

The load-bearing object is the ellipsoidal thin-wall bubble: a surface of constant Boyer-Lindquist radius $r=R(\tau)$ for every polar angle $\theta$, whose induced metric has spheroidal angular parts, with major axis $\sqrt{a^2+R^2}$ and minor axis $R$. The Israel junction conditions, applied in the zero-angular-momentum form of the Kerr metric, turn the wall's dynamics into an equation $\dot{R}^2+V(R,\theta)=0$ with an effective potential $V(R,\theta)$; the Euclidean action is assembled from horizon, interior, exterior, and wall pieces with a conical-deficit regularization adapted to axisymmetry. This machinery converts the question of whether spin changes the decay rate into a computation of how the roots of $V=0$ and the on-shell action shift with the spin parameter $a$.

What would settle it

A numerical search over all Euclidean bounce shapes around a Kerr black hole, without fixing $r=R(\tau)$, would settle the central question: if the least-action bubble is not the ellipsoid of constant Boyer-Lindquist radius, the claimed spin suppression is not established.

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Extended reading notes

Core claim

The paper's central claim is that black-hole spin weakens the catalyzing effect of a black hole on false-vacuum decay. Working in the thin-wall approximation, it models the nucleated bubble as a time-like ellipsoidal shell of constant Boyer-Lindquist radius, with major axis $\sqrt{a^2+R^2}$ and minor axis $R$, and matches the interior and exterior Kerr-family metrics through the Israel junction conditions. The resulting Euclidean bounce action for a spinning seed is larger than for a Schwarzschild seed of the same mass; for $a=0.99\,GM$ and a light wall, the action is about 85 percent larger, and for near-extremal spin it can exceed the Coleman-de Luccia action, so a false vacuum is more stable than in the no-black-hole decay. The authors also find that the critical seed mass below which a growing bubble exists rises with spin, by up to roughly 30 percent, while a heavier or denser wall reduces the spin dependence. The conclusion is conditional: it holds provided this ellipsoidal wall is the least-action bounce, which the paper asserts but does not derive.

Load-bearing premise

The result stands on the assumption that the nucleated bubble is a thin ellipsoid of constant Boyer-Lindquist radius and that this ellipsoid is the least-action bounce; the paper states this condition but does not derive it.

Editorial extensions

If this is right

For a fixed seed mass below the critical mass, a spinning black hole has a lower vacuum decay rate than a non-spinning one, so spin acts as a stabilizer.
A near-extremal Kerr black hole can make a false vacuum more stable than the Coleman-de Luccia empty-space bubble predicts, meaning vacuum decay can be shut off locally around such holes.
The mass window in which a black hole can catalyze decay widens with spin: the critical mass rises by up to about 30 percent at high spin.
A dense bubble wall suppresses the spin dependence, so spin corrections matter most for light walls.
Existing constraints on primordial black hole abundance from vacuum decay, which have used non-spinning holes, are not invalidated by spin; the paper offers this as supporting evidence.

Reading between the lines

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

If a full non-spherical bounce calculation ever shows that the least-action bubble is not the constant-Boyer-Lindquist-radius ellipsoid, the spin suppression could shrink or reverse; the paper leaves that check open.
The $\theta$-dependence of the effective potential $V(R,\theta)$ means the constant-$R$ wall ansatz is only approximate at high spin, so a genuine least-action bubble could have a different spheroidal shape and different numerical action.
Applied to Higgs-vacuum metastability, the widened critical-mass window suggests spinning primordial black holes slightly above the non-spinning critical mass could still catalyze decay, so abundance constraints may need to cover a wider mass range even though each hole's rate is lower.
The same junction-condition method could be extended to Kerr-Newman or higher-dimensional Kerr seeds, where it is not yet known whether charge or extra dimensions change the spin suppression.

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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

Colleague,

This is a genuine first: nobody has tried a non-spherical bubble around a spinning black hole, and the authors do it carefully. They derive the extrinsic curvature of an ellipsoidal wall in Kerr(-AdS) using ZAMO coordinates, get junction conditions that allow anisotropic wall stress, and compute the Euclidean action with a conical-deficit regularization extended to axisymmetric metrics. That part is solid and useful.

