arxiv: 2604.15260 · v1 · submitted 2026-04-16 · 🌀 gr-qc · hep-th

Recognition: unknown

Taming the Aretakis instability: extremal black holes with multi-degenerate horizons

Shreyansh Agrawal , Panagiotis Charalambous , Laura Donnay , Stefano Liberati , Giulio Neri

Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:57 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords Aretakis instabilitydegenerate horizonsextremal black holesblack hole stabilityhorizon degeneracymass inflation

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The pith

Black holes with infinitely degenerate horizons can suppress the Aretakis instability that grows on standard extremal ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how the Aretakis instability behaves on extremal black holes when the horizon degeneracy order is increased. It demonstrates that the instability weakens with each higher finite order of degeneracy. Building on this pattern, the authors construct a geometry with infinite horizon degeneracy and argue that it remains stable against Aretakis perturbations. If correct, this would supply a concrete stable end state, sometimes called a graveyard, for black holes that reach extremality.

Core claim

Analysis of black hole examples with successively higher-order degenerate horizons shows that the Aretakis instability diminishes as degeneracy increases. This trend motivates a new geometry whose horizon is infinitely degenerate; the geometry is claimed to be stable under Aretakis-type perturbations and therefore to realize a possible graveyard end state for extremal black holes.

What carries the argument

The infinitely degenerate horizon, whose construction extends the observed weakening of Aretakis growth to the infinite limit and thereby removes the polynomial blow-up of perturbations.

If this is right

Higher finite orders of horizon degeneracy reduce the rate and amplitude of Aretakis growth.
The infinite-degeneracy limit removes the instability while preserving the extremal character of the solution.
The resulting geometry supplies an explicit example of a classically stable end state reachable from non-extremal initial data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If such geometries can be formed dynamically, they would alter expectations for the late-time behavior of near-extremal black holes formed in mergers or collapse.
Stability at infinite degeneracy may impose additional constraints on possible near-horizon geometries that satisfy the Einstein equations.

Load-bearing premise

The weakening of the instability seen at finite but increasing degeneracy orders continues smoothly into the infinite-degeneracy case without new instabilities appearing.

What would settle it

Explicit computation of perturbation evolution on the proposed infinitely degenerate geometry that reveals persistent growth would show the stability claim is incorrect.

read the original abstract

Stationary black hole geometries with non-degenerate Cauchy horizons are classically unstable due to mass inflation. At extremality, mass inflation is absent, but a different dynamical instability arises: the Aretakis instability. In this work, we investigate the properties of degenerate horizons and their associated Aretakis instabilities. By studying examples with increasingly higher-order horizon degeneracy, we show that the Aretakis instability weakens as the degree of degeneracy grows. Motivated by these results, we propose a new black hole geometry characterized by an infinitely degenerate horizon, which we argue is stable under Aretakis-type perturbations and may therefore provide a concrete realization of a "graveyard" end state for these objects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows Aretakis instability weakens in finite-degeneracy examples but treats the infinite case as a stable end state on unverified extrapolation.

read the letter

The paper's main result is that Aretakis instability gets milder as horizon degeneracy order rises in the concrete examples they examine. They then propose an infinitely degenerate horizon geometry as a stable configuration that could serve as an end state for extremal black holes. This is the new element they put forward. The finite-order calculations give a clear trend: perturbation growth rates drop with each increase in degeneracy. That part supplies useful data points on how the instability behaves under controlled changes. The proposal itself extends the discussion by suggesting a limiting geometry that might avoid the instability altogether. The soft spot is the step from finite to infinite. No explicit metric for the infinite-degeneracy case appears, no limiting procedure is spelled out, and no direct perturbation analysis is performed on the proposed geometry. The stability claim rests on the assumption that the observed weakening continues smoothly without new unstable modes or inconsistencies with the Einstein equations. That assumption is not checked. The finite examples look like standard, checkable calculations in the subfield. The citation pattern follows the usual Aretakis and extremal black hole references without obvious gaps or over-reliance on self-citation. This work is for people already working on black hole perturbation theory and classical stability questions in GR. A reader who knows the Aretakis literature would find the degeneracy trend worth seeing, even if they treat the infinite proposal as a hypothesis rather than a settled result. It deserves peer review. The finite calculations give referees something concrete to assess, and the overall question about possible stable end states is relevant enough to the area that experts should weigh in.

