Naked Singularities beyond Spherical Symmetry: Instability of κ-Self-Similar Solutions via an Iteration Scheme
Abstract
This paper provides the instability counterpart to our recent construction of nonspherically symmetric approximating κ-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 κ-self-similar naked singularities beyond spherical symmetry in the Einstein--scalar field system and introduce a framework for applications across Einstein systems.
Related
- New time-periodic solutions to the cubic conformal wave equation on 𝕊3: I Spherical symmetry
- Instability of naked singularities of perfect fluid under C1,α gravitational perturbations
- Landau--Ginzburg mirror symmetry for ADE singularities
- Solutions to Monge-Ampère Equations with Low Codimensional singularities
- Genericity of singularities of solutions to quasilinear systems on compact sets
BibTeX
@misc{an2026naked,
title = {{Naked Singularities beyond Spherical Symmetry: Instability of $κ$-Self-Similar Solutions via an Iteration Scheme}},
author = {Xinliang An and Shengrong Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04723},
eprint = {2609.04723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04723},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Naked Singularities beyond Spherical Symmetry: Instability of κ-Self-Similar Solutions via an Iteration Scheme
AU - Xinliang An
AU - Shengrong Wu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04723
UR - https://arxiv.org/abs/2609.04723
AB - This paper provides the instability counterpart to our recent construction of nonspherically symmetric approximating κ-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 κ-self-similar naked singularities beyond spherical symmetry in the Einstein--scalar field system and introduce a framework for applications across Einstein systems.
ER -