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Naked Singularities beyond Spherical Symmetry: Instability of $\kappa$-Self-Similar Solutions via an Iteration Scheme

Xinliang An, Shengrong Wu

arXiv:2609.04723v1gr-qcmath-phmath.APmath.DGmath.MP

TL;DR

Nonspherical κ-self-similar naked-singularity solutions have angularly dependent borderline terms that make instability arguments developed for less singular settings insufficient. The paper uses an iteration scheme to control these terms and proves that arbitrarily small scale-critical anisotropic outgoing perturbations form trapped surfaces and corresponding MOTSs.

  • Problem

    Nonspherical κ-self-similar backgrounds couple angularly dependent shifts nonlinearly to the lapse and scalar field, while the earlier method does not produce an adequate existence region in this setting.

    Source: §1.1 · p. 3

  • Method

    The authors iteratively construct approximate double-null geometries, freezing nonlinear coefficients at the preceding geometry and solving the resulting equations in triangular order.

    Source: §1.3 · p. 6

  • Results

    Arbitrarily small scale-critical anisotropic perturbations of the outgoing data produce a trapped surface and a corresponding MOTS on every sufficiently early incoming null hypersurface.

    Source: §1.2 · p. 5

  • Takeaways & Limitations

    A matter-focusing condition, in which scalar-field focusing dominates shear and trace-free curvature contributions, supports a mean-curvature-flow construction of the MOTS.

    Source: §1.1 · p. 4

    The stated initial-data setting requires scale-invariant smallness after factoring out the leading self-similar powers on H0.

    Source: §1.2 · p. 4, §2.2 · p. 12

Abstract

from arXiv · show

This paper provides the instability counterpart to our recent construction of nonspherically symmetric approximating $\kappa$-self-similar naked-singularity solutions for the Einstein--scalar field system. These singular solutions contain pervasive nonspherical borderline terms, and to prove instability the delicate renormalization procedure developed in [2] does not extend to the more singular setting considered here. To overcome these difficulties, we introduce a new iteration scheme adapted to singular backgrounds whose leading-order geometry depends on the angular variables. At each step, the nonlinear coefficients are frozen using the preceding double-null geometry, and the resulting equations are solved in a triangular order. In this way, the nonspherical borderline terms are incorporated into the approximate geometry rather than treated as perturbative errors, yielding successively sharper estimates. After sufficiently many iterations, the scheme controls the singular angular structure and produces an approximate spacetime whose difference from the exact solution satisfies the required bounds. In particular, these bounds provide an existence region large enough to carry out the instability argument. We further prove that anisotropic perturbations of the outgoing data, arbitrarily small in a scale-critical norm, lead to the formation of a trapped surface. We also formulate and verify a matter-focusing condition under which a parabolic flow argument guarantees the existence of a corresponding marginally outer trapped surface (MOTS). Together, these results establish the nonlinear instability of $\kappa$-self-similar naked singularities beyond spherical symmetry in the Einstein--scalar field system and introduce a framework for applications across Einstein systems.

1. Introduction

The section introduces an iteration scheme for handling angularly dependent singular terms in the Einstein–scalar field system and establishing an existence region. It also outlines how anisotropic focusing, flow, and null-comparison methods are used to address trapped-surface and MOTS formation from small scale-critical perturbations.

1.1. Background.

  • Angularly dependent borderline singularities make prior energy-estimate and renormalization frameworks inadequate; the scheme absorbs these terms into iterated approximate geometries.At each step, nonlinear coefficients are frozen using the preceding double-null geometry and the equations are solved in triangular order, improving the residual by an additional small factor.Source: §1.1 · p. 3
  • A matter-focusing condition ensures that scalar-field focusing along the incoming direction dominates shear and trace-free curvature terms, enabling a mean-curvature-flow construction of a MOTS.The paper also derives a second-variation formula for area that yields coercive positivity underlying MOTS uniqueness in less singular regimes.Source: §1.1 · p. 4, §1.2 · p. 5

1.2. Main results.

  • For κ>0, under the stated (ϵ,N1)-small scale-invariant H0 bound, Ω²(u,0)≲(−u)^κ, and continuous finite-norm data on H−1, solutions exist for n−1≤u<0 and 0≤v<ϵ1(−u)^{1+max{δ,κ/N}}, with sufficiently small ϵ1.The proof uses 4N iteration steps and takes the 4N-th iterate as reference spacetime; κ/N records the iteration gain, while δ accounts for the small loss needed to close the exact estimates.Source: §1.2 · p. 5
  • Arbitrarily small scale-critical anisotropic perturbations of outgoing data produce trapped surfaces and corresponding MOTSs within the established existence region.The perturbation may be localized to an angular sector; a deformation of the sphere turns the resulting localized focusing into a closed trapped surface.Source: §1.2 · p. 5