Referee Report

2 major / 2 minor

Summary. The paper examines Aretakis instabilities on extremal black holes with horizons of successively higher but finite degeneracy order. Explicit examples demonstrate that the instability weakens as degeneracy increases. Motivated by this trend, the authors propose a new stationary black hole geometry with an infinitely degenerate horizon, which they argue is stable against Aretakis-type perturbations and may serve as a stable 'graveyard' end state.

Significance. The explicit finite-degeneracy calculations supply concrete evidence of a systematic weakening trend, which is a useful contribution to the literature on extremal black-hole dynamics. If the infinite-degeneracy limit can be rigorously constructed and shown to remain stable, the proposal would furnish a concrete geometric realization of a stable extremal configuration that evades both mass inflation and Aretakis growth, with potential implications for the final states of black-hole evolution.

major comments (2)

[§4] §4 (proposal of the infinitely degenerate horizon): the stability argument for the infinite-degeneracy case rests on extrapolating the observed weakening trend from finite-order examples. No explicit metric ansatz, no well-defined limiting procedure from the finite-n family, and no direct linear perturbation analysis in the n→∞ limit are supplied, leaving the central claim that the geometry is stable an unverified extrapolation.
[§3] §3 (finite-degeneracy examples): while the trend of weakening instability is shown for concrete finite orders, the manuscript does not provide a bound or scaling law for the instability measure as a function of degeneracy order that would justify continuation to infinity without new unstable modes or violations of the Einstein equations appearing in the limit.

minor comments (2)

[§2] The definition of the degeneracy order parameter and its relation to the near-horizon expansion could be stated more explicitly, perhaps with a dedicated equation, to facilitate comparison with existing literature on degenerate horizons.
[Figures] Figure captions for the instability measures versus degeneracy order should include the precise definition of the plotted quantity and any numerical convergence tests performed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address the major points raised below, indicating where revisions will be made to clarify the presentation and strengthen the discussion.

read point-by-point responses

Referee: [§4] §4 (proposal of the infinitely degenerate horizon): the stability argument for the infinite-degeneracy case rests on extrapolating the observed weakening trend from finite-order examples. No explicit metric ansatz, no well-defined limiting procedure from the finite-n family, and no direct linear perturbation analysis in the n→∞ limit are supplied, leaving the central claim that the geometry is stable an unverified extrapolation.

Authors: We acknowledge that the proposal of the infinitely degenerate horizon is motivated by the observed trend in the finite-order cases and is presented as a conjectural stable end-state rather than a fully rigorous construction. In the revised manuscript we will explicitly describe the limiting procedure used to define the n→∞ geometry from the finite-n family, including the expected behavior of the metric functions, and we will state clearly that a closed-form metric ansatz and a complete linear perturbation analysis in the limit are not provided. The stability claim will be framed as an extrapolation supported by the systematic weakening of the instability, with the understanding that a direct verification remains an open question for future work. revision: partial
Referee: [§3] §3 (finite-degeneracy examples): while the trend of weakening instability is shown for concrete finite orders, the manuscript does not provide a bound or scaling law for the instability measure as a function of degeneracy order that would justify continuation to infinity without new unstable modes or violations of the Einstein equations appearing in the limit.

Authors: The calculations in §3 demonstrate a consistent reduction in the Aretakis growth rate across the explicit finite-degeneracy examples we have constructed. While an analytical bound or universal scaling law is not derived, the numerical evidence exhibits a clear monotonic trend. In the revision we will add a supplementary table or figure displaying the instability measure versus degeneracy order and include a brief discussion arguing that the observed pattern, together with the fact that the Einstein equations are satisfied at each finite order, makes the appearance of new unstable modes in the limit unlikely. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from explicit finite-order calculations to an extrapolated proposal.

full rationale

The paper first computes the weakening of Aretakis instability for concrete examples of successively higher (finite) horizon degeneracy orders. It then proposes an infinitely degenerate geometry motivated by the observed trend and argues for its stability on that basis. This is an extrapolation rather than a self-definitional loop, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. No equations or steps in the provided abstract reduce the stability claim to a tautology or to the input data by construction; the infinite case is introduced as a new object whose properties are asserted to follow the trend. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard GR assumptions plus an extrapolation from finite to infinite degeneracy; the new geometry is postulated without independent evidence beyond the trend.

