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 $\phi$, 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 $\phi$ 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 $\phi$, we also carry out detailed modified scale-critical BV area estimates with renormalized quantities to deal with $Q$ and $\phi$. 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.

§ INTRODUCTION

In general relativity, the validity of the weak cosmic censorship is a central unresolved issue. It is conjectured that for generic initial data, the singularities formed in the gravitational collapse would be confined within black-hole regions and cannot be seen by distant observers. In 1969 Penrose [44] wrote: We are thus presented with what is perhaps the most fundamental unanswered question of general-relativistic collapse theory, namely: does there exist a “cosmic censor" who forbids the appearance of naked singularities, clothing each one in an absolute event horizon? More precisely, as written by Christodoulou [22], the weak cosmic censorship conjecture for Einstein's equations states:

Generic asymptotically flat initial data have a maximal future development possessing a complete future null infinity.

Christodoulou [22] further explained:  In heuristic terms, this means that, if we disregard exceptional initial conditions, no singularities are observed from infinity, even though observations from infinity are allowed to continue indefinitely. So far, only for the spherically symmetric Einstein-(real) scalar field system, this censorship was rigorously established and it was proved by Christodoulou in 1990s through a series of breakthrough results [18]-[21]. To generalize this celebrated conclusion to the more physical scenario, one would consider adding the joint evolution of the electric charge $Q$, often viewed as the poor man's angular momentum, to the Einstein field equations. This motivates the study of the spherically symmetric Einstein-Maxwell-charged scalar field system. However, proving the weak cosmic censorship for this system remains open for more than 20 years.

Returning to Christodoulou's proof in [18]-[21], integrated arguments with five indispensable steps are established. In each of these steps, he explored the delicate structures of the Einstein-(real) scalar field system and established sharp results. Based on these, he designed a spectacular blueprint for the proof. All these steps and the corresponding sharp results are later ingeniously incorporated into one piece, hence proving the final conclusion. In his proof, various monotonic formulas tied to the special structures of the Einstein-(real) scalar field system are critically employed. Taking the evolution of $Q$ into account, the subtle structure changes. The given monotonic formulas for the uncharged case are no longer present, and numerous crucial arguments in [18]-[21] have to be circumvented or modified.

In this article, we prove that, within spherical symmetry, the weak cosmic censorship holds for the gravitational collapse of the Einstein-Maxwell-charged scalar field system. Here we consider the $3+1$ dimensional spacetime  $(\mathcal{M},g,\phi,F)$ with $\mathcal{M}$ being the $3+1$ dimensional manifold, $g$ being the Lorenztian metric, $\phi$ being the complex-valued function on $\mathcal{M}$ and $F$ being a differential $2$-form on $\mathcal{M}$. The system of equations governing these quantities takes the form:

\begin{equation}\label{Intro1}
\begin{split}
R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R &= 8 \pi T_{\mu \nu}^{SF} + 8\pi T_{\mu \nu}^{EM},\\
T_{\mu \nu}^{SF} &= \frac{1}{2}(D_{\mu} \phi)(D_{\nu} \phi)^{\dagger} + \frac{1}{2}(D_{\nu} \phi)(D_{\mu} \phi)^{\dagger} - \frac{1}{2}g_{\mu \nu}g^{\alpha \beta} D_{\alpha}\phi (D_{\beta}\phi)^{\dagger}, \\
T_{\mu \nu}^{EM} &= \frac{1}{4\pi}(g^{\alpha \beta}F_{\alpha \mu}F_{\beta \nu} - \frac{1}{4}g_{\mu \nu}F^{\alpha\beta}F_{\alpha\beta}) \\
\end{split}
\end{equation}

with $^\dagger$ standing for the complex conjugate and $D_\mu:= \partial_{\mu} + \e i A_{\mu}$ being the gauge covariant derivative, where $\e$ is the coupling constant,  and $A_\mu$ is the electromagnetic potential. Note that, via the second Bianchi identity, the divergence-free property of the Einstein tensor also implies an evolution equation for the electromagnetic field, i.e.,

\begin{equation}\label{Intro2}
\nabla^{\nu}F_{\mu\nu} = 2 \pi \e i \big(\phi(D_{\mu}\phi)^{\dagger} - \phi^{\dagger}(D_{\mu}\phi)\big).
\end{equation}

§.§ Main Theorem

In this study, we conduct all the proofs with double null coordinates. Under spherical symmetry, the spacetime metric $g$ now takes the form:

\begin{equation}
g =  - \Omega^2(u,v) \; \mathrm{d}u\mathrm{d}v + r^{2}(u,v) \; (\mathrm{d}\theta^2 + \sin^2 \theta \mathrm{d} \phi^2)
\end{equation}

with $\Omega(u,v)$ called the lapse function and $r(u,v)$ called the radial function. Here we employ $u$ and $v$ as coordinates and they are optical functions toward the future satisfying $g^{\mu\nu}\partial_{\mu}u\partial_{\nu}u=0$ and $g^{\mu\nu}\partial_{\mu}v\partial_{\nu}v=0$. Furthermore, constant $u$ and constant $v$ hypersurfaces correspond to the outgoing and incoming null cones, respectively.

Our main conclusion is as follows:

(rough version) In the setting of gravitational collapse for the spherically symmetric Einstein-Maxwell-charged scalar field system, for generic[Here the term “generic” means that the subset of initial conditions leading to the formation of naked singularities has positive co-dimensions. ] asymptotically flat initial data prescribed along $C^+_0$, the singularities formed in the evolution would be covered by a black-hole region.

[-] (0,0) node[anchor=north]$\mathcal{O}$ –  (0,8) node[anchor = east]$\mathcal{O}'$;
[-] (0,0) – (8,8);
[rotate=45] at (5.2,4.8) $C_0^+$;
[] at (-0.3,5.0) $\Gamma$;

[gray!50] (0,8) to (1.5,9.5) to[out = 45,in=135] (6,10) to[out = -135, in = 45] (0,8);
[] (0,8) – (4,4);
[densely dashed] (0,4) – (6,10);
[rotate=45] at (4.2,7.8) $\mathcal{EH}$;
[] (6,10) – (8,8);
[rotate = -45] at (7.2,9.2)$\mathcal{I}^+$;
[thick, densely dotted] (0,8) to (1.5,9.5);
[thick, loosely dotted] (1.5,9.5) to [out = 45,in=135] (6,10);
[fill=white] (1.5,9.5) circle (2pt);
[rotate=12] at (3.6,10.9) $\mathcal{B} \setminus \mathcal{B}_0$;
[rotate=45] at (0.55,8.95) $\mathcal{B}_0$;
[thick, dashdotted] (0,8) to[out = 45, in = - 135] (6,10);
[] at (3.2,8.8) $\mathcal{A}$;
[] at (3.7,9.8) $\mathcal{BH}$;

[anchor = west] at (9.8,10.2) Legend:;
[anchor = west] at (11,9.5) $\mathcal{B}_0$;
[thick, densely dotted] (10,9.3) to (11,9.7);
[anchor = west] at (11,9) $\mathcal{B} \setminus \mathcal{B}_0$;
[thick, loosely dotted] (10,8.8) to (11,9.2);
[anchor = west] at (11,8.5) $\mathcal{A}$;
[thick, dashdotted] (10,8.3) to (11,8.7);
[fill=white] (0,8) circle (2pt);
[fill=white] (6,10) circle (2pt);

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.

The precise version of Theorem <ref> combines our statements in Theorem <ref>, Theorem <ref>, Theorem <ref>, Theorem <ref>, Theorem <ref> and Theorem <ref>.
The conclusion in Theorem <ref> also holds for generic asymptotically flat Cauchy initial data. For the convenience of observing the instability mechanism, we state our main theorem with characteristic initial data.

