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Naked Singularity Censoring with Anisotropic Apparent Horizon

Xinliang An

TL;DR

The paper proves that Christodoulou’s naked singularity in the Einstein-scalar field system becomes censored under arbitrarily small anisotropic perturbations by generating an anisotropic apparent horizon, using a refined, scale-critical short-pulse approach without large auxiliary parameters. It develops two complementary MOTS-elliptic ansatzes and a two-region hyperbolic bootstrap to handle borderline terms and degeneracies, leading to existence and regularity of a MOTS family M_{ul{u}} and an achronal apparent horizon. In addition to trapped-surface formation, the authors establish high codimension nonlinear instability results demonstrating sensitivity to initial data in BV and C^0 norms along outgoing characteristics. The work advances censorship theory beyond spherical symmetry, clarifying how anisotropic perturbations trigger horizon formation and preventing naked singularities in a broad dynamical regime. Overall, the paper combines hyperbolic energy methods, elliptic MOTS analysis, and renormalized geometric quantities to deliver scale-critical results with potential applicability to other Einstein-matter systems.

Abstract

Employing the Einstein-scalar field system, we demonstrate an approach for proving high co-dimensional nonlinear instability of naked-singularity solutions as constructed by Christodoulou in [18]. We further investigate the censorship of Christodoulou's naked singularity and show that a tiny anisotropic perturbation arising from the outgoing characteristic initial data would lead to the emergence of an anisotropic apparent horizon, which covers and censors the naked singularity. Our approach advances the hyperbolic short-pulse method by not requiring the aid of additional large parameters, by permitting the use of initial perturbations for the shear tensor and the derivative of scalar field to be with finite $BV$ and $C^0$ norms, and by allowing the initial perturbation to be arbitrarily small in scale-critical norms. New elliptic arguments based on non-perturbative methods are also developed.

Naked Singularity Censoring with Anisotropic Apparent Horizon

TL;DR

The paper proves that Christodoulou’s naked singularity in the Einstein-scalar field system becomes censored under arbitrarily small anisotropic perturbations by generating an anisotropic apparent horizon, using a refined, scale-critical short-pulse approach without large auxiliary parameters. It develops two complementary MOTS-elliptic ansatzes and a two-region hyperbolic bootstrap to handle borderline terms and degeneracies, leading to existence and regularity of a MOTS family M_{ul{u}} and an achronal apparent horizon. In addition to trapped-surface formation, the authors establish high codimension nonlinear instability results demonstrating sensitivity to initial data in BV and C^0 norms along outgoing characteristics. The work advances censorship theory beyond spherical symmetry, clarifying how anisotropic perturbations trigger horizon formation and preventing naked singularities in a broad dynamical regime. Overall, the paper combines hyperbolic energy methods, elliptic MOTS analysis, and renormalized geometric quantities to deliver scale-critical results with potential applicability to other Einstein-matter systems.

Abstract

Employing the Einstein-scalar field system, we demonstrate an approach for proving high co-dimensional nonlinear instability of naked-singularity solutions as constructed by Christodoulou in [18]. We further investigate the censorship of Christodoulou's naked singularity and show that a tiny anisotropic perturbation arising from the outgoing characteristic initial data would lead to the emergence of an anisotropic apparent horizon, which covers and censors the naked singularity. Our approach advances the hyperbolic short-pulse method by not requiring the aid of additional large parameters, by permitting the use of initial perturbations for the shear tensor and the derivative of scalar field to be with finite and norms, and by allowing the initial perturbation to be arbitrarily small in scale-critical norms. New elliptic arguments based on non-perturbative methods are also developed.
Paper Structure (38 sections, 63 theorems, 664 equations, 6 figures)

This paper contains 38 sections, 63 theorems, 664 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Formation of Trapped Surface
  4. Emergence of Apparent Horizon
  5. Difficulties and New Ingredients in the Proof
  6. Hyperbolic Part
  7. Elliptic Part
  8. Instability Results.
  9. Related Results
  10. Trapped Surfaces, MOTS and Short-Pulse Method
  11. Weak Cosmic Censorship within Spherical Symmetry and Naked Singularity Formation
  12. Organizations of the Paper
  13. Acknowledgement
  14. Setting
  15. Double Null Coordinates
  16. ...and 23 more sections

Key Result

Theorem 1.1

Consider the characteristic initial value problem for the Einstein-scalar field system 1.1. Assigning Christodoulou's naked-singularity initial data in Chr.2 along $\underline{H}_0$ with $-1\leq u \leq 0$ and prescribing perturbed initial data along $H_{-1}$ satisfying then for each $B$ there exist a sufficient small $\delta=\delta(B)$ and the Einstein-scalar field system admits a unique regular

Figures (6)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (121)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1.3
  • Remark 5
  • Remark 6
  • ...and 111 more