The new physics: spin suppresses the decay rate relative to a Schwarzschild seed, the critical black hole mass goes up by roughly 30%, and near-extremal holes can stabilize the false vacuum relative to Coleman-de Luccia. These are concrete, checkable claims, and the numerics are presented cleanly. The authors also state their limitations plainly, which I appreciate.

The stress-test concern is real and it lands. The wall is assumed to be r = R(τ) with no θ-dependence, and the effective potential V(R,θ) from Eq. (3.29) depends on θ. So the equation Ṙ² + V(R,θ) = 0 cannot be satisfied at all θ by a θ-independent R. The authors acknowledge this and estimate the error numerically, but that estimate only measures the spread of V within the assumed family. It does not minimize the action over θ-dependent profiles, so the true bounce could deform, lower the action, and possibly erase the suppression. The conclusion in Sec. 4 is explicitly conditional on the ellipsoid being the least-action shape. That is honest, but it leaves the central claim as a conjecture supported by a restricted calculation.

Is this fatal? No. As a first exploration it is a legitimate contribution. The paper is well organized, the derivation is transparent, and the caveats are front and center. The soft spots are the unproven ansatz, the assumed wall equation of state (pθ+pψ)/2 = −σ, and the absence of error estimates on the action values. The numerical conclusion that spin suppresses decay is provisional.

Who should read it: people working on vacuum decay around compact objects, PBH abundances, and Higgs metastability. A serious referee should engage with it—the right referee will ask for either a least-action proof, a variational upper bound, or a broader ansatz allowing θ-dependent walls. It deserves peer review, not desk rejection.

Referee Report

3 major / 4 minor

Summary. The paper studies false-vacuum decay seeded by a Kerr black hole using the thin-wall approximation and Israel junction conditions. It assumes that the nucleated vacuum bubble is an ellipsoid whose shape is fixed by the angular metric components of the Kerr background, with a wall at constant Boyer-Lindquist radius r=R(tau). The authors derive the effective potential V(R,theta), classify the bubble solutions, compute the Euclidean action including a conical-deficit regularization, and find that the Euclidean action increases with the spin parameter a, so that spin suppresses the decay rate relative to the non-spinning case. They also report that the critical black hole mass below which decay is possible grows with spin by up to about 30%, and that for near-extremal spins the false vacuum can be more stable than in the Coleman-de Luccia solution.

Significance. If the central claims were established, this would be the first quantitative analysis of vacuum decay around a spinning black hole and would affect phenomenological constraints on primordial black hole abundances and Higgs metastability. The paper contains a careful derivation of the junction conditions in the ZAMO frame, an explicit conical-deficit regularization for axisymmetric Euclidean metrics in Appendix A, and reproducible numerical evaluations in Figs. 5, 9, and 10. However, as the authors explicitly acknowledge in Sec. 4, the headline result is conditional on the unproven least-action property of the ellipsoidal wall shape, and the theta-dependence of V(R,theta) in Eq. (3.29) makes that assumption load-bearing rather than a harmless technical detail.

major comments (3)

[Sec. 3.2, Eq. (3.29)] The effective potential V(R,theta) depends on theta, so the equation dot(R)^2 + V(R,theta) = 0 cannot be satisfied for all theta by a theta-independent R(tau). Thus the wall r=R(tau) introduced in Eq. (3.10) is not an exact stationary point of the junction-condition action. The error estimate in Sec. 3.4 and the gray band in Fig. 10 only quantify the theta-spread of V for the assumed one-parameter family; they do not bound the reduction in the Euclidean action that could arise from theta-dependent wall profiles r=R(tau,theta). A variational check over such profiles, or at least a perturbative stability analysis of the ellipsoidal wall, is required before the claimed ~85% suppression for a=0.99GM can be accepted.
[Sec. 4, Conclusion] The conclusion states the main physical claim 'provided that the ellipsoidal vacuum bubble characterized by (3.10) gives the least action,' but no proof or numerical evidence toward the least-action property is supplied. Because the first Israel junction condition constrains the analysis to M+ = M- and a+ = a-, the spin dependence of the Euclidean action is largely inherited from the assumed Kerr geometry of the wall surface; the computation demonstrates that the ansatz yields a spin-dependent action, but it does not demonstrate that the true instanton has the same behavior. The authors should either supply a variational argument that the ellipsoidal shape is the saddle point, or restrict the claims to a narrower statement about the action of that specific trial shape.
[Sec. 3.2, Eq. (3.39)] The junction conditions (3.23) and (3.24) only determine the difference p_theta - p_psi, while the final expression for the Euclidean action uses the additional assumption (p_theta + p_psi)/2 = -sigma. This is an extra model input that is not a consequence of the Israel conditions. Please justify this choice from a microphysical wall model or demonstrate that the conclusions are insensitive to it; as stated, the action depends on an undetermined combination of the anisotropic pressures.

