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

Abstract: This is the second part of a series of papers proving the nonlinear stability of a one-parameter family of continuous self-similar $C<sup>{1,α}$ naked singularity solutions, with $0&lt;α\ll1$, 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.

• The paper proves linearized stability of continuously self-similar naked singularity solutions under threshold regularity in the Einstein-scalar field system.
• It employs weighted energy estimates and scattering-theoretic analysis to manage low-regularity nonlinear couplings near the singular horizon.
• Results refute strong cosmic censorship by showing that small regular perturbations bypass the blue-shift instability.

Linearized Stability of Continuously Self-Similar Naked Singularities in the Einstein-Scalar Field System

Introduction

This work undertakes the linear stability analysis of a distinguished one-parameter family of continuously self-similar $C^{1,\alpha}$ naked singularity solutions in spherical symmetry for the Einstein-scalar field (ESF) equations, originally constructed by Christodoulou for $0 < k^2 < 1/3$. Prior results have established that these backgrounds are unstable to rough perturbations due to the blue-shift mechanism, but remain stable to regular ones in the exterior [(2605.16095) and references therein]. This paper proves the linearized stability of these solutions within their minimal interior regularity, thereby both clarifying the regularity threshold at which the blue-shift instability is triggered, and forming the analytical foundation for the companion nonlinear stability theorem.

Background: Einstein-Scalar Field Equation and Self-Similar Spacetimes

The ESF system describes a Lorentzian manifold $(\mathcal{M}, g)$ coupled to a real scalar field $\phi$ via

$$Ric[g]_{\mu\nu} = 2\partial_\mu\phi \partial_\nu\phi, \qquad \Box_g \phi = 0.$$

A family of spherically symmetric, continuously self-similar naked singularity solutions parameterized by $k$ can be written in double-null coordinates as

$$g = -\tfrac{1}{2}\Omega^2 du\,dv - \tfrac{1}{2}\Omega^2 dv\,du + r^2 d\sigma^2,$$

with $r, \Omega, \phi$ solving ODEs in a similarity variable $z = v/(-u)^{1-k^2}$. These "k-self-similar" spacetimes exhibit a naked singularity at $(u, v) = (0, 0)$; initial data is smooth away from the singular horizon but only $0 < k^2 < 1/3$0 at the intersection, with higher regular derivatives allowed to diverge polynomially at $0 < k^2 < 1/3$1.

Stability and Instability Phenomena: Blue-Shift versus Regularity Effects

Previous works, notably Christodoulou [1999], Liu-Li, and Singh, established that for initial perturbations rougher than the background's minimal regularity, the solutions are nonlinearly unstable to trapped surface formation via the blue-shift instability. Conversely, perturbations that respect the background regularity or are confined to the exterior yield stability, both linearly and nonlinearly [see references and Singh 2024, Singh-Zheng].

A schematic mechanism:

• Blue-shift instability: For data with low regularity (e.g., with nonzero jump in first derivatives), the focusing of ingoing null rays near the singular horizon amplifies perturbations, producing growth rates in derivatives exceeding the self-similar background.
• High-regularity stabilization: For data with regularity at or above threshold, the blue-shift amplification cannot be triggered; energy methods and scattering approaches reveal decay rates at or above the self-similar rates.

The precise threshold is dictated by the Hölder exponent $0 < k^2 < 1/3$2. Initial data above this regularity avoid the blue-shift-induced instability.

Linearization and Technical Obstacles

For nonlinear stability, it is necessary to analyze the linearized Einstein-scalar field system about the background. This introduces formidable technical complications absent in the scalar wave toy model:

• Loss of regularity: The coupling introduces terms with only minimal Hölder regularity, e.g., from $0 < k^2 < 1/3$3, which is $0 < k^2 < 1/3$4 and blows up polynomially at $0 < k^2 < 1/3$5.
• Resonant modes and translation symmetry: Invariance under field translations induces a nontrivial mode proportional to the area radius $0 < k^2 < 1/3$6, which decays only as quickly as the background itself; this obstructs improved decay estimates unless it is projected out.
• Coupling-induced resonances: In the ODE system for the Laplace transform of the linearized fields, spurious "fake" resonances can appear from the coupling between metric and scalar, complicating the scattering resonance analysis necessary for asymptotic control.