Our proofs in this paper are divided into six main steps:

* Find Novel Estimates for $Q$ and $\phi$.
In Section <ref>, we derive new estimates for $\phi$ and $Q$ and develop a novel strategy crucially employed throughout the paper to control related charged terms.

* Obtain the Sharp Criterion for Formation of Trapped Surfaces via a New Approach.
In Section <ref>, via a novel approach, we prove the desired trapped-surface-formation criterion for the charged scenario and state it as Theorem <ref>. Specifically, restricted to the $Q=0$ case, in Appendix <ref>, we give a 2-page new proof of Christodoulou's sharp criterion for the Einstein-(real) scalar field system.

* Establish Scale-Critical $BV$ Area Estimates with Renormalizations. In this step, the objective is to establish technical BV area estimates that are crucial for the entire regularity argument.  Section <ref> is dedicated to proving Theorem <ref> for the charged case. Incorporating the evolution of $Q(u,v)$ into the system breaks the transitional symmetry of $\phi(u,v)$ and complicates the system. These require us to introduce new renormalizations and to conduct heavy intricate calculations. In the process, the complex structure of the system is carefully explored and utilized.

* Present a New $C^1$ Extension Principle related to the Blueshift Effect. To connect with the instability arguments, we propose and certify a new $C^1$ extension principle (Theorem <ref>) based on the Doppler exponent $\gamma$, which resembles the Beale-Kato-Majda type breakdown criterion, requiring the boundness of the integration of a certain physical quantity. The proof of an equivalent $C^1$ extension principle for the uncharged scalar field is also listed as Appendix <ref>.

* Prove the First Instability Theorem with Double Null Foliations. In Section <ref>, we prove and summarize the first-instability conclusions as Theorem <ref> with double null foliations. Christodoulou employed Bondi formalism in his original arguments. Our proofs for both the first and second instability theorems inject new ingredients about the charge and associated estimates that are central and delicate to the overall argument.

* Vindicate the Second Instability Theorem with Double Null Foliations. In Section <ref>, we further confirm the second instability mechanism and prove Theorem <ref>. All the aforementioned components are interconnected, constituting an integrated argument. These eventually lead to Theorem <ref> in Section <ref> describing conclusions of genericness.

To prove the weak cosmic censorship conjecture for the spherically symmetric Einstein-Maxwell-charged scalar field system, we incorporate these steps with the following coherent progression of logical arguments:

- Near the center $\Gamma$ portrayed below, we start to consider the characteristic initial value problem with initial data prescribed along $C_0^+$. Applying Theorem <ref>, we extend the BV existence region toward the future, with scale-critical BV area estimates controlled.

- Assume that $O'$ is the first singular point along $\Gamma$. For any regular point $R$ along $\Gamma$ before $O'$, we consider a triangular shadowed region with the future tip at $R$. Within this region, a blueshift factor $e^\gamma$ (referred to as the Doppler factor, where $\gamma$ is termed the Doppler exponent) is bounded. Physically, the Doppler factor $e^\gamma$ measures the blue shift of a photon emitted backward from any $(u,v)$ in the shadowed region and measured by an observer located at $(0,v)$ along $C_0^+$. As $R$ approaches $O'$ along $\Gamma$, by employing Theorem <ref>, along the backward cone $v=v_0$ emitting from $\mathcal{O}'$, we have $\gamma\rightarrow+\infty$ as it approaches $O'$.

- The fact that $\gamma\rightarrow +\infty$ along $v=v_0$ triggers the blueshift-driven instability mechanism. We then employ the first and the second instability theorems (together with the trapped surface formation criterion) in the region between $v=v_0$ and $v=v_1$ to arrive at the main conclusion.

[gray!50] (0,0) to (1.75,1.75) to (0,3.5) to (0,0);
(0,0) node[anchor=north]$\mathcal{O}$ –  (0,5);
[->] (0,0) – (4,4) node[anchor = west]$C_0^+$;
at (-0.35,3.5) $R$;
at (-0.3,4.3) $\Gamma$;
at (-0.3,5.3) $\mathcal{O}'$;
(0,3.5) – (1.75,1.75);
[thick,dotted](0,4) – (2,2);
[thick,dashed](0,5) – (2.5,2.5);
at (0,3.5)[circle,fill,inner sep=1.5pt];
at (0,4)[circle,inner sep=1.5pt];
[rotate=-45] at (1.4,4) $v = v_0$;
[fill=white] (0,5) circle (2pt);

(7,0) node[anchor=north]$\mathcal{O}$ –  (7,5);
[->] (7,0) – (11,4) node[anchor = west]$C_0^+$;
at (6.7,4) $\Gamma$;
at (6.7,5.3) $\mathcal{O}'$;
[thick](7,5) – (9.5,2.5);
at (7,4)[circle,inner sep=1.5pt];
[rotate=-45] at (8.4,4) $v = v_0$;
[fill=white] (7,5) circle (2pt);
[thick,dashed](7,5) – (8,5);
[thick](8,5) – (10,3);
[rotate=-45] at (9.4,4) $v = v_1$;
[] at (7.5,5.3) $\mathcal{AH}$;

§.§ Equations
Before we explain our proof strategy, we review several physical quantities and state their evolution equations below. The Hawking mass $m(u,v)$ for sphere $S(u,v)$ on $\mathcal{Q}$ is defined via

\begin{equation}\label{Setup5}
1 - \frac{2m}{r} = g(\nabla r, \nabla r) = - 4\Omega^{-2}\du r \dv r.
\end{equation}

The mass ratio $\mu(u,v)$ for sphere $S(u,v)$ is further set to be

\begin{equation}\label{Setup6}
\mu(u,v) := \frac{2m(u,v)}{r(u,v)}.
\end{equation}
Given the electromagnetic $2$-form $F(u,v)$ in (<ref>), we obtain the charge $Q(u,v)$ contained in a sphere $S(u,v)$ implicitly by solving

\begin{equation}\label{Setup7}
F(u,v) =  F_{uv}\; \mathrm{d}u \wedge \mathrm{d}v =\frac{Q(u,v)}{2r^2(u,v)}\Omega^2(u,v) \; \mathrm{d}u \wedge \mathrm{d}v.
\end{equation}

Equivalently, an explicit expression for $Q(u,v)$ is given by

\begin{equation}\label{Setup8}
Q(u,v) := 2r^2 \Omega^{-2}F_{uv}(u,v).
\end{equation}

At the same time, notice that each electromagnetic field $F_{uv}$ is associated with a $1$-form $A_u$, called the electromagnetic potential. Furthermore, $F_{uv}$ can be rewritten as

\begin{equation}\label{Setup9}
F_{uv} = \du A_v - \dv A_u.
\end{equation}

As explained in [35], when $\e \neq 0$, the system (<ref>)-(<ref>) is invariant under the following local $U(1)$ gauge transformations:

\begin{equation}\label{Setup10}
\begin{aligned}
\phi \rightarrow e^{-\e \ii f}\phi, \quad  A_\mu \rightarrow A_\mu + \partial_\mu f
\end{aligned}
\end{equation}

with $f$ being a smooth real-valued function to be fixed. In this study, we impose the gauge condition

$$A_v = 0$$

by choosing the appropriate $f$. With this gauge choice, we reduce (<ref>) and (<ref>) to

\begin{equation}\label{Setup11}
\begin{aligned}
F_{uv} = -\dv A_u \quad \mbox{ and } \quad Q(u,v) = -2r^2\Omega^{-2} \dv A_u.
\end{aligned}
\end{equation}

