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High regularity waves on self-similar naked singularity interiors: decay and the role of blue-shift

Jaydeep Singh

Abstract

We consider solutions to the linear wave equation $\Box_{g}\varphi = 0$ on a class of approximately $k$-self-similar naked singularity interiors. This equation models the blue-shift effect, an instability exploited by Christodoulou in the proof of low-regularity weak cosmic censorship. Using a combination of resonance expansions and multiplier estimates, we find in the small-mass regime $k^2 \ll 1$ that the asymptotics of solutions are strongly sensitive to the regularity assumed on outgoing, characteristic initial data across the past light-cone of the singularity. Above a threshold regularity set by the $k$-self-similar scalar field, solutions are shown to always obey self-similar bounds, indicating that the blue-shift instability competes with the stabilizing influence of high regularity. We conclude that a proper statement of weak cosmic censorship, as well as an understanding of the role of naked singularities in phenomena such as critical collapse, may depend on the topology of initial data.

High regularity waves on self-similar naked singularity interiors: decay and the role of blue-shift

Abstract

We consider solutions to the linear wave equation on a class of approximately -self-similar naked singularity interiors. This equation models the blue-shift effect, an instability exploited by Christodoulou in the proof of low-regularity weak cosmic censorship. Using a combination of resonance expansions and multiplier estimates, we find in the small-mass regime that the asymptotics of solutions are strongly sensitive to the regularity assumed on outgoing, characteristic initial data across the past light-cone of the singularity. Above a threshold regularity set by the -self-similar scalar field, solutions are shown to always obey self-similar bounds, indicating that the blue-shift instability competes with the stabilizing influence of high regularity. We conclude that a proper statement of weak cosmic censorship, as well as an understanding of the role of naked singularities in phenomena such as critical collapse, may depend on the topology of initial data.
Paper Structure (36 sections, 52 theorems, 371 equations, 6 figures, 1 table)

This paper contains 36 sections, 52 theorems, 371 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Rough statement of results
  3. Exterior perturbations and the blue-shift
  4. Proof outline
  5. Outlook
  6. Related works
  7. Preliminaries
  8. Spherically symmetric spacetimes with scalar field
  9. $k$-self-similarity and self-similar gauge
  10. Renormalized gauge
  11. Notation and conventions
  12. Admissible spacetimes
  13. Coordinate systems in $\widetilde{\mathcal{Q}}^{(in)}$
  14. The wave equation (\ref{['eq:1']}) and separation of variables
  15. Properties of the potentials $V_k(z), L_{k,\ell}(z)$
  16. ...and 21 more sections

Key Result

Theorem 1

Fix a value of $k$ sufficiently small, and let $(\mathcal{M},g,\phi)$ denote a naked singularity interior that is "sufficiently close" to $(\mathcal{M}, g_k, \phi_k)$. Assume spherically symmteric, characteristic initial data $\varphi_0(v): \{u\!=\!-1\} \rightarrow \mathbb{R}$ is given for (eq:1), w

Figures (6)

  • Figure 1: Penrose diagram representation of a spherically symmetric, globally naked singularity. To the future of initial data, the spacetime is foliated by outgoing null hypersurfaces, each extending from the center ${(r=0)}$ to null infinity ${(r\rightarrow \infty)}.$ Within finite $u$-coordinate time all observers intersect the future light-cone of the first singularity $\mathcal{O},$ which lies at the center. Also shown are the conventions for double-null coordinates adopted in this paper; the initial data surface is $\{u\!=\!-1\}$, and the singularity is normalized to ${(u,v)=(0,0)}$.
  • Figure 2: Schematic representation of Step 1. The diagram should be read "towards the past," with the result being the construction of data along $\{t=0\}$. Also shown is an explicit example of null data $\widetilde{\psi}_0$, and a component of the associated spacelike data.
  • Figure 3: Domains in the complex plane for which various resolvent operators are defined. In the context of a resonance expansion, the imaginary part of $\sigma$ corresponds to growth/decay of waves at a rate $e^{(\Im \sigma) t}$ in regions $\{x \leq \text{const.}\}$. This diagram is specific to setting of Theorem \ref{['thm:introrough1']}(a).
  • Figure 4: The conclusion of the resonance expansion (active in light shaded area) and the multiplier estimates (active additionally in the darker shaded area).
  • Figure 5: Penrose diagram of the relevant subdomains of the interior and exterior regions. The interior region $\mathcal{Q}^{(in)}$ (lightly shaded) is a subset of $\{u\geq -1\}$, and the extended interior $\widetilde{\mathcal{Q}}^{(in)}$ (lightly shaded and darkly shaded) extends to $\{u > -\infty \}$.
  • ...and 1 more figures

Theorems & Definitions (109)

  • Definition 1.1
  • Theorem 1: Spherically symmetric solutions
  • Theorem 2: Non-spherically symmetric solutions
  • Proposition 1.1
  • Proposition 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.1: singh
  • ...and 99 more