minor comments (4)

[Eq. (3.26)] The typesetting of Eq. (3.26) appears corrupted in the manuscript (the 'radicaltp' and 'radicalvertex' artifacts), obscuring the definition of Xi_±; please correct the formula.
[Fig. 5 caption] The caption does not state the value of theta used to plot V(R,theta), unlike the footnote for Fig. 6; please add this information for reproducibility.
[Sec. 2.1, Eq. (2.1)] The notation for the interior and exterior metrics is not consistent across the paper: the sign conventions for f_± and the role of H_± could be stated more explicitly near Eq. (2.2), and the indices in Eq. (2.20) contain several typos (for example, 'tEtE' should be 'lambda_E lambda_E' in the Kerr section).
[Sec. 3.1] The paper states that the first Israel junction condition holds approximately for a+ = a-, M+ = M-, and a^2 << l^2, but the order of approximation is not tracked carefully through the subsequent derivation of the extrinsic curvature and the action. Please state explicitly which terms are dropped in Eqs. (3.12) and (3.13) and verify that the same approximation is used consistently in Sec. 3.4.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity: the spin suppression is a computed consequence of an explicitly stated ellipsoidal-wall ansatz, whose least-action status is an acknowledged limitation rather than a circular input.

full rationale

Walking the derivation chain, I find no circular step. The paper's input is an explicit thin-wall ansatz: the nucleated bubble is placed on the constant Boyer-Lindquist-radius surface r = R(τ) of the ZAMO-diagonalized Kerr metric (Eq. 3.10), which makes the induced metric spheroidal (Eqs. 3.12–3.13 and 3.14–3.15). The spin dependence of the final Euclidean action (Eq. 3.39) is computed, not fitted: the action contains competing horizon-area and extrinsic-curvature contributions, so the sign and size of S_E(a) − S_E(0) is a nontrivial result; for example, the larger ellipsoidal wall area by itself would tend to decrease the wall term, while the paper finds an overall increase. The conclusion is repeatedly and explicitly conditioned on the ansatz ('provided that the ellipsoidal vacuum bubble characterized by (3.10) gives the least action', Sec. 4), so the unproven least-action property is a limitation of the model, not an equation that reduces to its own output. The only self-citation, Ref. [34] (Oshita-Yamada-Yamaguchi), is used in the Introduction to note earlier work on compact-object catalysts and is not load-bearing for the Kerr calculation; no uniqueness theorem is imported from the authors' prior work. The comparison against the CDL action and the Schwarzschild limit is external. Accordingly, the central claim is ansatz-dependent but not circular.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The calculation rests on the thin-wall/Israel junction formalism, the specific ellipsoidal shape ansatz whose spin dependence is baked in, the simplifying wall equation of state (p_theta+p_psi)/2 = -sigma, and an unproven least-action property. The shape ansatz and wall equation of state are ad hoc to this paper; the junction-condition formalism is standard. No new physical entities are introduced.

free parameters (1)

Surface tension Sigma = 4*pi*G*sigma = Scanned values: 1.3e-5, 1.5e-5, 2e-5, 3e-5, 6e-5, 7e-5 M_Pl in numerical examples
The wall surface tension is a physical input of the vacuum transition, not fitted to data, but the strength of the spin-suppression conclusion depends on its value: for dense walls (Sigma = 7e-5 M_Pl) the spin dependence is suppressed (upper left panel of Fig. 10). The paper scans this parameter rather than predicting it.