Substituting (<ref>) and (<ref>) back to the field equations (<ref>) and (<ref>), we then derive the following evolution system:

\begin{equation}\label{Setup12}
r \dv \du r + \dv r \du r = -\frac{\Omega^2}{4}\ml 1- \frac{Q^2}{r^2}\mr,
\end{equation}

\begin{equation}\label{Setup13}
r^2 \du \dv \log(\Omega) = -2\pi r^2(D_u \phi (\dv \phi)^\dagger + \dv \phi(D_u \phi)^\dagger) - \frac{1}{2}\Omega^2 \frac{Q^2}{r^2} + \frac{1}{4}\Omega^2 + \du r \dv r,
\end{equation}

\begin{equation}\label{Setup14}
\du(\Omega^{-2}\du r) = - 4\pi r\Omega^{-2}|D_u\phi|^2,
\end{equation}

\begin{equation}\label{Setup15}
\dv(\Omega^{-2}\dv r) = - 4\pi r\Omega^{-2}|\dv \phi|^2,
\end{equation}

\begin{equation}\label{Setup16}
r\du \dv \phi + \du r \dv \phi + \dv r \du \phi + \e \ii \Psi(A) = 0,
\end{equation}

\begin{equation}\label{Setup17}
\Psi(A) = A_u \dv(r\phi) - \frac{\Omega^2}{4}\frac{Q}{r}\phi,
\end{equation}

\begin{equation}\label{Setup18}
\du Q = 4\pi \e r^2 \im{(\phi^\dagger D_u \phi)},
\end{equation}

\begin{equation}\label{Setup19}
\dv Q = -4\pi \e r^2 \im{(\phi^\dagger \dv \phi)}.
\end{equation}

Note that in the equations above we use the notation $D_u := \du + \ii \e A_u$ for the covariant derivative. Moreover,the coupling constant $\e$ here satisfies $0 \leq \e \in \mathbb{R}$. We also emphasize that, in the above system, $\phi$ is a complex-valued unknown function, while $r, A_u, \Omega^2, \im(\phi)$,  $\re(\phi)$ are real-valued unknown functions.

In our arguments, derived from (<ref>), (<ref>), (<ref>), (<ref>),  we also frequently employ the following equations:

\begin{equation}\label{Setup20}
\begin{aligned}
\dv \du r &= \frac{1}{r}\ml \frac{\mu - Q^2/r^2}{1-\mu}\mr \du r \dv r,
\end{aligned}
\end{equation}

\begin{equation}\label{Setup21}
\begin{aligned}
\du \ml \frac{1-\mu}{\dv r}\mr &= \frac{4 \pi r |D_u \phi|^2(1-\mu)}{(-\du r)(\dv r)},
\end{aligned}
\end{equation}

\begin{equation}\label{Setup22}
\begin{aligned}
\dv \ml \frac{1-\mu}{-\du r}\mr &= -\frac{4 \pi r |\dv \phi|^2(1-\mu)}{(-\du r)(\dv r)},
\end{aligned}
\end{equation}

\begin{equation}\label{Setup71}
\begin{aligned}
\Psi(A) &= A_u \dv(r\phi) + \frac{Q \phi (\du r) (\dv r)}{r(1-\mu)}.
\end{aligned}
\end{equation}

For the wave equation in (<ref>), by (<ref>), we also obtain and utilize the following equivalent forms:

\begin{equation}\label{Setup72}
\begin{aligned}
\du \ml r \dv \phi \mr + (\dv r)(\du \phi) &= - \ii \e A_u \dv (r\phi) - \ii \e \frac{Q \phi (\du r) (\dv r)}{r(1-\mu)}, \\
\dv \ml r \du \phi \mr + (\du r)(\dv \phi) &= - \ii \e A_u \dv (r\phi) - \ii \e \frac{Q \phi (\du r) (\dv r)}{r(1-\mu)},
\end{aligned}
\end{equation}

\begin{equation}\label{Setup73}
\begin{aligned}
D_u(r\dv\phi) + \dv r(D_u \phi) &= -\ii\e \frac{Q\phi(\du r)(\dv r)}{r(1-\mu)}, \\
\dv(r D_u \phi) + \du r(\dv \phi) &= \ii\e \frac{Q\phi(\du r)(\dv r)}{r(1-\mu)}.
\end{aligned}
\end{equation}

It is also instructive to compute the evolution equations for $m$ and for $\mu$. These equations are

\begin{equation}\label{Setup24}
\begin{aligned}
\du m
&= -\frac{2\pi r^2 (1-\mu) |D_u \phi|^2}{(-\du r)} -  \frac{Q^2 (-\partial_u r)}{2 r^2},
\end{aligned}
\end{equation}

\begin{equation}\label{Setup25}
\begin{aligned}
\dv m &= \frac{2\pi r^2 (1-\mu)|\dv \phi|^2}{\dv r} +  \frac{Q^2 \dv r}{2 r^2},
\end{aligned}
\end{equation}

\begin{equation}\label{Setup26}
\begin{aligned}
\du \mu
&= - \frac{4 \pi r (1-\mu) |D_u \phi|^2}{(-\du r)} + \frac{(-\du r)}{r}\ml \mu - \frac{Q^2}{r^2}\mr,
\end{aligned}
\end{equation}

\begin{equation}\label{Setup27}
\begin{aligned}
\dv \mu
&= \frac{4 \pi r (1-\mu) |\dv \phi|^2}{\dv r} - \frac{\dv r}{r} \ml \mu - \frac{Q^2}{r^2}\mr.
\end{aligned}
\end{equation}

In addition, the terms $\dv(Q^2/r^2)$, $\du(Q^2/r^2)$, $\dv A_u$ appear fairly often in later sections. We can obtain their expressions via using (<ref>), (<ref>),  and (<ref>) as follows:

\begin{equation}\label{Setup29}
\dv \ml \frac{Q^2}{r^2}\mr = -8 \pi \e Q \im(\phi^\dagger \dv \phi) - \frac{2 Q^2 \dv r}{r^3},
\end{equation}

\begin{equation}\label{Setup29.1}
\du \ml \frac{Q^2}{r^2}\mr = 8 \pi \e Q \im(\phi^\dagger D_u \phi) - \frac{2 Q^2 \du r}{r^3},
\end{equation}

\begin{equation}\label{Setup30}
\dv A_u = \frac{2 Q \du r \dv r}{r^2(1-\mu)}.
\end{equation}

A physical quantity of great interest in this paper is the Doppler exponent. For any $(u,v), (\overline{u},v) \in \mathcal{Q}$, we define $\gamma(u,v;\overline{u})$ to be

\begin{equation}\label{Setup90}
\gamma(u,v;\overline{u}) := \int_{\overline{u}}^u \frac{-\du r}{r}\frac{\mu - Q^2/r^2}{1-\mu}(u',v) \D u'.
\end{equation}

In particular, if we consider initial data prescribed along the $u = 0$ hypersurface, we denote $\gamma(u,v)$ to be the associated Doppler exponent at $(u,v)$, given by

\begin{equation}\label{gamma}
\gamma(u,v) := \int_{0}^u \frac{-\du r}{r}\frac{\mu - Q^2/r^2}{1-\mu}(u',v) \D u'.
\end{equation}

For the uncharged case, in [21, 22] Christodoulou employed this quantity to study the instability mechanism. In this paper, in addition to the instability theorems, we also use $\gamma(u,v)$ to establish the $C^1$ extension theorem. Note that the etymology of this term in this paper stems from the following Doppler identity derived from (<ref>):

\begin{equation}\label{Setup92}
(\dv r)(0,v) = (\dv r)(u,v) \exp \ml \int^u_0 \frac{-\du r}{r}\frac{\mu - Q^2/r^2}{1-\mu}(u',v) \D u' \mr = (\dv r)(u,v)e^{\gamma(u,v)}.
\end{equation}

