(Anti-)evaporation of Schwarzschild-de Sitter black holes revisited

Maciej Kolanowski

REVIEW 2 major objections 3 minor 18 references

(Anti-)evaporation of Schwarzschild-de Sitter black holes revisited

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The s-wave large-N effective action admits no Schwarzschild–de Sitter solution, so anti-evaporation is likely an artifact.

desk verdict Useful conceptual distinction, but the no-go theorem is unsupported by a sign error in the appendix. read the letter →

arxiv 1908.01716 v2 pith:PSIQZHBU submitted 2019-08-05 gr-qc

Maciej Kolanowski This is my paper · ORCID

classification gr-qc PACS 04.70.Dy04.50.Kd

keywords anti-evaporationSchwarzschild-deSitterblackholeNariaispacetimes-waveeffectiveactionsemiclassicalbackreactionextremalholescosmologicalconstant

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 the widely reported "anti-evaporation" of black holes in a universe with a positive cosmological constant is an artifact of conflating two distinct spacetimes: the Nariai solution, a product of a two-dimensional de Sitter space and a sphere, and the extremal Schwarzschild–de Sitter black hole. In the simplified semiclassical framework that produced the anti-evaporation claim—an effective action for spherically symmetric fluctuations with many scalar fields—the paper proves a no-go result: there is no solution with the large-radius behavior of Schwarzschild–de Sitter. Earlier work had identified the near-horizon limit of the extremal black hole with Nariai spacetime, so its conclusions about growing horizons do not apply to the physical black-hole spacetime. If the argument is right, the claimed slowdown or reversal of black hole evaporation in de Sitter space needs to be rethought.

What carries the argument

The load-bearing object is the effective action of [1]: one-loop equations of motion (9)–(13) for s-wave, large-N massless scalar fields coupled to gravity with Λ>0, in a metric of the form ds² = $e^{{2ρ}}$(-dt²+dx²)+$e^{{-2φ}}$dΩ². The proof specializes to the ansatz $e^{{-2φ}}$=r² with r and ρ depending only on t, then rewrites the metric in coordinates ds² = -dr²/h² + g² dx² + r² dΩ², where $e^{{ρ}}$=g(r) and dr/dt=g h(r). Imposing the falloff g=Ar+O($r^{{1-ε}}$), h=Br+O($r^{{1-ε}}$) and substituting into the φ-equation yields a leading term -A²(2B²+Λ)r²+O($r^{{2-ε}}$)=0, contradicting Λ>0. The asymptotic form (16) is what would guarantee the conformal boundary (scri) structure matching Schwarzschild–de Sitter.

What would settle it

Numerically integrate (9)–(13) for the metric ds² = -dr²/h² + g² dx² + r² dΩ² with Λ>0 and boundary conditions g~Ar, h~Br at large r; the leading-order equation forces 2B²+Λ=0, so no such solution exists. The claim would be falsified by exhibiting any solution with this Killing vector and a different large-r falloff that still reproduces the Schwarzschild–de Sitter conformal boundary.

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Extended reading notes

Core claim

The central claim is a theorem: for the effective equations (9)–(13), there is no static solution with a Killing vector ∂x and the large-sphere-radius asymptotics ds² = -dr²/(B² r²) + A² r² dx² + r² dΩ², with A and B nonzero constants and Λ>0. Under the ansatz $e^{{-2φ}}$=r², treating r as a time coordinate, and assuming the falloff g=Ar+O($r^{{1-ε}}$), h=Br+O($r^{{1-ε}}$), the leading-order substitution forces -(2B²+Λ)=0, which is impossible. Thus the framework that produced the anti-evaporation claim contains no Schwarzschild–de Sitter-like solution at all, not merely no extremal one. The paper concludes that previous treatments' identification of Nariai spacetime with the extremal Schwarzschild–de Sitter black hole is incorrect, and that the anti-evaporation effect is likely a mathematical artifact of that misidentification.

Load-bearing premise

The theorem depends on the s-wave large-N effective equations being the correct semiclassical description and on the assumption that every Schwarzschild–de Sitter-like solution must satisfy the falloff g=Ar+O($r^{{1-ε}}$), h=Br+O($r^{{1-ε}}$); if the true backreaction lies outside this class, the no-go result need not apply.

Editorial extensions

If this is right

Within the s-wave large-N effective action, there is no Schwarzschild–de Sitter-like solution at all, so anti-evaporation claims drawn from that framework lose their background solution.
The Nariai spacetime is only the near-horizon geometry of the black hole, not the extremal black hole itself; their global structures, including their conformal boundaries, are different.
For the extremal Schwarzschild–de Sitter metric, r behaves as a time coordinate outside the horizon, so a shift in the apparent "Schwarzschild radius" under a perturbation may amount to a time translation rather than a physical change in horizon area.
Repeating the analysis without the s-wave approximation and in modified gravity (f(R)) is needed; if the same obstruction appears there, the expected lifetime of primordial black holes could shorten.