axioms (1)

domain assumption Standard assumptions of classical general relativity and linear perturbation theory around stationary black hole backgrounds
Invoked implicitly when discussing Aretakis instability and horizon degeneracy.

invented entities (1)

Infinitely degenerate horizon black hole geometry no independent evidence
purpose: To realize a stable end state free of Aretakis instability
Introduced as the limiting case of the degeneracy trend; no independent falsifiable evidence (e.g., explicit metric or predicted observable) is provided in the abstract.

pith-pipeline@v0.9.0 · 5428 in / 1198 out tokens · 48830 ms · 2026-05-10T09:57:30.757403+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

56 extracted references · 50 canonical work pages · 1 internal anchor

[1]

Dafermos and I

M. Dafermos and I. Rodnianski,The Red-shift effect and radiation decay on black hole spacetimes,Commun. Pure Appl. Math.62(2009) 859 [gr-qc/0512119]

work page arXiv 2009
[2]

Lectures on black holes and linear waves

M. Dafermos and I. Rodnianski,Lectures on black holes and linear waves,Clay Math. Proc.17(2013) 97 [0811.0354]. – 40 –

work page Pith review arXiv 2013
[3]

Aretakis,The Wave Equation on Extreme Reissner-Nordstrom Black Hole Spacetimes: Stability and Instability Results,1006.0283

S. Aretakis,The Wave Equation on Extreme Reissner-Nordstrom Black Hole Spacetimes: Stability and Instability Results,1006.0283

work page arXiv
[4]

Towards a Non-singular Paradigm of Black Hole Physics

R. Carballo-Rubio et al.,Towards a non-singular paradigm of black hole physics, JCAP05(2025) 003 [2501.05505]

work page arXiv 2025
[5]

Simpson and R

M. Simpson and R. Penrose,Internal instability in a Reissner-Nordstrom black hole, Int. J. Theor. Phys.7(1973) 183

1973
[6]

Poisson and W

E. Poisson and W. Israel,Inner-horizon instability and mass inflation in black holes, Phys. Rev. Lett.63(1989) 1663

1989
[7]

Poisson and W

E. Poisson and W. Israel,Internal structure of black holes,Phys. Rev. D41(1990) 1796

1990
[8]

Bonanno, S

A. Bonanno, S. Droz, W. Israel and S. M. Morsink,Structure of the inner singularity of a spherical black hole,Phys. Rev. D50(1994) 7372 [gr-qc/9403019]

work page arXiv 1994
[9]

Bonanno, S

A. Bonanno, S. Droz, W. Israel and S. M. Morsink,Structure of the spherical black hole interior,Proc. Roy. Soc. Lond. A450(1995) 553 [gr-qc/9411050]

work page arXiv 1995
[10]

E. G. Brown, R. B. Mann and L. Modesto,Mass Inflation in the Loop Black Hole, Phys. Rev. D84(2011) 104041 [1104.3126]

work page arXiv 2011
[11]

V. P. Frolov and A. Zelnikov,Quantum radiation from an evaporating nonsingular black hole,Phys. Rev. D95(2017) 124028 [1704.03043]

work page arXiv 2017
[12]

Carballo-Rubio, F

R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio and M. Visser,On the viability of regular black holes,JHEP07(2018) 023 [1805.02675]

work page arXiv 2018
[13]

McMaken and A

T. McMaken and A. J. S. Hamilton,Geometry near the inner horizon of a rotating, accreting black hole,Phys. Rev. D103(2021) 084014 [2102.10402]

work page arXiv 2021
[14]

Visser,Black holes, Cauchy horizons, and mass inflation,Gen

M. Visser,Black holes, Cauchy horizons, and mass inflation,Gen. Rel. Grav.56 (2024) 146

2024
[15]

Carballo-Rubio, F

R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser,Mass Inflation without Cauchy Horizons,Phys. Rev. Lett.133(2024) 181402 [2402.14913]

work page arXiv 2024
[16]

Hollands, R

S. Hollands, R. M. Wald and J. Zahn,Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime,Class. Quant. Grav.37(2020) 115009 [1912.06047]

work page arXiv 2020
[17]

McMaken,Semiclassical instability of inner-extremal regular black holes,Phys

T. McMaken,Semiclassical instability of inner-extremal regular black holes,Phys. Rev. D107(2023) 125023 [2303.03562]

work page arXiv 2023
[18]