Furthermore, we will refer to $e^{\gamma(u,v)}$ as the Doppler factor, and $\gamma(u,v)$ as the associated Doppler exponent. Physically, we can view the quantity $\dv r(u,v)$ as a proxy for the local scale factor, which measures the wavelength of a photon emitted (to the past) from $(u,v)$. The Doppler identity in (<ref>) implies that an observer at $(0,v)$ would observe the same photon with its wavelength scaled by a factor of $e^{\gamma(u,v)}$. On the other hand, if a photon is emitted from $(0,v)$ towards $(u,v)$, an observer at $(u,v)$ would observe the same photon with its wavelength scaled by a factor of $e^{-\gamma(u,v)}$. In both cases, if $\gamma(u,v) > 0$, the former corresponds to the redshift effect, while the latter corresponds to the blueshift effect. The Doppler factor is thus a proxy for the extent in which the wavelength of a photon is being redshifted or blueshifted. A parallel argument using the notion of proper time and length to justify the redshift and blueshift effects is given in [28].

It is worth noting that along $\Gamma$ the Doppler exponent $\gamma(v,v)$ defined in (<ref>) is the integral of a dimensionless quantity. As we will see in Theorem <ref>, the breakdown of regularity is signaled by the divergence of $\gamma(v,v)$. This is consistent with Christodoulou's comment in [22] on finding an analogous quantity to the integration of vorticity in the breakdown criterion for the incompressible 3D Euler equation as in [13].

We can further compute the derivatives of $\gamma$ with respect to $u$ and $v$ as follows. For the former, we have

\begin{equation}\label{Setup93}
\du \gamma(u,v) = \frac{-\du r}{r}\frac{\mu - Q^2/r^2}{1-\mu}(u,v)
\end{equation}

if $\frac{-\du r}{r}\frac{\mu - Q^2/r^2}{1-\mu}(u,v)$ is an integrable function in its corresponding domain. For the latter, we  first calculate

\begin{equation}\label{Setup94}
\begin{aligned}
&\dv \ml \frac{-\du r}{r} \frac{\mu - Q^2/r^2}{1-\mu}\mr \\
=\; & \dv \ml \frac{-\du r}{r}\mr \ml \frac{\mu - Q^2/r^2}{1-\mu}\mr + \ml \frac{-\du r}{r}\mr \dv \ml \frac{1-Q^2/r^2}{1-\mu} - 1\mr \\
=\; & \frac{(-\du \dv r) r + (\dv r)(\du r)}{r^2}\ml \frac{\mu - Q^2/r^2}{1-\mu}\mr  + \ml \frac{-\du r}{r}\mr \ml \frac{-\dv(Q^2/r^2)}{1-\mu} + \frac{1-Q^2/r^2}{(1-\mu)^2} \dv \mu \mr.
\end{aligned}
\end{equation}

With (<ref>), we have

\begin{equation}\label{Setup95}
\frac{(-\du \dv r) r + (\dv r)(\du r)}{r^2}\ml \frac{\mu - Q^2/r^2}{1-\mu}\mr  = \frac{(\dv r)(\du r)}{r^2} \ml \ml \frac{\mu - Q^2/r^2}{1-\mu}\mr \ml 1 - \frac{\mu - Q^2/r^2}{1-\mu} \mr \mr.
\end{equation}

Employing (<ref>) and (<ref>), we also rewrite

\begin{equation}\label{Setup96}
\begin{aligned}
& \ml \frac{-\du r}{r}\mr \ml \frac{-\dv(Q^2/r^2)}{1-\mu} + \frac{1-Q^2/r^2}{(1-\mu)^2} \dv \mu \mr \\
=\; & \ml \frac{-\du r}{r}\mr \ml \frac{8 \pi \e Q \im(\phi^\dagger \dv \phi)}{1-\mu} + \frac{2 Q^2 \dv r}{r^3(1-\mu)} + \frac{1-Q^2/r^2}{(1-\mu)^2} \ml \frac{4 \pi r (1-\mu)|\dv \phi|^2}{\dv r} + \frac{- \dv r}{r}\ml \mu - \frac{Q^2}{r^2} \mr \mr \mr \\
=\; &\ml \frac{\du r \dv r}{r^2} \mr \ml - \frac{2 Q^2}{r^2(1-\mu)} + \frac{(\mu - Q^2/r^2)(1-Q^2/r^2)}{(1-\mu)^2}\mr \\
& \; + \ml \frac{- \du r}{r}\mr\ml  \frac{8 \pi \e Q \im(\phi^\dagger \dv \phi)}{1-\mu} \mr + \ml \frac{- \du r}{r}\mr \ml \frac{4 \pi r (1-Q^2/r^2)|\dv \phi|^2}{\dv r(1-\mu)} \mr.
\end{aligned}
\end{equation}

Combining (<ref>) and (<ref>) back to (<ref>), we hence obtain

\begin{equation}\label{Setup97}
\begin{aligned}
\dv \ml \frac{-\du r}{r} \frac{\mu - Q^2/r^2}{1-\mu}\mr = T_1 + T_2 + T_3
\end{aligned}
\end{equation}

\begin{equation}\label{Setup98}
T_1 := \frac{-\du r}{r}\ml \frac{8 \pi \e Q \im(\phi^\dagger \dv \phi)}{1-\mu}\mr, \; T_2 := \frac{-\du r}{r}\ml \frac{4\pi r(1-Q^2/r^2)|\dv \phi|^2}{\dv r(1-\mu)}\mr, \;
T_3 := \frac{2 \du r \dv r}{r^2}\ml \frac{\mu - 2Q^2/r^2}{1-\mu}\mr.
\end{equation}

Back to (<ref>), this implies

\begin{equation}\label{Setup99}
\dv \gamma(u,v) = \int_0^u (T_1 + T_2 + T_3)(u',v) \D u'
\end{equation}

if we can justify the interchange between the derivative and the integral.

§.§ Key Elements and New Ingredients
In this subsection, we outline some central ideas and remark on several new ingredients for each of the important steps.

§.§.§ Novel Estimates for $Q$ and $\phi$
In Section <ref>, we will show that $|Q|$ and $|Q\phi|$ enjoy enhanced estimates, which are heavily used throughout this paper. To explain this, we start by listing the estimate for $\phi$:

\begin{equation}\label{phi intro}
\begin{split}
|\phi|(u,v)\leq& \; |\phi|(\bar{u}, v)+\int_{\bar{u}}^u|D_u\phi|(u',v)du'\\
\leq& \; |\phi|(\bar{u},v)+\frac{1}{(4\pi)^{{\frac12}}}\bigg(\int_{\bar{{u}}}^u\frac{4\pi r|D_u\phi|^2}{-\partial_u r}(u',v)du'\bigg)^{\frac12}\bigg(\int_{\bar{u}}^u\frac{-\partial_u r}{r}(u',v)du'\bigg)^{\frac{1}{2}}\\
\leq& \; |\phi|(\bar{u},v)+\frac{1}{(4\pi)^{\frac12}}\bigg(\log \ml \frac{\frac{1-\mu}{\partial_v r}(u,v)}{\frac{1-\mu}{\partial_v r}(\bar{u},v)} \mr \bigg)^\frac12 \bigg(\log\ml \frac{r(\bar{u},v)}{r(u,v)} \mr \bigg)^\frac12.
\end{split}
\end{equation}

Here, data along $u=\bar{u}$ are considered given. The first inequality in (<ref>) utilizes Grönwall's inequality while  the second inequality employs
$$\partial_u \log \ml \frac{1-\mu}{\partial_v r} \mr =\frac{4\pi r|D_u\phi|^2}{-\partial_u r}.$$

Note that the above identity also implies $\du (\frac{\dv r}{1-\mu})\leq 0$, and this could give an upper bound for $\frac{\dv r}{1-\mu}$. However, it fails to offer an upper bound for $\frac{1-\mu}{\dv r}$ and $\log\frac{\frac{1-\mu}{\partial_v r}(u,v)}{\frac{1-\mu}{\partial_v r}(\bar{u},v)}$. Hence, potentially utilizing (<ref>) to control $\phi$ and related terms could lead to non-uniform bounds.

