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Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system

Dawei Shen, Jingbo Wan

Abstract

In this paper, we prove a scale-critical trapped surface formation result for the Einstein--Maxwell--charged scalar field (EMCSF) system, without any symmetry assumptions. Specifically, we establish a scale-critical semi-global existence theorem from past null infinity and show that the focusing of gravitational waves, the concentration of electromagnetic fields, or the condensation of complex scalar fields, each individually, can lead to the formation of a trapped surface. In addition, we capture a nontrivial charging process along past null infinity, which introduces new difficulties due to the abnormal behavior of the matter fields. Nevertheless, the semi-global existence result and the formation of a trapped surface remain valid.

Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system

Abstract

In this paper, we prove a scale-critical trapped surface formation result for the Einstein--Maxwell--charged scalar field (EMCSF) system, without any symmetry assumptions. Specifically, we establish a scale-critical semi-global existence theorem from past null infinity and show that the focusing of gravitational waves, the concentration of electromagnetic fields, or the condensation of complex scalar fields, each individually, can lead to the formation of a trapped surface. In addition, we capture a nontrivial charging process along past null infinity, which introduces new difficulties due to the abnormal behavior of the matter fields. Nevertheless, the semi-global existence result and the formation of a trapped surface remain valid.
Paper Structure (102 sections, 81 theorems, 486 equations, 3 figures)

This paper contains 102 sections, 81 theorems, 486 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Einstein--Maxwell--charged scalar field system
  3. Small data regime
  4. Trapped surface formation
  5. Einstein vacuum equations
  6. Einstein equations coupled with matter fields
  7. Einstein--Maxwell--charged scalar field
  8. Rough versions of the main theorems
  9. Main difficulty and new ingredients
  10. --weighted estimates
  11. Maxwell fields and complex scalar fields
  12. Electromagnetic potential
  13. Weyl components
  14. The Bianchi pair $(\alpha,\beta)$
  15. The Bianchi pair $(\beta,(\rho,-\sigma))$
  16. ...and 87 more sections

Key Result

Theorem 1.1

Let $(\mathcal{M},\mathbf{g},\mathbf{F},\psi)$ be a spacetime satisfying EMCSF, containing a non-compact Cauchy hypersurface. If $\mathcal{M}$ contains a compact trapped surface, then it is future causally geodesically incomplete.

Figures (3)

  • Figure 1: The initial conditions in Theorem \ref{['Maintheoremintro']} can lead to the formation of a trapped surface $S_{-\frac{a}{4},1}$ in the future of $H_{u_\infty}\cup{\underline{H}}_0$. Moreover, the Hawking mass and electric charge of the final sphere $S_{u_\infty,1}$ on $H_{u_\infty}$ satisfy $m(S_{u_\infty,1}) \simeq a$ and $|Q(S_{u_\infty,1})| \simeq \mathfrak{e} a$, respectively.
  • Figure 2: The initial conditions in Theorem \ref{['scalingintro']} can lead to the formation of a trapped surface $S_{-\frac{\delta a}{4},\delta}$ in the future of $H_{u_\infty}\cup{\underline{H}}_0$. Moreover, the Hawking mass and electric charge of the final sphere $S_{u_\infty,\delta}$ on $H_{u_\infty}$ satisfy $m(S_{u_\infty,\delta}) \simeq \delta a$ and $|Q(S_{u_\infty,\delta})| \simeq \delta^2\mathfrak{e} a$, respectively.
  • Figure 3: The initial conditions in Theorem \ref{['thmscaling']} lead to trapped surface ($S_{-\frac{\delta a}{4},\delta}$) formation in the future of the $H_{u_\infty}$ and ${\underline{H}}_0$.

Theorems & Definitions (182)

  • Theorem 1.1: Penrose Penrose
  • Theorem 1.2: Main Theorem (first version)
  • Remark 1.3
  • Theorem 1.4: Trapping Mechanism and Charging Process (first version)
  • Theorem 1.5: Main Theorem (rescaled version)
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 172 more