Main Results

Let $0 < k^2 < 1/3$7 denote the localized Hölder spaces matching the interior regularity of the background, and $0 < k^2 < 1/3$8.

Theorem (Linear Stability at the Threshold)

For fixed small $0 < k^2 < 1/3$9, consider initial data for the linearized system in

$(\mathcal{M}, g)$0

for $(\mathcal{M}, g)$1, $(\mathcal{M}, g)$2. Then the linearized solution $(\mathcal{M}, g)$3 exists globally, remains $(\mathcal{M}, g)$4 at the singular horizon, and satisfies: $(\mathcal{M}, g)$5 with analogous decay for $(\mathcal{M}, g)$6 and the coefficient $(\mathcal{M}, g)$7 controlled by the initial norm.

Theorem (Perturbations Above Threshold)

If initial scalar data are more regular, all derivatives of the solution decay strictly faster than the background self-similar rate, with explicit exponents given in terms of the data regularity $(\mathcal{M}, g)$8.

Key Analytical Innovations

• Second-order energy estimates with weighted multipliers: This enables control of quantities whose regularity is weaker near the singular horizon.
• Scattering-theoretic analysis: A backward Laplace transform and detailed spectral analysis show that for admissible data, no nontrivial growing resonances are present apart from the background translation mode.
• Resolution of low-regularity singularities in the linearized system: By leveraging the relative regularity of time-like versus null derivatives and the precise structure of the Christodoulou background, the authors produce uniform $(\mathcal{M}, g)$9-holomorphicity in the Laplace-transformed ODE system, despite the apparent singularities.

Implications and Contradictions to Cosmic Censorship

These results directly establish that the Christodoulou naked singularity backgrounds are linearly stable against spherical perturbations at the threshold regularity—not just in the exterior but also in the interior. Since nonlinear stability at this regularity is proved in the companion work, this conclusively demonstrates the failure of the Weak Cosmic Censorship conjecture for initial data at this threshold in the ESF system: globally naked singularities can arise stably from small regular data of the appropriate class.

The analysis also shows that the blue-shift mechanism is not universally triggered by the presence of a Cauchy horizon or naked singularity; it is intimately tied to the roughness of initial data relative to the background. The apparent dichotomy—instability below threshold, stability at and above—sharpens the understanding of the interplay between PDE regularity class and strong-field gravitational phenomena.

Theoretical and Practical Consequences

• Foundational insight: The sharp dichotomy in stability as a function of data regularity provides a canonical example of threshold phenomena in nonlinear hyperbolic PDEs and gravitational collapse.
• Techniques: The Laplace transform and scattering resonance methods applied to a singular PDE-ODE system set a template for future study of threshold stability of dynamically singular spacetimes.
• Physical implications: For the ESF system, small data evolution can lead to naked singularity formation in an open set of initial data, contradicting the strong form of cosmic censorship in its most optimistic interpretations.

Anticipated future work includes:

• Extension of these techniques to less symmetric settings and other matter models.
• Quantitative analysis of the stability/instability boundaries in other systems with dynamically formed Cauchy horizons.

Conclusion

This paper supplies a complete linear stability theory for continuously self-similar naked singularity solutions to the spherically symmetric ESF equations at and above their critical regularity. The blue-shift instability mechanism is shown to be regularity-sensitive: while rough perturbations provoke instability, threshold or smoother data do not. This foundational result underpins the overall program falsifying cosmic censorship for low regularity data, and refines the analytical toolkit for probing stability of dynamically singular solutions in hyperbolic geometric PDEs.