arxiv: 2605.16235 · v1 · pith:OGUKBF5Snew · 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 I: main results

Weihao Zheng This is my paper

Pith reviewed 2026-05-20 16:35 UTC · model grok-4.3

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

keywords naked singularitiesEinstein-scalar field equationsnonlinear stabilityself-similar solutionsweak cosmic censorshipHölder topologyspherical symmetry

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The pith

Continuously self-similar naked singularities remain stable under small perturbations of the same Hölder regularity as the background.

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

The paper establishes the nonlinear stability of a one-parameter family of continuously self-similar naked singularity solutions to the spherically symmetric Einstein-scalar field equations. These C^{1,α} solutions, for small positive α, were previously shown to be unstable to black hole formation under rough perturbations. Here they are proven stable when the initial data perturbations lie in a small neighborhood measured in a localized Hölder topology. The argument relies on a linearized stability result from the companion paper. The outcome indicates that whether these naked singularities violate weak cosmic censorship depends on the precise function space chosen for the perturbations.

Core claim

The central claim is that the one-parameter family of continuously self-similar C^{1,α} naked singularity spacetimes, with 0 < α ≪ 1, is nonlinearly stable to the spherically symmetric Einstein-scalar field equations under general perturbations of matching regularity lying in a small open neighborhood of the background data in the localized Hölder topology.

What carries the argument

The continuously self-similar naked singularity solutions together with the localized Hölder topology used to control small perturbations around the background data.

If this is right

Naked singularities persist rather than collapsing into black holes for all sufficiently small perturbations of the given regularity.
Weak cosmic censorship fails to hold inside this Hölder regularity class.
The stability conclusion is restricted to small values of the parameter α.
The functional framework chosen for the perturbations determines whether cosmic censorship is verified or violated.

Where Pith is reading between the lines

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

The same stability mechanism might extend to other matter models or to non-spherically symmetric settings with analogous regularity.
Numerical evolution of the perturbation equations for concrete small α could provide independent checks of the analytic result.
Reformulating cosmic censorship statements with explicit regularity thresholds may clarify apparent contradictions between different instability and stability theorems.

Load-bearing premise

Linearized stability must hold and the perturbations must remain small enough in the localized Hölder topology when α is sufficiently small.

What would settle it

An explicit initial perturbation inside the small Hölder neighborhood that produces a trapped surface and black hole formation instead of preserving the naked singularity would falsify the stability statement.

Figures

Figures reproduced from arXiv: 2605.16235 by 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: The numerical expectation for initial data of the form [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Figure 3. Figure 3: The rigorous result for initial data of the form [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figure 4. Figure 4: The rigorous result for initial perturbations of the form [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

This is the first part of a series of papers proving the nonlinear stability of a one-parameter family of continuously self-similar $C^{1,\alpha}$ naked singularity solutions, with $0<\alpha\ll1$, to the spherically symmetric Einstein-scalar field equations. The stability holds for initial perturbations lying in a small open neighborhood of the data generating these naked singularity solutions, measured in a localized H\"older topology. These continuously self-similar naked singularity spacetimes were previously constructed by Christodoulou [D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. of Math. 140 (1994), 607--653], who also proved their instability to black hole formation under sufficiently rough perturbations [D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. 149 (1999), 183--217], thereby verifying weak cosmic censorship within a rough functional framework. In complete contrast, in this paper, we obtain the first nonlinear stability of these naked singularity spacetimes under general perturbations of the same regularity as the background. We rely on the linearized stability result established in the companion paper [J. Singh and W. Zheng, Nonlinear stability of continuously self-similar naked singularities for the Einstein--scalar field equations II: linearized stability]. Our result underscores the decisive role of the functional framework in formulating the Weak Cosmic Censorship conjecture.

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

This paper claims the first nonlinear stability for Christodoulou's self-similar naked singularities under same-regularity perturbations, but it rests on the companion paper's linear analysis and is restricted to small alpha.

read the letter

This paper claims the first nonlinear stability for Christodoulou's self-similar naked singularities under same-regularity perturbations, but it rests on the companion paper's linear analysis and is restricted to small alpha. The main result is that for small enough alpha these C^{1,alpha} solutions stay stable when you add small perturbations measured in the localized Holder topology. That is the contrast with Christodoulou's earlier work, which found instability once the perturbations were rougher. The paper reduces the nonlinear Einstein-scalar system to a perturbation equation and uses linear decay estimates to absorb the quadratic and higher terms. The small-alpha regime is key for closing those estimates. The setup and the statement of the theorem are laid out clearly, and the dependence on the companion linearized stability paper is stated up front. The work gives proper credit to the original constructions and the rough-perturbation instability results. The central limitation is that nothing here verifies the linear spectral properties independently. If the companion paper has a zero mode or a resonance that the weighted Holder space does not control, the bootstrap for the nonlinear estimates will not close. The result is also local in alpha and in the size of the perturbation, which is honest but narrows the scope. This is for specialists in mathematical general relativity who follow self-similar solutions and the weak cosmic censorship conjecture. A reader who already knows the Christodoulou papers will see the point about how the function space changes the stability picture. The paper shows structured thinking and direct engagement with the prior literature, so it deserves a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript is the first part of a series on the nonlinear stability of a one-parameter family of continuously self-similar C^{1,α} naked singularity solutions (0 < α ≪ 1) to the spherically symmetric Einstein-scalar field equations. The central claim is that these backgrounds are stable under sufficiently small perturbations lying in a small open neighborhood in the localized Hölder topology, in contrast to Christodoulou's instability results for rougher data. The result is obtained by reducing the nonlinear system to a perturbation equation controlled by the linearized operator analyzed in the companion paper.

