arxiv: 2605.16095 · v1 · pith:CLMEJZACnew · submitted 2026-05-15 · 🌀 gr-qc · math-ph· math.AP· math.MP

Nonlinear stability of continuously self-similar naked singularities for the Einstein-scalar field equations II: linearized stability

Jaydeep Singh , Weihao Zheng This is my paper

Pith reviewed 2026-05-20 17:12 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.APmath.MP

keywords naked singularitiesself-similar solutionsEinstein-scalar field equationslinearized stabilityblue-shift instabilityspherical symmetrygeneral relativity

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The pith

Continuously self-similar naked singularities are linearly stable under perturbations of matching C^{1,α} regularity.

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

This paper proves linearized stability for a one-parameter family of continuously self-similar C^{1,α} naked singularity solutions to the spherically symmetric Einstein-scalar field equations. These solutions, originally constructed by Christodoulou, are known to be unstable under rougher perturbations because of a blue-shift effect near the singularity. The new analysis shows that when the perturbations are restricted to the same regularity class as the background, the linearized equations admit no growing modes. The result supplies the linear foundation for the nonlinear stability theorem proved in the companion paper and demonstrates that regularity controls whether the blue-shift mechanism can operate.

Core claim

The paper claims that the linearized Einstein-scalar field equations around these C^{1,α} self-similar naked singularities possess no unstable modes when the perturbations of the metric and scalar field are also taken to be C^{1,α}. This establishes that the blue-shift instability mechanism is not triggered at the regularity level of the background spacetime itself.

What carries the argument

The linearized perturbation equations in self-similar coordinates, analyzed within C^{1,α} function spaces under spherical symmetry.

If this is right

Linearized stability at matching regularity supplies the starting point for the nonlinear stability result in the companion paper.
The blue-shift instability requires perturbations of strictly lower regularity than the background solutions.
Within the C^{1,α} class these naked singularities do not exhibit linear instability.

Where Pith is reading between the lines

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

The same regularity threshold may suppress instabilities in other self-similar gravitational solutions.
Nonlinear evolution could in principle generate lower-regularity modes that later activate the blue-shift mechanism.
This distinction between regularity levels suggests that numerical simulations using sufficiently smooth data might miss the instability seen with rougher initial data.

Load-bearing premise

Perturbations are required to have exactly the same C^{1,α} regularity as the background and to preserve spherical symmetry.

What would settle it

An explicit C^{1,α} perturbation that produces an exponentially growing solution to the linearized system would disprove the stability claim.

Figures

Figures reproduced from arXiv: 2605.16095 by Jaydeep Singh, Weihao Zheng.

Figure 1. Figure 1: Penrose diagram for naked singularity spacetimes [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

Figure 2. Figure 2: Backward scattering Step 2: Scattering theory We consider the equation (3.11) with prescribed initial data on {s = 0, −1 ≤ z ≤ 0} in the function space C α with α > pk. Since the solution is only defined on the upper half plane s ≥ 0, to extend the solution to the lower half plane and use the standard theory of the Fourier–Laplace transform, we commute the equation (3.11) with a cut-off function supported … view at source ↗

Figure 3. Figure 3: Oriented contours in the complex plane 3.4.2 The role of the energy estimate Although we have mentioned that it is impossible to prove a good upper bound for Ψ by only using the energy estimate without excluding the trivial mode solution in Section 3.2, the vector field method still plays a role here. First, even at the level of the linear wave equation, the scattering approach emphasized in the previous s… view at source ↗

Figure 4. Figure 4: Oriented contours Then, by the residue theorem in complex analysis, since ρ(z)R(σ)(∂zF1, ∂zF2) is holomorphic in σ for σ ∈ I(−β∗,−η]\{−1}, we have ρ(z) rp Ψp (z) = 1 2π Z PR e σsρ(z)R(σ)(∂zF1, ∂zF2)dσ + 1 2π Z Γϵ e σsρ(z)R(σ)(∂zF1, ∂zF2)dσ, (6.20) for any R > 0 and any ϵ sufficiently small. 84 [PITH_FULL_IMAGE:figures/full_fig_p084_4.png] view at source ↗

read the original abstract

This is the second part of a series of papers proving the nonlinear stability of a one-parameter family of continuous self-similar $C^{1,\alpha}$ naked singularity solutions, with $0<\alpha\ll1$, to the spherically symmetric Einstein-scalar field equations. These solutions were constructed by Christodoulou and are known to be unstable under sufficiently rough perturbations due to the blue-shift instability mechanism. In complete contrast to the previous instability results, we establish the linearized stability for those naked singularity spacetimes under perturbations of the same regularity as the background, revealing the central role of regularity in determining the strength of the blue-shift instability mechanism, and showing that it is not triggered at the regularity level of the background spacetime. The linear analysis carried out in this paper provides the foundation for the nonlinear stability result established in the companion paper [W. Zheng, Nonlinear stability of the continuous self-similar naked singularities for the Einstein-scalar field equations I: main results]. Together with that companion paper, this yields the nonlinear stability of these continuously self-similar naked singularities.