Reading between the lines

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

If the no-go result extends to the full one-loop effective action, the s-wave large-N framework would be left without any Schwarzschild–de Sitter background on which to define Hawking radiation, making evaporation calculations in that framework questionable.
A reader could test whether the obstruction is an artifact of the s-wave reduction by repeating the leading-order calculation in the full four-dimensional effective action; a surviving Schwarzschild–de Sitter solution would locate the problem in the approximation rather than in the Nariai/SdS identification.
The theorem motivates reformulating anti-evaporation searches gauge-invariantly: define horizon growth by geometric quantities such as apparent-horizon area or null geodesic expansion, rather than by changes in coordinate radius, since in these coordinates radius is not spacelike.

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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

The first thing to know: the paper's conceptual point is right, but its central theorem is not established by the printed calculation. The distinction between Nariai and extremal Kottler is real, and the authors are right that anti-evaporation claims built on identifying them deserve scrutiny. What is new is the claim that within Bousso–Hawking effective equations there is no Schwarzschild–de Sitter-like solution with the assumed asymptotic. That would be an important correction if it held. The paper reads the global-structure literature well, correctly notes the different scri limits, and is honest about the effective-action, s-wave, large-N caveats.

The soft spot is the appendix, and it is load-bearing. With the convention (14), for t-dependent fields (∂φ)² = -φdot², so the second term in (9) is +2φdot², not -2φdot² as written in (A1). Redoing the substitution with the sign fixed gives the leading r² coefficient A²(3B² - Λ)r², which vanishes exactly at B² = Λ/3—the Schwarzschild–de Sitter falloff the theorem intends to exclude. So the contradiction is an artifact of the sign error. I also see a mismatch between (A3) and (A4) that I did not chase; the sign issue is decisive on its own. As printed, the theorem is unsupported.

That said, the paper is still worth engaging. The geometric distinction is solid and supported by the cited work, and the conclusion that anti-evaporation may be a mathematical artifact could be right even if this particular derivation fails. There is no parameter fitting and no self-citation issue. The paper would be more effective as a short comment making the geometric point and arguing that previous stability analyses need revisiting; the no-go theorem should be either repaired or dropped. The right audience is people working on semiclassical black holes, Bousso–Hawking effective actions, or primordial black hole lifetimes. I would send it to a serious referee, but with the expectation of major revision. If the authors fix the sign and the theorem still fails, fine; if it does not fail, they need to say so in the revision.

Referee Report

2 major / 3 minor

Summary. The paper argues that the Nariai spacetime and the extremal Schwarzschild--de Sitter (Kottler) solution are distinct spacetimes, and that previous anti-evaporation studies which identify them are based on an incorrect assumption. After reviewing the geometry of both spacetimes, the paper states a theorem that no solution of the Bousso--Hawking effective equations (9)--(13) admits a Killing vector ∂_x and the large-radius asymptotic behavior (16) characteristic of Schwarzschild--de Sitter. From this it concludes that the effective action approach cannot describe Schwarzschild--de Sitter backreaction and that the anti-evaporation effect may be an artifact of the Nariai identification. The proof of the theorem is relegated to Appendix A.

Significance. If the theorem were correct, it would be an important caveat to the existing anti-evaporation literature and would support the paper's conceptual point that the Nariai and extremal Kottler spacetimes should not be conflated. The paper is also useful in emphasizing the global-structure differences between these spacetimes and in noting that perturbations which appear small near the horizon can become large asymptotically. However, the main technical result is not established: the appendix derivation contains a sign error that reverses the leading-order conclusion. The conceptual distinction alone is not enough to support the paper's strong no-go claim.

major comments (2)

[Appendix A, Eq. (A1)] The reduction from Eq. (9) to Eq. (A1) is inconsistent with the notation defined in Eq. (14). For fields depending only on t, (∂φ)² = φ'² − φ̇² = −φ̇², so the second term in Eq. (9) contributes +2φ̇² to Eq. (A1), not −2φ̇² as written. This sign error is then carried through Eq. (A3) into the leading r² coefficient in Eq. (A4).
[Section III, Theorem and Appendix A, Eq. (A4)] With the corrected sign, the leading r² term in the reduced equation is A²(3B² − Λ)r², not −A²(2B² + Λ)r². This term vanishes precisely at B² = Λ/3, which is the Schwarzschild--de Sitter falloff (16) with A² = B² = Λ/3. Therefore the claimed contradiction does not follow from the calculation. The theorem as stated is not proven, and the conclusions in Section III.A and Section IV that the effective action admits no Schwarzschild--de Sitter-like solution are unsupported. If the sign in (A1) is instead taken as authoritative, then either (9) or (14) would need correction; in either case the printed proof does not establish the theorem.

minor comments (3)

[Throughout] The manuscript contains several typographical errors, including 'frutiful disscusions', 'prelimnary', and 'empasize', which should be corrected.
[Appendix A, Eq. (A3)] The displayed equation (A3) is difficult to parse because the parentheses and derivative notation are not fully clear; please rewrite it with explicit definitions of all terms.
[Section II.C] The statement that the extremal Kottler spacetime has scri I⁻ ≅ S³ would benefit from a brief explanation or reference, since this is not obvious from the metric (2).