Significance. If established, the result would be the first nonlinear stability theorem for these naked singularities under perturbations of matching regularity, underscoring that the formulation of weak cosmic censorship depends on the choice of function space. The work builds directly on Christodoulou's constructions and instability theorems, providing a clear example of how a refined functional framework can yield stability where coarser norms do not.

major comments (2)

[§1] §1 (main theorem statement): The nonlinear stability claim is obtained by reducing to the linearized operator from the companion paper, but this manuscript provides no explicit summary of the spectral properties, decay rates, or absence of unstable modes in the weighted Hölder space that are used to absorb quadratic and higher terms. Since the small-α regime is invoked precisely to close the nonlinear estimates via these linear bounds, the reduction step is load-bearing and requires at least a brief outline of the key linear estimates.
[Theorem 1.1] Theorem 1.1 (or equivalent main result): The stability holds only for sufficiently small α ≪ 1 and perturbations in a small neighborhood whose size depends on α. The manuscript does not indicate whether the smallness of α is required solely for the background construction or also for controlling the nonlinear remainder terms, leaving open whether the result can be extended beyond this regime.

minor comments (2)

[Abstract] The abstract and introduction could include a short roadmap clarifying which parts of the nonlinear bootstrap are treated in this paper versus the companion and subsequent works in the series.
Notation for the localized Hölder topology and the precise meaning of 'general perturbations of the same regularity' should be cross-referenced to the definitions used in the companion paper for consistency.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and have revised the manuscript accordingly to improve clarity on the linear estimates and the role of small α.

read point-by-point responses

Referee: §1 (main theorem statement): The nonlinear stability claim is obtained by reducing to the linearized operator from the companion paper, but this manuscript provides no explicit summary of the spectral properties, decay rates, or absence of unstable modes in the weighted Hölder space that are used to absorb quadratic and higher terms. Since the small-α regime is invoked precisely to close the nonlinear estimates via these linear bounds, the reduction step is load-bearing and requires at least a brief outline of the key linear estimates.

Authors: We agree that an explicit summary would strengthen the presentation. In the revised manuscript we have added a short subsection (now §1.3) outlining the key spectral properties of the linearized operator in the weighted Hölder space, including the absence of unstable modes, the spectral gap, and the decay rates proved in the companion paper. These bounds are then used to absorb the quadratic and higher-order terms for sufficiently small α, making the reduction step self-contained for readers of this part. revision: yes
Referee: Theorem 1.1 (or equivalent main result): The stability holds only for sufficiently small α ≪ 1 and perturbations in a small neighborhood whose size depends on α. The manuscript does not indicate whether the smallness of α is required solely for the background construction or also for controlling the nonlinear remainder terms, leaving open whether the result can be extended beyond this regime.

Authors: The smallness of α is required for both the construction of the background family and for the linear estimates that close the nonlinear argument. Specifically, the spectral gap and decay rates hold only for α sufficiently small; for larger α the linearized operator may lose these properties, preventing control of the nonlinear remainder. We have inserted a clarifying remark immediately after Theorem 1.1 stating this dependence and noting that extension to larger α would require a separate analysis. The size of the perturbation neighborhood is chosen depending on α to ensure the bootstrap closes. revision: yes

Circularity Check

1 steps flagged

Nonlinear stability rests on linearized result from companion paper with overlapping authors

specific steps

self citation load bearing [Abstract]
"We rely on the linearized stability result established in the companion paper [J. Singh and W. Zheng, Nonlinear stability of continuously self-similar naked singularities for the Einstein--scalar field equations II: linearized stability]."

The nonlinear stability theorem is justified solely by citing the linearized stability from a companion paper sharing an author (W. Zheng); the present manuscript contains no independent verification of the linear spectral properties or decay estimates, so the central claim reduces to acceptance of this self-cited result.

full rationale

The derivation chain begins from Christodoulou's external construction of the background self-similar solutions and reduces the nonlinear stability claim to control of perturbations via the linearized operator analyzed in the companion paper. This is a load-bearing self-citation with author overlap, but the companion provides an independent linear analysis rather than a tautological redefinition or fitted-parameter prediction within this manuscript. No self-definitional loops, ansatz smuggling, or renaming of known results occur; the small-α restriction and Hölder topology are stated explicitly as assumptions. The overall structure remains non-circular, with the central claim retaining independent content once the companion linear estimates are granted.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior existence of the background solutions and the linearized stability result from the companion paper, together with the smallness condition on α and the choice of localized Hölder topology for perturbations.

free parameters (1)

α = small positive number much less than 1
The one-parameter family is restricted to 0 < α ≪ 1 to ensure the solutions are C^{1,α} and the stability analysis applies.

axioms (2)

domain assumption Existence of the continuously self-similar naked singularity solutions as constructed by Christodoulou
The stability analysis is performed around these pre-existing background solutions.
domain assumption Linearized stability holds as established in the companion paper
The nonlinear stability result is built directly on this prior linearized analysis.

pith-pipeline@v0.9.0 · 5805 in / 1484 out tokens · 105671 ms · 2026-05-20T16:35:21.166337+00:00 · methodology

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Lean theorems connected to this paper

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IndisputableMonolith.Foundation.RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We rely on the linearized stability result established in the companion paper... the nonlinear stability will hold for a larger range of k... bootstrap or iteration scheme used to close the nonlinear estimates
IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The stability holds for initial perturbations lying in a small open neighborhood... measured in a localized Hölder topology

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

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