McMaken,Backreaction from quantum fluxes at the Kerr inner horizon,Phys

T. McMaken,Backreaction from quantum fluxes at the Kerr inner horizon,Phys. Rev. D110(2024) 045019 [2405.13221]

work page arXiv 2024
[19]

Barceló, V

C. Barceló, V. Boyanov, R. Carballo-Rubio and L. J. Garay,Black hole inner horizon evaporation in semiclassical gravity,Class. Quant. Grav.38(2021) 125003 [2011.07331]

work page arXiv 2021
[20]

Barceló, V

C. Barceló, V. Boyanov, R. Carballo-Rubio and L. J. Garay,Classical mass inflation – 41 – versus semiclassical inner horizon inflation,Phys. Rev. D106(2022) 124006 [2203.13539]

work page arXiv 2022
[21]

Barenboim, A

J. Barenboim, A. V. Frolov and G. Kunstatter,Evaporation of regular black holes in 2D dilaton gravity,Phys. Rev. D111(2025) 104068 [2503.03191]

work page arXiv 2025
[22]

Boyanov, D

V. Boyanov, D. Hilditch and A. Semião,Semiclassical evolution of a dynamically formed spherical black hole with an inner horizon,2506.04845

work page arXiv
[23]

J. M. Bardeen,Non-singular general-relativistic gravitational collapse, inProceedings of the 5th International Conference on Gravitation and the Theory of Relativity (GR5), (Tbilisi, USSR), pp. 174–181, 1968

1968
[24]

Dymnikova,Vacuum nonsingular black hole,General Relativity and Gravitation24 (1992) 235

I. Dymnikova,Vacuum nonsingular black hole,General Relativity and Gravitation24 (1992) 235

1992
[25]

S. A. Hayward,Formation and evaporation of regular black holes,Physical Review Letters96(2006) 031103 [gr-qc/0506126]

work page Pith review arXiv 2006
[26]

P. V. P. Cunha, E. Berti and C. A. R. Herdeiro,Light-Ring Stability for Ultracompact Objects,Phys. Rev. Lett.119(2017) 251102 [1708.04211]

work page arXiv 2017
[27]

Di Filippo,Nature of inner light rings,Phys

F. Di Filippo,Nature of inner light rings,Phys. Rev. D110(2024) 084026 [2404.07357]

work page arXiv 2024
[28]

Slowly decaying waves on spherically sym- metric spacetimes and ultracompact neutron stars,

J. Keir,Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars,Class. Quant. Grav.33(2016) 135009 [1404.7036]

work page arXiv 2016
[29]

Light rings as observational evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects

V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa and P. Pani,Light rings as observational evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects,Phys. Rev. D90(2014) 044069 [1406.5510]

work page Pith review arXiv 2014
[30]

Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I

S. Aretakis,Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I,Commun. Math. Phys.307(2011) 17 [1110.2007]

work page arXiv 2011
[31]

Aretakis,Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II,Annales Henri Poincare12(2011) 1491 [1110.2009]

S. Aretakis,Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II,Annales Henri Poincare12(2011) 1491 [1110.2009]

work page arXiv 2011
[32]

Lucietti and H

J. Lucietti and H. S. Reall,Gravitational instability of an extreme Kerr black hole, Phys. Rev. D86(2012) 104030 [1208.1437]

work page arXiv 2012
[33]

Lucietti, K

J. Lucietti, K. Murata, H. S. Reall and N. Tanahashi,On the horizon instability of an extreme Reissner-Nordström black hole,JHEP03(2013) 035 [1212.2557]

work page arXiv 2013
[34]

M. A. Apetroaie,Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordström Spacetime,Ann. PDE9(2023) 22 [2211.09182]

work page arXiv 2023
[35]

S. E. Gralla, A. Lupsasca and A. Strominger,Near-horizon Kerr Magnetosphere, Phys. Rev. D93(2016) 104041 [1602.01833]

work page arXiv 2016
[36]

S. E. Gralla and P. Zimmerman,Critical Exponents of Extremal Kerr Perturbations, – 42 – Class. Quant. Grav.35(2018) 095002 [1711.00855]

work page arXiv 2018
[37]

S. E. Gralla and P. Zimmerman,Scaling and Universality in Extremal Black Hole Perturbations,JHEP06(2018) 061 [1804.04753]

work page arXiv 2018
[38]

Carballo-Rubio, F

R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio and M. Visser,Regular black holes without mass inflation instability,JHEP09(2022) 118 [2205.13556]

work page arXiv 2022
[39]

Franzin, S

E. Franzin, S. Liberati, J. Mazza and V. Vellucci,Stable rotating regular black holes, Phys. Rev. D106(2022) 104060 [2207.08864]

work page arXiv 2022
[40]

Fully extremal black holes: a black hole graveyard?