1.3. New ingredients.

  • High-order approximate spacetimes reduce the exact-solution estimates to differences with higher-order residuals, allowing transport and curvature energy estimates to close systematically.The paired estimates use four variable pairs rather than Weyl curvature components as primary energy variables, making control of differences after iteration more systematic.Source: §1.3 · pp. 6–8
  • Under the stated parameter condition, lower-order terms are dominated and tr χ < 0 for −u sufficiently small.Source: §1.3 · p. 8
  • A barrier adapted to the shrinking transported angular region converts localized anisotropic focusing into a closed trapped surface.The strict threshold gap supplies a δ1-gain that survives comparison with the exact solution; the trapped surface is constructed in the exact spacetime.Source: §1.3 · pp. 8–9
  • An anisotropic trapped surface serves as an inner barrier in a mean-curvature-flow argument establishing existence of a corresponding MOTS.The stated aim in this singular setting is existence of a MOTS arising from anisotropic instability.Source: §1.3 · p. 9
  • The second-variation formula for outgoing expansion under incoming-null variation underpins MOTS uniqueness and the null comparison principle introduced by An and He [4].The comparison argument uses graph deformation and a maximum-principle condition.Source: §1.3 · pp. 9–10
  • In the naked-singularity-censoring setting [2], the comparison conditions yield an alternative proof of MOTS uniqueness and apparent-horizon achronality.Source: §1.3 · p. 10

1.4. Outline of the paper.

  • The paper proceeds from double-null formalism and self-similar data to approximate spacetimes, exact-solution estimates, and trapped-surface and MOTS arguments.Source: §1.4 · p. 10

2. Preliminaries

The preliminaries establish the double-null framework and construct characteristic initial data for the Einstein–scalar field system. They treat κ-self-similar data on an ingoing cone and short-pulse perturbations on an outgoing cone.

2.1. Equations.

  • The section collects general-Lorentzian double-null equations, including null propagation equations and first-order derivative equations for Ricci coefficients.Source: §2 · p. 10, §2.1 · pp. 10–12

2.2. κ-self-similarity and initial data on an ingoing cone.

  • An iteration scheme constructs κ-self-similar initial data on the ingoing cone, with the iterates converging in their respective norms.The construction uses the sphere propagation equation and determines the torsion and Ω⁻¹χ along H0.Source: §2.2 · pp. 12–14
  • For the constructed incoming data, the Ricci coefficients satisfy the stated estimates and the data satisfy the Einstein–scalar field constraint equations.Source: §2.2 · pp. 12, 15

2.3. Short-pulse initial data on an outgoing cone.

  • On the outgoing cone H−1, Ω is prescribed first, and the shift b then follows from transport equations.The data prescription includes (g, ϕ) along H−1.Source: §2.3 · p. 15
  • The short-pulse perturbation is required to be sufficiently strong for trapped-surface formation, and the stated requirement is satisfied.Source: §2.3 · p. 15

3. Construction of Approximating Spacetimes

This section constructs approximate spacetimes by iteratively freezing coefficients against the preceding geometry and solving the resulting equations in sequence. It develops estimates for the approximations and their Einstein–scalar field residues, then combines estimates for Ricci and scalar-field quantities to complete the proof of Theorem 3.6.

3.1. Construction.

  • The construction iterates approximate metrics and scalar fields, solving equations sequentially with contractions taken in the preceding metric; the equations are ODEs with known right-hand sides.Induction estimates control the approximations and their initial values along H−1, supporting convergence toward a solution of the Einstein–scalar field system.Source: §3.1 · pp. 15, 17

3.2. First-step estimates.

  • When v/(−u)^(1+δ) ≪ 1, Proposition 3.4 estimates ζ, Ωtr χ, ΩD3ϕ, and ∇ϕ; derivative loss is allowed because the argument solves a system of ODEs.Source: §3.2 · pp. 19–20

3.3. Iteration estimates.

  • Theorem 3.6 establishes inductive estimates for the Einstein residues, the scalar wave operator, differences in outgoing Ricci coefficients, and auxiliary quantities.The stated estimates apply when v ≪ (−u)^(1+δ); the derivative count N0 − 4i is not optimal but suffices to solve the ODE system.Source: §3.3 · pp. 21–22

3.3.1. Estimates for γ, ψ, ψ.

  • Inductive estimates compare successive iterates for the metric, connection coefficients, and related fields, completing the stated difference bounds despite derivative loss in ∇log Ω.The proof uses the preceding step’s estimates and transport equations to derive sharper bounds for the next iterate.Source: §3.3.1 · pp. 23, 25

3.3.2. Estimates for

  • The proof turns to Einstein-residue and scalar-wave estimates, deriving Ricci-component identities from the construction equations to complete Theorem 3.6.The provided passage begins with the ΩRic34 identity and then treats ΩRic4A.Source: §3.3 · p. 21, §3.3.2 · pp. 25–27
  • The estimates cover d RicAB, Ric34, Ric4A, and □ϕ, while Ω²Ric44 vanishes identically by construction.The passage then invokes equations (3.36) and (3.38) to obtain a further conclusion, but does not include that conclusion here.Source: §3.3.2 · p. 27
  • The authors state that this completes the proof of Theorem 3.6.Source: §3.3.2 · p. 28

4. The A Priori Estimates for Einstein Scalar-Field Equations

This section establishes local existence for the characteristic initial value problem and develops a priori estimates for the Einstein scalar-field equations. It uses elliptic and transport estimates to control non-top-order and top-order quantities and improve the bootstrap bounds.