In this paper, we overcome this problem and similar difficulties by spotting and employing a new strategy. Take the estimate for $|Q|$ as an example. We first consider the expression for $\int_u^v \partial_v Q(u, v')dv'$. Applying the above estimate (<ref>) for $\phi$ and the Hölder's inequality, as shown in (<ref>)-(<ref>), for each give $\chi\in(0,1)$, we obtain

\begin{equation*}
\begin{aligned}
|Q|&(u,v)\leq \int^v_u 4\pi \e r^{2}|\phi||\dv \phi| (u,v') \mathrm{d}v' \\
\leq& \; 4\pi \e  \int^v_u \frac{r^2(u,v')}{(4\pi)^{\frac{1}{2}}}\ml  \log \ml \frac{\frac{1 - \mu}{ \dv r}(u,v')}{\frac{1 - \mu}{ \dv r}(\overline{u},v')}\mr  \mr^{\frac{1}{2}}\ml \log \ml \frac{r(\overline{u},v')}{r(u,v')}\mr \mr^{\frac{1}{2}}|\dv \phi| (u,v') \mathrm{d}v'+l.o.t.  \\
\leq & \; \frac{1}{(8\pi^2)^{\frac{1}{2}}}\ml \int^v_u \frac{2\pi r^2(1-\mu)|\dv \phi|^2}{\dv r} (u,v') \mathrm{d}v'\mr^{\frac{1}{2}} \\
&\times \ml \int^v_u \frac{ r^2 \dv r}{1 - \mu}(u,v') \cdot \ml \log \ml \frac{\frac{1 - \mu}{ \dv r}(u,v')}{\frac{1 - \mu}{ \dv r}(\overline{u},v')}\mr  \mr \cdot \ml \log \ml \frac{r(\overline{u},v')}{r(u,v')}\mr \mr \mathrm{d}v'\mr^{\frac{1}{2}}+l.o.t. \\
\leq & \; \frac{m^{\frac12}(u,v)}{(8\pi^2)^{\frac{1}{2}}} \ml \int^v_u r^2 (u,v') \cdot \ml \frac{\dv r}{1 - \mu}(\overline{u},v') \mr \cdot
\ml \frac{\frac{\dv r}{1 - \mu}(u,v')}{\frac{\dv r}{1 - \mu}(\overline{u},v')}\mr \cdot \ml \log \ml \frac{\frac{1 - \mu}{ \dv r}(u,v')}{\frac{1 - \mu}{ \dv r}(\overline{u},v')}\mr  \mr \cdot \ml \log \ml \frac{r(\overline{u},v')}{r(u,v')}\mr \mr \mathrm{d}v'\mr^{\frac{1}{2}}+l.o.t. \\
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\leq & \; C_1(\bar{u}, v; \chi) \mu^{\oh}(u,v)r^{\oh}(u,v) \cdot \ml \int^v_u r^{2-\chi} (u,v') \cdot r^{\chi}(\overline{u},v') \cdot \ml \frac{r^{\chi}(u,v')}{r^{\chi}(\overline{u},v')}\mr \cdot
\ml \log \ml \frac{r(\overline{u},v')}{r(u,v')}\mr \mr \mathrm{d}v'\mr^{\frac{1}{2}}+l.o.t. \\
\leq & \; C_1(\bar{u}, v; \chi) \mu^{\oh}(u,v) \cdot r^{\frac{3}{2} - \frac{\chi}{2}}(u,v) \cdot r^{\frac{\chi}{2}}(\overline{u},v) \cdot (v - u)^{\oh} \\
\end{aligned}
\end{equation*}

with constant $C_1(\bar{u}, v; \chi)$ provided in (<ref>). Here, to get rid of the dangerous $\bigg(\log\bigg(\frac{\frac{1-\mu}{\partial_v r}(u,v)}{\frac{1-\mu}{\partial_v r}(\bar{u},v)} \bigg) \bigg)^\frac12$ term in the second line, we crucially utilize the fact that
after applying the Hölder's inequality for $|\partial_v\phi|$, there is an additional $\frac{\dv r}{1-\mu}$ introduced in the expression (as in the third inequality). By viewing $\frac{\partial_v r}{1-\mu}$ as $x$, we then employ the inequality that, for any $x \in (0,1)$ and $c > 0$, we have

$$x^c \log \ml \frac{1}{x}\mr \leq \frac{1}{ce}.$$

Hence, a uniform upper bound is obtained, and the worrisome term in the logarithm is thus controlled. Note that this strategy is employed throughout this paper. With this strengthened bound for $|Q|$, we arrive at yet another frequently used estimate, as follows:

\begin{equation}
\bigg(\mu-\frac{Q^2}{r^2}\bigg)(u,v)\geq \mu(u,v) \ml 1-C_1^2(\bar{u}, v; \chi)r^{1-\chi}(u,v)r^{\chi}(\bar{u},v)(v-u) \mr >0.
\end{equation}

Besides $|Q|$, we also derive an improved estimate for $|Q\phi|$. This is done by utilizing $\partial_v Q=-4\pi e r^2 \mbox{Im}(\phi^{\dagger}\partial_v\phi)$ together with (<ref>)-(<ref>) to obtain

\begin{equation*}
\begin{aligned}
|Q\phi|^{2}(u,v)  \leq&\; \int^v_u 2 |Q\phi| \dv(Q\phi) (u,v') \D v' \\
\leq& \; \int^v_u 2  (4\pi \e r^2 |\phi|^{2} |Q|+ |\phi||Q|^{2})|\dv \phi| (u,v') \D v' \\
\leq& \; 2  \ml \int^v_u \frac{\dv r}{2\pi r^2 (1-\mu)}(4\pi \e r^2 |\phi|^{3} |Q| + |\phi||Q|^{2})^2  \D v' \mr^{\frac{1}{2}} \ml \int^v_u  \frac{2 \pi r^2(1-\mu)|\dv \phi|^2}{\dv r} \D v'\mr^{\frac{1}{2}}\\
\leq& \; 2  \ml \int^v_u \frac{\dv r}{2\pi r^2 (1-\mu)}(4\pi \e r^2 |\phi|^{3} |Q| + |\phi||Q|^{2})^2  (u,v') \D v' \mr^{\frac{1}{2}} m(u,v)^{\frac{1}{2}}\\
\leq& \; \mu(u,v)^{\frac12} r^{2-\frac{\chi}{2}}(u,v).
\end{aligned}
\end{equation*}

For the final inequality, we use (<ref>) to control $|\phi|$ and apply our new strategy with the gained factor $\frac{\partial_v r}{1-\mu}$. Note that $|Q\phi|$ obeys a better bound than the product of individual bounds for $|Q|$ and $|\phi|$.