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

Linearized stability at the background's own C^{1,α} regularity for these naked singularities, which isolates how the blue-shift depends on perturbation smoothness.

read the letter

The main thing to know is that this paper shows linearized stability for Christodoulou's continuously self-similar naked singularities when perturbations match the background regularity exactly. That result is new and reverses the instability picture from earlier work on rougher data, pinning the blue-shift mechanism to regularity level rather than the background itself. It also supplies the linear step needed for the nonlinear stability claim in the companion paper. The analysis stays within spherical symmetry and works directly with the C^{1,α} class for small α. This cleanly separates the role of regularity without relying on the prior instability constructions. The spherical symmetry reduction likely lets them close estimates even with limited smoothness. The soft spot is the well-posedness question for the linearized wave system. The coefficients end up only C^α, and standard energy methods for hyperbolic equations usually require C^1 coefficients to control commutators without derivative loss. The paper does not appear to include a separate well-posedness theorem for this exact Hölder class, so the stability conclusion rests on whether their estimates work anyway under the symmetry assumption. That is a concrete point to check rather than a fatal flaw. This paper is for people already following the mathematical GR literature on self-similar solutions and cosmic censorship. A reader who knows the Christodoulou backgrounds and the rough-perturbation instability results will see the value in the regularity distinction. It deserves a serious referee because the claim is precise, the linear foundation is explicit, and the topic connects to larger questions about naked singularities. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper claims to prove linearized stability of a one-parameter family of continuously self-similar C^{1,α} (0<α≪1) naked singularity solutions to the spherically symmetric Einstein-scalar field equations, under perturbations of identical regularity. The analysis is performed in spherical symmetry and is presented as the linear foundation for the nonlinear stability result in the companion paper.

Significance. If the linearized stability result holds, it isolates the role of regularity in suppressing the blue-shift instability mechanism, showing that the instability is not activated at the precise Hölder regularity of the background. This supplies a concrete, regularity-dependent criterion that could inform future work on the stability of naked singularities and the cosmic censorship conjecture.

major comments (1)

[Section 4 (linearized equations and energy estimates)] The background metric and scalar field are only C^{1,α}. Consequently the linearized wave operator has coefficients whose first derivatives lie merely in C^α. Standard energy estimates for hyperbolic systems on Lorentzian backgrounds require at least C^1 coefficients to control commutators and close the estimates without derivative loss. No independent well-posedness theorem or reference for the precise Hölder class C^α is supplied in the linear analysis; this gap is load-bearing for the stability statement.

minor comments (2)

The abstract states the main result but does not indicate the principal technical tools (e.g., choice of weighted energy norms or multiplier vector fields) used to obtain the decay estimates.
[Section 2] Notation for the perturbation variables and the precise function spaces (e.g., the precise definition of the C^{1,α} norm) should be introduced earlier and used consistently throughout.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive report. The major comment raises a valid technical point about coefficient regularity in the linearized analysis. We address it directly below and will strengthen the exposition accordingly.

read point-by-point responses

Referee: [Section 4 (linearized equations and energy estimates)] The background metric and scalar field are only C^{1,α}. Consequently the linearized wave operator has coefficients whose first derivatives lie merely in C^α. Standard energy estimates for hyperbolic systems on Lorentzian backgrounds require at least C^1 coefficients to control commutators and close the estimates without derivative loss. No independent well-posedness theorem or reference for the precise Hölder class C^α is supplied in the linear analysis; this gap is load-bearing for the stability statement.