Circularity Check

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No circularity: the no-go theorem is derived from the Bousso–Hawking equations without parameter fitting, self-citations, or importing the target result.

full rationale

The paper's central claim is a no-go theorem in Section III: no solution exists to the Bousso–Hawking effective equations (9)–(13) with a Killing vector ∂x and the Schwarzschild–de Sitter asymptotic falloff (16). The derivation is self-contained: the field equations are taken from the literature [1], the reduction to the ansatz e^{−2φ} = r^2 with r and ρ depending on t only is stated explicitly, and the contradiction in Appendix A follows from substituting the assumed large-r behaviour g = Ar + O(r^{1−ε}), h = Br + O(r^{1−ε}) into the reduced equation. No free parameter is fitted to data, no later result is assumed in deriving the contradiction, and no load-bearing step is justified by a self-citation. The paper's strongest vulnerability is the apparent sign inconsistency in equation (A1) relative to the definitions (14); a sign correction would change the leading-order coefficient in (A4) and could invalidate the theorem. That is an algebraic correctness concern, not circularity. The ansatz and the characterization of the SdS boundary are assumptions, but they are not equivalent to the theorem they are used to prove. The paper is therefore not circular; its result is either a valid consequence of the effective equations or a miscalculation, but in neither case is it a disguised restatement of its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters and no invented entities. The paper's contribution is a no-go statement conditional on the adopted effective action and asymptotic ansatz. The main unresolved point is algebraic consistency of the proof, not the introduction of new structure.

assumptions (4)

domain assumption Equations (9)-(13) from Bousso and Hawking are the correct one-loop s-wave large-N effective equations for spherically symmetric semiclassical gravity.
The no-go theorem is proved inside this framework; the paper does not rederive or independently justify these equations.
domain assumption The asymptotic form (16) with constant A and B characterizes the conformal boundary of Schwarzschild-de Sitter spacetime.
Section III and Appendix A use this asymptotic to identify SdS-like solutions, citing [15] for the scri structure.
domain assumption A solution with Killing vector partial_x and with r and rho depending only on t covers the SdS background in the metric ansatz (8).
Section III assumes e^{-2 phi} = r^2 and all functions depend only on t; this excludes other coordinate or perturbation sectors.
standard math The classical Kottler metric (2) is the correct background and Nariai spacetime is its near-horizon limit.
Section II uses the classical geometry to argue the physical distinction between the two extremal limits.

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Cite this review

Pith. "Pith review of (Anti-)evaporation of Schwarzschild-de Sitter black holes revisited." pith.science (2026). https://pith.science/paper/PSIQZHBU

@misc{pith2026190801716,
  author       = {Pith},
  title        = {Pith review of: (Anti-)evaporation of Schwarzschild-de Sitter black holes revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSIQZHBU}},
  note         = {Machine review of arXiv:1908.01716}
}

read the original abstract

It is widely believed that in the presence of a positive cosmological constant, heavy black holes can exhibit non-standard behaviour, namely there is a possibility that such objects would grow instead of evaporating. We point out that all those results (obtained in different frameworks) rely heavily upon the identification of the Nariai spacetime with the Schwarzschild--de Sitter (Kottler) black hole. In this note we argue that it is an incorrect assumption. As a result, previous treatments need revisiting. In particular, we show that within effective action approach, there is no solution corresponding to the Schwarzschild--de Sitter black hole.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 8 canonical work pages

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Pith tools

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

[1] [1]

Bousso and S

R. Bousso and S. W. Hawking, Phys. Rev. D57, 2436 (1998), arXiv:hep-th/9709224 [hep-th]

arXiv 1998

[2] [2]

Nojiri and S

S. Nojiri and S. D. Odintsov, Int. J. Mod. Phys. A14, 1293 (1999), arXiv:hep-th/9802160 [hep-th]

arXiv 1999

[3] [3]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Rev. D59, 044026 (1999), arXiv:hep-th/9804033 [hep-th]

arXiv 1999

[4] [4]

Nojiri and S

S. Nojiri and S. D. Odintsov, Int. J. Mod. Phys. A16, 1015 (2001), arXiv:hep-th/0009202 [hep-th]

arXiv 2001

[5] [5]

Nojiri and S

S. Nojiri and S. D. Odintsov, Class. Quant. Grav. 30, 125003 (2013), arXiv:1301.2775 [hep-th]

arXiv 2013

[6] [6]

Evaporation and anti-evaporation instability of a Schwarzschild-de Sitter braneworld: the case of five-dimensional F(R) gravity

A. Addazi, S. Nojiri, and S. Odintsov, Phys. Rev. D95, 124020 (2017), arXiv:1705.03265 [gr-qc]

work page Pith review arXiv 2017

[7] [7]

D. V. Singh and N. K. Singh, Annals Phys. 383, 600 (2017), arXiv:1704.01831 [physics.gen-ph]

work page Pith review arXiv 2017

[8] [8]

Y. Fang, Z. Huang, H. Miao, and N. K. Singh, Phys. Rev. D99, 044011 (2019), arXiv:1811.11547 [gr-qc]. 9

work page Pith review arXiv 2019