assumptions (6)

domain assumption Thin-wall approximation: the vacuum bubble is a singular hypersurface and Israel junction conditions apply.
Used throughout Sec. 2 and Sec. 3 to replace the field configuration by a thin wall with surface stress-energy. Standard for vacuum decay calculations since Coleman and De Luccia, but it is a modeling restriction.
ad hoc to paper Bubble wall surface is r = R(tau) in Boyer-Lindquist coordinates, i.e., an ellipsoid with shape fixed by the Kerr angular metric.
Introduced in Sec. 3.1, Eq. (3.10). The shape is chosen to match the Kerr geometry, not derived from minimizing the action. The paper explicitly conditions its conclusion on this shape giving the least action.
domain assumption First Israel junction condition holds with a+ = a-, M+ = M-, and a^2 << l^2 (spin much smaller than the AdS length scale).
Sec. 3.1 states that exact matching of the induced metrics is only approximate under these conditions; the analysis is restricted to small vacuum-energy changes and slowly spinning (compared to l^-1) black holes.
ad hoc to paper The bubble wall stress-energy tensor is Sab = diag(sigma, p_theta h_theta_theta, p_psi h_psi_psi), and to compute the action the combination (p_theta + p_psi)/2 = -sigma is assumed.
The anisotropic pressure form is introduced in Sec. 3.2, and the additional assumption (p_theta+p_psi)/2 = -sigma is stated after Eq. (3.39) without physical justification. This is needed to reduce the action to the displayed form.
domain assumption The semiclassical decay rate is Gamma ~ D_pre e^{-S_E} and the dominant Euclidean saddle is the O(3)-symmetric-in-time oscillating bubble solution.
Standard Euclidean bounce formalism from Coleman and De Luccia, applied in Sec. 2.2 and Sec. 3.4. The paper assumes this remains valid with gravity and a non-spherical wall.
ad hoc to paper The ellipsoidal bubble (3.10) gives the least Euclidean action among all possible bubble shapes.
The authors explicitly state that their conclusion holds 'provided that the ellipsoidal vacuum bubble characterized by (3.10) gives the least action' (Sec. 4). No proof or comparison with other shapes is given.

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Cite this review

Pith. "Pith review of Vacuum decays around spinning black holes." pith.science (2026). https://pith.science/paper/LKOKK6NO

@misc{pith2026190901378,
  author       = {Pith},
  title        = {Pith review of: Vacuum decays around spinning black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LKOKK6NO}},
  note         = {Machine review of arXiv:1909.01378}
}

read the original abstract

We investigate a vacuum decay around a spinning seed black hole by using the Israel junction condition and conclude that the spin of black hole would suppress a vacuum decay rate compared to that for a non-spinning case, provided that the surface of vacuum bubble has its ellipsoidal shape characterized by the Kerr geometry. We also find out that in the existence of a near-extremal black hole, a false vacuum state can be more stabilized than the case of the Coleman-de Luccia solution. A few necessary assumptions to carry the calculations are discussed.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

I. Yu. Kobzarev, L. B. Okun, and M. B. Voloshin, Sov. J. Nuc l. Phys. 20, 644 (1975), [Yad. Fiz.20,1229(1974)]

work page 1975

[2] [2]

S. R. Coleman, Phys. Rev. D15, 2929 (1977) , [Erratum: Phys. Rev.D16,1248(1977)]

work page 1977

[3] [3]

C. G. Callan, Jr. and S. R. Coleman, Phys. Rev. D16, 1762 (1977)

1977

[4] [4]

S. R. Coleman and F. De Luccia, Phys. Rev. D21, 3305 (1980)

1980

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S. R. Coleman, V. Glaser, and A. Martin, Commun. Math. Phys. 58, 211 (1978)

work page 1978

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work page 1981

[7] [7]

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work page 1982

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Susskind, , 247 (2003), arXiv:hep-th/0302219 [hep-th]

L. Susskind, , 247 (2003), arXiv:hep-th/0302219 [hep-th]

arXiv 2003