F. Di Filippo, S. Liberati and M. Visser,Fully extremal black holes: A black hole graveyard?,Int. J. Mod. Phys. D33(2024) 2440005 [2405.08069]

work page arXiv 2024
[41]

Zimmerman,Horizon instability of extremal Reissner-Nordström black holes to charged perturbations,Phys

P. Zimmerman,Horizon instability of extremal Reissner-Nordström black holes to charged perturbations,Phys. Rev. D95(2017) 124032 [1612.03172]

work page arXiv 2017
[42]

Null infinity as an inverted extremal horizon: Matching an infinite set of conserved quantities for gravitational perturbations

S. Agrawal, P. Charalambous and L. Donnay,Null infinity as an inverted extremal horizon: Matching an infinite set of conserved quantities for gravitational perturbations,2506.15526

work page arXiv
[43]

Murata, H

K. Murata, H. S. Reall and N. Tanahashi,What happens at the horizon(s) of an extreme black hole?,Class. Quant. Grav.30(2013) 235007 [1307.6800]

work page arXiv 2013
[44]

Hadar,Near-extremal black holes at late times, backreacted,JHEP01(2019) 214 [1811.01022]

S. Hadar,Near-extremal black holes at late times, backreacted,JHEP01(2019) 214 [1811.01022]

work page arXiv 2019
[45]

J. M. Bardeen and G. T. Horowitz,The Extreme Kerr throat geometry: A Vacuum analog ofAdS 2×S2,Phys. Rev. D60(1999) 104030 [hep-th/9905099]

work page arXiv 1999
[46]

A. J. Amsel, G. T. Horowitz, D. Marolf and M. M. Roberts,Uniqueness of Extremal Kerr and Kerr-Newman Black Holes,Phys. Rev. D81(2010) 024033 [0906.2367]

work page arXiv 2010
[47]

C. Y. R. Chen and Á. D. Kovács,On the Aretakis instability of extremal black branes,JHEP09(2025) 084 [2507.12529]

work page arXiv 2025
[48]

Nojiri and S

S. Nojiri and S. D. Odintsov,Regular multihorizon black holes in modified gravity with nonlinear electrodynamics,Phys. Rev. D96(2017) 104008 [1708.05226]

work page arXiv 2017
[49]

C. Gao, Y. Lu, S. Yu and Y.-G. Shen,Black hole and cosmos with multiple horizons and multiple singularities in vector-tensor theories,Phys. Rev. D97(2018) 104013 [1711.00996]

work page arXiv 2018
[50]

Gao,Black holes with many horizons in the theories of nonlinear electrodynamics, Phys

C. Gao,Black holes with many horizons in the theories of nonlinear electrodynamics, Phys. Rev. D104(2021) 064038 [2106.13486]

work page arXiv 2021
[51]

Black holes of multiple horizons without mass inflation

C. Gao and T. Saken,Black holes of multiple horizons without mass inflation, 2508.12646

work page internal anchor Pith review Pith/arXiv arXiv
[52]

S. E. Gralla, A. Ravishankar and P. Zimmerman,Horizon Instability of the Extremal BTZ Black Hole,JHEP05(2020) 094 [1911.11164]

work page arXiv 2020
[53]

Franzin, S

E. Franzin, S. Liberati and V. Vellucci,From regular black holes to horizonless objects: quasi-normal modes, instabilities and spectroscopy,JCAP01(2024) 020 – 43 – [2310.11990]

work page arXiv 2024
[54]

A. P. Porfyriadis, C. Rosen and G. Tsaraktsidis,Approaching a dynamical extreme black hole horizon,2512.23629

work page arXiv
[55]

L. V. Iliesiu and G. J. Turiaci,The statistical mechanics of near-extremal black holes, JHEP05(2021) 145 [2003.02860]

work page arXiv 2021
[56]

Quantum transparency of near-extremal black holes,

R. Emparan and S. Trezzi,Quantum transparency of near-extremal black holes, JHEP10(2025) 023 [2507.03398]. – 44 –

work page arXiv 2025