§.§.§ Trapped Surface Formation Criterion
In this paper, we take a new approach to establish the desired trapped surface formation criterion for the charged case. With these new arguments, in the appendix, we also give a direct proof for the uncharged case to retrieve Christodoulou's sharp criterion in [18]. Let us begin by first explaining some key calculations in [18]. With the mass ratio $\eta=\frac{m_2-m_1}{r_2}$ introduced by Christodoulou, for the uncharged case, we can compute the following:

\begin{equation}\label{eta intro1}
\begin{split}
\frac{\D \eta}{\D x} = \frac{\frac{\mathrm{d}\eta}{\mathrm{d}u}}{\frac{\mathrm{d}x}{\mathrm{d}u}} =-\frac{\eta}{x}-\frac{16\pi\dv r_2 \Omega_2^{-2}}{x\du r_2}\bigg(r_2^2|\du \phi_2|^2-\frac{\Omega_1^{-2}\dv r_1}{\Omega_2^{-2}\dv r_2}r_1^2|\du \phi_1|^2 \bigg).
\end{split}
\end{equation}

Based on the particular structures of the Einstein-scalar field system, via using $\Theta:=r_2\du \phi_2-r_1\du \phi_1$ and
several ingenious lemmas, one could further prove

\begin{equation}\label{eta intro2}
\begin{split}
r_2^2|\du \phi_2|^2-\frac{\Omega_1^{-2}\dv r_1}{\Omega_2^{-2}\dv r_2}r_1^2|\du \phi_1|^2\leq&\; r_2^2|\du \phi_2|^2-e^{\eta}r_1^2|\du \phi_1|^2\\
\leq&\; \Theta^2+2\Theta\du \phi_1 r_1+(1-e^{\eta})r_1^2|\du \phi_1|^2\\
\leq&\; \bigg(1+\frac{1}{e^{\eta}-1}\bigg)\Theta^2\leq\bigg(1+\frac{1}{\eta}\bigg)\Theta^2\\
\leq&\; \bigg(1+\frac{1}{e^{\eta}-1}\bigg)\cdot\frac{\du r_2}{8\pi\Omega_2^{-2}\dv r_2}(m_2-m_1)\bigg(\frac{1}{r_2}-\frac{1}{r_1}\bigg).
\end{split}
\end{equation}

Together with (<ref>), this gives

\begin{equation}\label{eta intro2.5}
\begin{split}
\frac{d\eta}{d x}\leq&-\frac{\eta}{x}-\frac{2}{x}\bigg(1+\frac{1}{\eta}\bigg)\bigg(\frac{1}{r_2}-\frac{1}{r_1}\bigg)(m_2-m_1)\\
\end{split}
\end{equation}

Solving this ODE inequality then leads to a desired criterion for the formation of trapped surfaces. The first author and Lim tried to generalize this approach in [18] to the charged scenario. However, for the charged case, a few inequalities in (<ref>) cannot be as sharp as for the uncharged case. Hence, the result obtained by An-Lim in [6] is slightly weaker than the desired one, making it incompatible with the instability arguments we seek to employ.

We then state our new approach in this paper as follows. We begin by demonstrating it for the uncharged case.[The complete proof for the uncharged case is in the appendix.]
With the mass ratio $\eta$, we compute and rewrite the expression of its derivative as

\begin{equation}\label{eta intro3}
\frac{\D \eta}{\D x} = \frac{\frac{\mathrm{d}\eta}{\mathrm{d}u}}{\frac{\mathrm{d}x}{\mathrm{d}u}} = \frac{\du \ml \frac{2(m_2 - m_1)}{r_2} \mr}{\frac{\du r_2}{r_2(u_0)}} = - \frac{\eta}{x} - \frac{2}{x(\du r_2)}\int^{v_2}_{v_1} -\du \dv m \; \D v'.
\end{equation}

Employing $\dv m = 2\pi r^2(1-\mu)(\dv \phi)^2/\dv r$, we further calculate

\begin{equation*}
\begin{aligned}
- \du\dv m  &= -\du \ml \frac{2\pi r^2(1-\mu)(\dv \phi)^2}{\dv r} \mr \\
&= -2 \pi \ml \du\ml \frac{(1-\mu)}{\dv r}\mr r^2(\dv \phi)^2 + \ml \frac{2(1-\mu)}{\dv r}\mr r (\dv \phi) \du (r\dv \phi) \mr \\
&= 8\pi^2 r^3 (1-\mu) \ml  (\du r)(\dv r) \frac{(\du \phi)^2 (\dv \phi)^2}{(\du r)^2(\dv r)^2} - \frac{1}{2\pi r^2} (\du r)(\dv r) \frac{(\dv \phi)(\du \phi)}{(\dv r)(\du r)} \mr.
\end{aligned}
\end{equation*}

Utilizing the elementary inequality $-x^2 + ax \leq {a^2}/{4}$ for $x, a\in \mathbb{R}$, we immediately obtain

\begin{equation*}
\begin{aligned}
\int^{v_2}_{v_1} -\du \dv m \; \D v' &= \int^{v_2}_{v_1} 8\pi^2 r^3 (1-\mu)(-\du r)(\dv r) \ml  -\frac{(\du \phi)^2 (\dv \phi)^2}{(\du r)^2(\dv r)^2} + \frac{1}{2\pi r^2} \frac{(\dv \phi)(\du \phi)}{(\dv r)(\du r)} \mr \; \D v' \\
&\leq \int^{v_2}_{v_1} 8\pi^2 r^3 (1-\mu)(-\du r)(\dv r) \ml \frac{1}{16\pi^2 r^4}\mr \; \D v' \\
&\leq \frac{(-\du r)_2}{2}\int^{v_2}_{v_1} \frac{\dv r}{r} \D v' = \frac{(-\du r)_2}{2}\log \ml 1 + \frac{r_2(u) - r_1(u)}{r_1(u)}\mr\leq \frac{(-\du r)_2}{2}\delta(u).
\end{aligned}
\end{equation*}

Here, we define $\delta(u):=(r_2(u)-r_1(u))/r_1(u)$, and we further denote $\delta_0$ to be the corresponding initial value.

Back to (<ref>), this leads to an ODE inequality of the form

\begin{equation*}
\begin{aligned}
\frac{\D \eta}{\D x} &= - \frac{\eta}{x} + \frac{\delta(u)}{x} \leq - \frac{\eta}{x} + \frac{\delta_0}{x(x(1+\delta_0)-\delta_0)}.
\end{aligned}
\end{equation*}

Upon solving the ODE inequality above, we then get the desired result. In this new approach, we tend to use identities instead of inequalities, which relies on fewer requirements of the structures of equations.

We now generalize this new approach to the charged scenario. We first define a new quantity

\begin{equation*}
\eta^*(u) := \frac{2(m(u,v_2) - m(u,v_1))}{r(u,v_2)} - \frac{2}{r(u,v_2)}\int^{v_2}_{v_1} \frac{Q^2(\dv r)}{2r^2}(u,v')\mathrm{d}v'.
\end{equation*}

Note that outside the trapped region, this quantity is less or equal to the standard mass ratio, and we refer to it  as the reduced mass ratio, denoted by $\eta^*(u)$. We conclude that, for the charged case, if it satisfies the condition:

\begin{equation}\label{TSF2'}
\eta^*(u_0) > C_{24} \frac{\delta(u_0)}{1+\delta(u_0)} \log \ml \frac{1}{\delta(u_0)}\mr + C_{23} \ml \frac{\delta(u_0)r_1(u_0)}{1-\mu^*(u_0)}\mr
\end{equation}

with the constants $C_{23}$, $C_{24}$ given in (<ref>), (<ref>), and $\mu^*(u_0)$ being the supremum of $\mu(u,v)$ along $u=u_0$, then a MOTS or a trapped surface is guaranteed to form in the evolution.  We emphasize that this is a crucial criterion. The factor $\frac{1}{1-\mu^*(u_0)}$ is related to the blueshift effect, and the additional $r_1(u_0)$ term will play a pivotal role in establishing the corresponding instability theorems.