4.1. Main theorem and bootstrap assumption.

  • For sufficiently small ε, the characteristic initial value problem has a solution in the stated region for the given small κ-self-similar data.The existence region is {−1 ≤ u < 0, 0 ≤ v ≤ ε₁(−u)^(1+κN)} for suitably small ε₁.Source: §4.1 · p. 28

4.2. Calculation of differences and useful estimates.

  • Scale-invariant Sobolev, product, difference, transport, energy, and perturbative elliptic estimates furnish tools for controlling nonlinear terms and differences.The elliptic estimate uses induction on commutations; curvature terms containing the curvature difference are controlled by the corresponding energy bound.Source: §4.2 · pp. 29, 31, 32, 33, 35, 36

4.3. Non-top-order estimates.

  • The section establishes non-top-order estimates for metric components, curvature components, Ricci coefficients, and scalar-field derivatives.The metric and curvature estimates precede estimates for the connection components and Dϕ.Source: §4.3 · pp. 36–37

4.4. Energy estimates for top-order terms.

  • Applying the model energy estimate to the listed Bianchi pairs yields estimates used to close the top-order bootstrap argument.The passage identifies the Bianchi pairs as the quantities of interest for this step.Source: §4.4 · p. 39
  • The remaining non-top-order estimate for η in (4.13) follows as a direct consequence of the preceding estimates.The subsection also obtains further bounds using elliptic estimates.Source: §4.4 · pp. 39–40
  • For top-order terms, the argument treats the ∇3 equations for tr χ and divη − K carefully and uses transport estimates with non-top-order bounds.It avoids using ∇(Ωtr χ) to prevent top-order derivatives of ω.Source: §4.4 · pp. 39–40
  • These estimates complete the a priori estimates.Source: §4.4 · p. 41

5. Trapped Surface Formation and Marginally Outer Trapped Surfaces

The section proves trapped-surface formation for isotropic and anisotropic perturbations and establishes MOTS existence under a matter-focusing condition. It also develops comparison and variation-based results on MOTS uniqueness and apparent-horizon achronality, relating strict positivity to quantities in a null foliation.

5.1. Isotropic trapped surface.

  • If 0 < λ < κ(2N−3) 2(2N+3κ), then along v0(u)=ϵ1(−u)^(1+3κ/(2N)), for sufficiently small ϵ1 and −u, tr χ(u,v0(u),θ)<0 for each θ, establishing existence of a trapped surface.Source: §5.1 · pp. 41, 43

5.2. Anisotropic trapped surface.

  • Under a local lower-bound assumption on an angular ball and the stated λ restriction, anisotropic perturbations yield a trapped surface.The constructed graph has ℓ+ < 0 everywhere when δ1 > 2δ and −U1 is sufficiently small.Source: §5.2 · pp. 44, 46
  • A matter-focusing condition and a parabolic flow on each incoming null cone containing a trapped surface produce a corresponding MOTS.The trapped surface and a round outer untrapped section serve as barriers; the flow converges to a section with ℓ+ = 0.Source: §5.2 · p. 46

5.3. More remarks on MOTS.

  • The paper derives a second variation formula for the outgoing null expansion under variation along e3 in a general incoming-null-cone setting.The general setting uses a geodesic foliation based on a spacelike sphere.Source: §5.3 · p. 47
  • If two MOTSs are joined by the stated interpolation and 2K_w(t) + 1/2 > 0, then they coincide.This is the uniqueness criterion in Proposition 5.9.Source: §5.3 · p. 48
  • Under the stated curvature condition along an interpolation, a surface with ℓ+ ≤ 0 lies at or outside the MOTS in the null comparison principle.The conclusion is w1 ≥ w0.Source: §5.3 · p. 49
  • The comparison condition involves a positive-definite matrix and the quantity (1 + a) tr χℓ+(w(t)) + 2K′.Source: §5.3 · p. 49
  • A negative minimum of h would make the relevant right-hand side strictly positive, yielding a contradiction.Source: §5.3 · p. 50
  • In the setting of [2], the null comparison principle gives the desired comparison between the corresponding graph spheres.The positivity hypothesis along the interpolation w(t) is verified using the asymptotic estimates of Remark 5.11.Source: §5.3 · p. 50
  • This comparison provides an alternative proof of MOTS uniqueness and apparent-horizon achronality in the setting of [2].Source: §5.3 · p. 50
  • The remarks formulate the argument using the null foliation x(t, θ) = (w(t, θ), θ) and denote the corresponding quantities by ψ′.Source: §5.3 · p. 50
  • In that foliation, the strict-positivity condition in Proposition 5.12 is expressed as an equivalent condition on the corresponding quantities.Source: §5.3 · p. 50
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