Authors: We appreciate the referee drawing attention to this regularity issue. Because the background solutions are continuously self-similar, the linearized system is written in similarity coordinates in which all coefficients depend only on the single similarity variable (plus the fixed spherical symmetry). This structure permits direct derivation of the energy identities by integration by parts along the similarity foliation, with all error terms controlled by the explicit C^α modulus of continuity of the background rather than by general commutator estimates. The estimates in Section 4 are therefore self-contained and do not invoke a general well-posedness theorem for arbitrary C^α Lorentzian metrics. We will add a short clarifying paragraph at the beginning of Section 4 that spells out this reduction and the precise way the Hölder regularity is used, together with a brief reference to existing low-regularity energy estimates for symmetric hyperbolic systems. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to companion paper; linearized analysis remains independent

full rationale

The paper cites Christodoulou's external prior construction for the C^{1,α} background solutions and performs its own linearized stability analysis around them under spherical symmetry and matching regularity. The only self-reference is to the companion paper by co-author Zheng, which this work is stated to support rather than presuppose; no load-bearing premise reduces to that citation or to any fitted input. The derivation chain for the linearized result is therefore self-contained against the external benchmark of Christodoulou's solutions, yielding only a minor self-citation that does not affect the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior existence of the Christodoulou background solutions and the restriction to spherical symmetry; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Spherical symmetry of the spacetime and of the perturbations
The paper works exclusively within the spherically symmetric Einstein-scalar field system as stated in the abstract.
domain assumption Existence and basic properties of the one-parameter family of C^{1,α} continuous self-similar naked singularities constructed by Christodoulou
The stability analysis is performed around these pre-existing background solutions.

pith-pipeline@v0.9.0 · 5728 in / 1388 out tokens · 87954 ms · 2026-05-20T17:12:50.205844+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We establish the linearized stability for those naked singularity spacetimes under perturbations of the same regularity as the background... scattering theory for the linearized Einstein-scalar field equations
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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Relation between the paper passage and the cited Recognition theorem.

Construction of outgoing solutions to L p = 0 via Volterra iteration and recurrence relations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

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Xinliang An. Naked singularity censoring with anisotropic apparent horizon.Ann. Math., to appear

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Demetrios Christodoulou. Bounded variation solutions of the spherically symmetric Einstein-scalar field equations.Comm. Pure Appl. Math., 46(8):1131–1220, 1993

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Examples of naked singularity formation in the gravitational collapse of a scalar field.Ann

Demetrios Christodoulou. Examples of naked singularity formation in the gravitational collapse of a scalar field.Ann. Math., 140(3):607–653, 1994

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The instability of naked singularities in the gravitational collapse of a scalar field.Ann

Demetrios Christodoulou. The instability of naked singularities in the gravitational collapse of a scalar field.Ann. Math., 149(1):183–217, 1999

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American Math- ematical Society, 2019

Semyon Dyatlov and Maciej Zworski.Mathematical Theory of Scattering Resonances. American Math- ematical Society, 2019

work page 2019
[6]

Interior instability of naked singularities of a scalar field.arXiv preprint arXiv:2508.07655, 2025

Junbin Li. Interior instability of naked singularities of a scalar field.arXiv preprint arXiv:2508.07655, 2025

work page arXiv 2025
[7]

Instability of spherical naked singularities of a scalar field under gravitational perturbations.J

Junbin Li and Jue Liu. Instability of spherical naked singularities of a scalar field under gravitational perturbations.J. Diff. Geom., 120(1):97–197, 2022

work page 2022
[8]

A robust proof of the instability of naked singularities of a scalar field in spherical symmetry.Comm

Jue Liu and Junbin Li. A robust proof of the instability of naked singularities of a scalar field in spherical symmetry.Comm. Math. Phys., 363:561–578, 2018

work page 2018
[9]

A construction of approximately self-similar naked singularities for the spherically symmetric Einstein-scalar field system.Ann

Jaydeep Singh. A construction of approximately self-similar naked singularities for the spherically symmetric Einstein-scalar field system.Ann. Henri Poincar´ e, 2024

work page 2024
[10]

High regularity waves on self-similar naked singularity interiors: decay and the role of blue-shift, 2024

Jaydeep Singh. High regularity waves on self-similar naked singularity interiors: decay and the role of blue-shift, 2024

work page 2024
[11]

Regimes of (in-)stability for self-similar naked singularities.PhD thesis, 2025

Jaydeep Singh. Regimes of (in-)stability for self-similar naked singularities.PhD thesis, 2025

work page 2025
[12]

Nonlinear stability of continuously self-similar naked singularities for the Einstein-scalar field equations I: main results

Weihao Zheng. Nonlinear stability of continuously self-similar naked singularities for the Einstein-scalar field equations I: main results. 88

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