For the proof of this conclusion, we proceed as in (<ref>) and obtain

\begin{equation}\label{eta intro4}
\begin{split}
\frac{d\eta^*}{dx}=\frac{\frac{d\eta^*}{du}}{\frac{dx}{du}}=-\frac{\eta^*}{x}-\frac{2}{x(\du r_2)}\bigg(\int_{v_1}^{v_2}[-\du \dv m+\du \ml \frac{Q^2\dv r}{2r^2}\mr] \bigg).    \end{split}
\end{equation}

Applying the equations for $m$ and for $Q$, we get

\begin{equation}\label{eta intro5}
\begin{aligned}
&- \du\dv m + \du \ml \frac{Q^2 \dv r}{2r^2} \mr \\ =& \; (\dv r)(-\du r) \ml -8\pi^2 r^3 (1-\mu) \ml \frac{|D_u \phi| |\dv \phi|}{(-\du r)(\dv r)} \mr^2  - 4\pi r (1-\mu) \frac{\re{((\dv \phi)^\dagger (D_u \phi))}}{\dv r (-\du r)}\mr\\
&\; + 4\pi \e Q (-\du r) \im{(\phi^\dagger (\dv \phi))} \\
\leq&\; 8\pi^2 r^3 (\dv r)(-\du r)(1-\mu) \ml - \ml \frac{|D_u \phi| |\dv \phi|}{(-\du r)(\dv r)} \mr^2  + \frac{1}{2\pi r^2} \frac{|D_u \phi||\dv \phi|}{(-\du r)(\dv r)}\mr + 4\pi \e |Q| (-\du r) |\phi||\dv \phi|. \\
\end{aligned}
\end{equation}

Back to (<ref>), we then integrate the first two terms on the right of (<ref>) and obtain

\begin{equation*}
\begin{aligned}
&\int^{v_2}_{v_1} 8\pi^2 r^3 (\dv r)(1-\mu) \ml - \ml \frac{|D_u \phi| |\dv \phi|}{(-\du r)(\dv r)} \mr^2  + \frac{1}{2\pi r^2} \frac{|D_u \phi||\dv \phi|}{(-\du r)(\dv r)}\mr(u,v') \mathrm{d}v' \\
\leq&\; \int^{v_2}_{v_1} 2\pi^2 r^3 (\dv r)(1-\mu) \ml \frac{1}{2\pi r^2}\mr^2(u,v') \mathrm{d}v' \\
\leq&\; \frac{1}{2} \int^{v_2}_{v_1}   \frac{\dv r}{r}(u,v') \mathrm{d}v' = \frac{1}{2}\log\ml \frac{r(u,v_2)}{r(u,v_1)}\mr  = \frac{1}{2}\log \ml 1 + \frac{r(u,v_2) - r(u,v_1)}{r(u,v_1)}\mr \leq \frac{1}{2}\delta(u).
\end{aligned}
\end{equation*}

To bound the integration of the last term in (<ref>), by applying the obtained estimates for $|Q\phi|$, with $\chi \in \left( 0, \frac{1}{3}\right]$ we derive

\begin{equation*}
\begin{aligned}
\int^{v_2}_{v_1} 4\pi\e|Q||\phi||\dv \phi| (u,v') \mathrm{d}v'
&\leq \sqrt{4\pi} \e {\eta^*}^{\frac{1}{2}} r_2^{\frac{1}{2}}(u) \ml \int^{v_2}_{v_1} \frac{\dv r}{1-\mu} \frac{|Q|^2|\phi|^2}{r^2}(u,v') \mathrm{d}v'\mr^{\frac{1}{2}} \\
&\leq \sqrt{4\pi} \e C_{5} {\eta^*}^{\frac{1}{2}} r_2^{\frac{1}{2}}(u)  \ml \int^{v_2}_{v_1} \frac{\dv r}{1 - \mu}(u_0,v') r^{\frac{1}{2}-\frac{3\chi}{2}}(u,v') \mathrm{d}v'\mr^{\frac{1}{2}} \\
&\leq \frac{\sqrt{4\pi} \e C_{5}}{(1-\mu_0^*)^{\frac{1}{2}}} {\eta^*}^{\frac{1}{2}} r_2^{\frac{3}{4}-\frac{3\chi}{4}}(u)  r_1^{\frac{1}{2}}(u) \delta(u_0)^{\frac{1}{2}}.
\end{aligned}
\end{equation*}

These together with $(-\du r)(u,v') \leq (-\du r)_2(u)$ for $v' \in [v_1,v_2]$ give

\begin{equation*}
\begin{aligned}
\frac{\mathrm{d}\eta^*}{\mathrm{d}x} &\leq - \frac{\eta^*}{x} - \frac{2}{x(\du r_2)} \ml \int^{v_2}_{v_1} -\du\dv m + \du \ml \frac{Q^2 \dv r}{2r^2}\mr \mathrm{d}v'\mr \\
&\leq - \frac{\eta^*}{x} + \frac{\delta}{x} + \frac{1}{x} \frac{4\sqrt{\pi} \e C_{5}}{(1-\mu_0^*)^{\frac{1}{2}}} {\eta^*}^{\frac{1}{2}} r_2^{\frac{3}{4}-\frac{3\chi}{4}}(u)r_1^{\frac{1}{2}}(u_0) \delta^{\frac{1}{2}}(u_0) \\
&\leq - \frac{\eta^*}{x} + \frac{\delta}{x} + C_{21}x^{-\frac{1}{4}-\frac{3\chi}{4}}\ml \frac{2{\eta^*}^{\frac{1}{2}}\delta_0^{\frac{1}{2}}(r_1)_0^{\frac{1}{2}}}{(1-\mu_0^*)^{\frac{1}{2}}} \mr \\
&\leq - \frac{\eta^*}{x} + \frac{\delta}{x} + \frac{C_{21}}{x^{\frac{1}{2}}} \ml \eta^* + \frac{\delta_0\cdot(r_1)_0}{1-\mu_0^*} \mr \\
\end{aligned}
\end{equation*}

with $C_{21} = 2\sqrt{\pi} \e C_{5}r(\overline{u},v_3)^{\frac{3}{4}-\frac{3\chi}{4}}$. Solving this ODE then leads us to the desired trapped surface formation criterion for the charged case.

§.§.§ BV Area Estimates

In Section <ref>, we establish the scale-critical BV area estimates in a region $\mathcal{D}(0,v_0)$ with $v_0$ small. We start by recalling the definition for the area of a pair of functions and it is given by

$$A[f,g] = \int_{\mathcal{D}(0,v_0)} \mlm \frac{\p f}{\p u} \frac{\p g}{\p v} - \frac{\p f}{\p v} \frac{\p g}{\p u}\mrm \D u \D v.$$

Notice that, unlike the non-charged case, our system (<ref>) - (<ref>) is not invariant under the transformation $r \mapsto r$, $\Omega \mapsto \Omega$, $\phi \mapsto \phi + c$.[This is translation transformation for $\phi$.] We can no longer assume $\phi(0,0)=0$ without loss of generality as in [19] for the uncharged case. Henceforth, in this paper, we conduct renormalizations for terms involving $\phi$ and meticulously define the following quantities:

\begin{equation*}
\begin{aligned}
\phi_0 := \phi(0,0), \quad \tp := \phi - \phi_0, \quad \lambda := \dv r, \quad \nu := \du r, \\
\tilde{\theta} := r \dv \tp, \quad \quad \quad
\tilde{\zeta} := r \du \tp, \quad \quad \quad
\ttz := r D_u \phi, \\
\mfa := \dv(r\phi), \quad \mfb := \du(r\phi), \quad \alpha := \frac{\mfa}{\lambda}, \quad \beta := \frac{\mfb}{\nu}, \\
\tmfa := \dv(r\tp), \quad \tmfb := \du(r\tp), \quad \tilde{\alpha} := \frac{\alpha}{\lambda}, \quad \tilde{\beta} := \frac{\beta}{\nu},  \\
A:= \sup_{\mathcal{D}(0,v_0)} |\tmfa|, \quad B:= \sup_{\mathcal{D}(0,v_0)} |\tmfb|, \quad P:= \sup_{\mathcal{D}(0,v_0)} |\tp|.
\end{aligned}
\end{equation*}

