A Proof of Weak Cosmic Censorship Conjecture for the Spherically Symmetric Einstein-Maxwell-Charged Scalar Field System

Abstract

Under spherical symmetry, we show that the weak cosmic censorship holds for the gravitational collapse of the Einstein-Maxwell-charged scalar field system. Namely, for this system, with generic initial data, the formed spacetime singularities are concealed inside black-hole regions. This generalizes Christodoulou's celebrated results to the charged case. Due to the presence of charge $Q$ and the complexification of the scalar field $φ$, multiple delicate features and miraculous monotonic properties of the Einstein-(real) scalar field system are not present. We develop a systematical approach to incorporate $Q$ and the complex-valued $φ$ into the integrated arguments. For instance, we discover a new path, employing the reduced mass ratio, to establish the sharp trapped surface formation criterion for the charged case. Due to the complex structure and the absence of translational symmetry of $φ$, we also carry out detailed modified scale-critical BV area estimates with renormalized quantities to deal with $Q$ and $φ$. We present a new $C^1$ extension criterion by utilizing the Doppler exponent to elucidate the blueshift effect, analogous to the role of integrating vorticity in the Beale-Kato-Madja breakdown criterion for incompressible fluids. Furthermore, by utilizing only double-null foliations, we establish the desired first and second instability theorems for the charged scenarios and identify generic initial conditions for the non-appearance of naked singularities. Our instability argument requires intricate generalizations of the treatment for the uncharged case via analyzing the precise contribution of the charged terms and its connection to the reduced mass ratio.

Figures (15)

• Figure 1: In above picture, $\Gamma$ corresponds to the set of fixed points under the group action $SO(3)$; $\mathcal{O}'$ stands for the first singularity formed along $\Gamma$ in the evolution; $\mathcal{O}'$ and the singular boundary $\mathcal{B}$ are censored by the black-hole region ($\mathcal{BH}$); $\mathcal{A}$ is the apparent horizon; $\mathcal{EH}$ is the event horizon; $\mathcal{I}^+$ is the future null infinity.
• Figure 2: The coordinate system in $\mathcal{Q}$
• Figure 3:
• Figure 4: For a given point $p = (u,v) \in \mathcal{R}$, suppose that we can find some $\overline{p} = (\overline{u},v) \in \mathcal{R}$ with $0 \leq \overline{u} < u$ such that the incoming null curve $\overline{p}p$ is in $\mathcal{R}$, then the estimates \ref{['Estimates1']} holds. Here the supremums in \ref{['Bandmu*']} are obtained along the outgoing null curve originating from $\Gamma$ which intersects with $\overline{p}$. Even though the dotted line from $\overline{p}$ extends to the past along an incoming null curve may intersect $\mathcal{A}$ and transits into $\mathcal{T}$ before intersecting $C_0^+$, this does not affect our lemma here as it suffices to only find a point $\overline{p}$ along a past-directed incoming null curve from $p$ such that $\overline{p}p$ is in $\mathcal{R}$. If the dotted line from $\overline{p}$ does not intersect $\mathcal{A}\cup\mathcal{T}$, we can set $\overline{u} = 0$.
• Figure 5: For a given point $p = (u,v) \in \mathcal{R}$, let $\overline{p} = (\overline{u},v)$ with $0 \leq \overline{u} < u$ be also a point in $\mathcal{R}$. Denote $p'p$ and $\overline{p}'\overline{p}$ as outgoing null curves emitting from $p'$ and $\overline{p}'$ along $\Gamma$ respectively. We then have that the shaded region indicated above must be in $\mathcal{R}$. To derive pointwise estimates at $p = (u,v)$, we consider the data along $\overline{p}'\overline{p}$ as the initial data.

Theorems & Definitions (72)

• Theorem 1.1
• Remark 1
• Remark 2
• Remark 3
• Proposition 2.1
• proof
• Definition 2.2
• Remark 4
• Remark 5
• Definition 2.3