We further set

\begin{equation*}
\begin{aligned}
\ttmfb := \tmfb + \ii \e A_u r \phi, \quad X := \sup_{u\in(0,v_0)}\int_{u}^{v_0}|\dv \tp|(u,v') \D v', \quad Y := \sup_{v \in (0,v_0)} \int_{v}^{0}|\du \tp|(u',v) \D u',
\end{aligned}
\end{equation*}

and let

$$D := TV_{\{0\} \times (0,v_0)}[\mfa] = TV_{\{0\} \times (0,v_0)}[\tmfa].$$

Note that in $\ttz, \alpha, \beta, \tilde{\alpha}, \tilde{\beta}$, we exclude the use of $\tp$, and the form $\ttmfb$ is specifically tailored for the charged scenario. Additionally, notice that the size of the initial data is measured in $D$ and is small. Our obtained area estimates are summarized in the theorem below.

Theorem. ($BV$ Area Estimates.)
Assume that
\begin{equation}\label{Area1'}
\inf_{\mathcal{D}(0,v_0)} \dv r \geq \frac{5}{12}, \quad \sup_{\mathcal{D}(0,v_0)} (-\du r) \leq \frac{2}{3}, \quad \sup_{\mathcal{D}(0,v_0)} \mu \leq \frac{1}{3}
\end{equation}

hold for $v_0$ sufficiently small.[In the regular region, employing a spacetime rescaling as conducted by Christodoulou in Section 6 of [19] can verify these assumptions.] Then there exists a constant $\varepsilon_0 \in (0,1]$, such that if the initial data satisfy

\begin{equation*}
Z = \max\{X,Y\} \leq \varepsilon_0 \quad \text{ and } \quad D := TV_{\{0\} \times (0,v_0)}[\tmfa]  \leq \varepsilon_0,
\end{equation*}

then we have $Z \lesssim D$, the total variations satisfy

\begin{equation}\label{Area40'}
\begin{aligned}
\sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\tmfa] &\lesssim D,\\
\sup_{v \in (0,v_0)}TV_{(0,v) \times \{v\}}[\ttmfb] &\lesssim D,\\
TV_{(0,v_0)}[\tmfa|_{\Gamma}]  &\lesssim D,
\end{aligned} \quad
\begin{aligned}
\sup_{u \in (0,v_0)} TV_{ \{u\} \times (u,v_0) } [\log \lambda] &\lesssim D^2, \\
\sup_{v\in(0,v_0)} TV_{(0,v) \times \{v\}}[\log |\nu|] &\lesssim D^2, \\
TV_{(0,v_0)}[\log(\lambda|_{\Gamma})] &\lesssim D^2, \\
\end{aligned}
\end{equation}

and the area estimates obey

\begin{equation}\label{Area41'}
\begin{aligned}
A[\tp^\dagger ,\tmfa/\lambda] &\lesssim D^2, \\
A[\tp^\dagger ,\ttmfb/\nu] &\lesssim D^2, \\
\end{aligned} \quad
\begin{aligned}
A[\lambda, \tp] &\lesssim D^3, \\
A[\nu, \tp] &\lesssim D^3. \\
\end{aligned}
\end{equation}

Prior to computing the relevant area estimates, we first establish a couple of lemmas in Section <ref> which we will described below.

Lemma 1. For all $(u,v) \in \mathcal{D}(0,v_0),$ we have

$$\int^{v_0}_u |\dv \tilde{\phi}|(u,v') \D v' \leq TV_{\{u\} \times (u,v_0)} [\tilde\alpha] \quad \text{ and } \quad \int^{v}_0 |\du \tilde{\phi}|(u',v) \D u' \leq TV_{(0,v) \times \{v\} } [\tilde\beta].$$

We prove this lemma by utilizing integration by parts and leveraging the fact that the relevant boundary terms vanish along the center $\Gamma$. To proceed, in Section <ref>, we then prove

Lemma 2. Assuming $D \leq 1$, $v_0$ being sufficiently small and (<ref>) holds, then we get

$$A \leq 4D, \quad B  \leq 8D, \quad P  \leq 12 D.$$

The proof of the lemma is via a bootstrapping argument. We first assume that for a fixed $(\tilde{u},\tilde{v}) \in \mathcal{D}(0,v_0),$

$$\sup_{\mathcal{D}(0,\tilde{u},\tilde{v})} |\tmfa| \leq 4D, \quad \sup_{\mathcal{D}(0,\tilde{u},\tilde{v})} |\tp|  \leq 12 D,$$

and attempt to improve on such an estimate. Since $\tp = \frac{1}{r(u,v)}\int^v_u \tmfa (u,v') \D v'$, we obtain

\begin{equation}\label{Area2'}
\sup_{\mathcal{D}(0,\tu,\tv)}|\tp| \leq \sup_{\mathcal{D}(0,\tu,\tv)}|\tmfa| \sup_{\mathcal{D}(0,\tu,\tv)}\mlm \frac{v - u}{r(u,v)}\mrm \leq 3 \sup_{\mathcal{D}(0,\tu,\tv)}|\tmfa|.
\end{equation}

To control $\tmfa$, we employ the following equation:

\begin{equation}\label{Area10'}
\du\ml \dv \ml r\tp \mr \mr = \tp \du \dv r- \ii \e  A_u \dv (r\tp) - \ii \e \phi_0 A_u \dv r  - \ii \e
\frac{\du r \dv r }{1 - \mu}\frac{Q\phi}{r}.
\end{equation}

By viewing this as a first-order ODE in $\dv(r\tp)$, with the help of an integrating factor, we deduce that

\begin{equation}\label{Area3'}
\begin{aligned}
|\dv(r\tp)|(u,v) \leq&\; |\dv (r\tp)|(0,v) + \int^u_0 |\tp|(-\du \dv r) (u',v)\D u'   \\
&+ \e \int^u_0 \frac{(-\du r)(\dv r)}{r(1-\mu)}|Q||\phi| (u',v)\D u'  + \e |\phi_0| \int^u_0 |A_u| \dv r (u',v) \D u'.
\end{aligned}
\end{equation}

To close the bootstrap, we will use $\tmfa$ to bound $|Q|$ and $|A_u|$. From (<ref>), we have

\begin{equation}\label{Area5'}
\begin{aligned}
|Q|(u,v) &\leq 4 \pi \e \int^v_u r^2 |\phi||\dv \phi|(u,v') \D v' \leq 4 \pi \e \ml |\phi_0| + \sup_{\mathcal{D}(0,\tu,\tv)}|\tp| \mr r(u,v) \int^v_u r |\dv \phi|(u,v') \D v' \\
&\leq  4 \pi \e \ml |\phi_0| + \sup_{\mathcal{D}(0,\tu,\tv)}|\tp| \mr r(u,v) \int^v_u |\dv (r\tp)|(u,v') + |\tp|(\dv r)(u,v') \D v' \\
&\leq  4 \pi \e \ml |\phi_0| + \sup_{\mathcal{D}(0,\tu,\tv)}|\tp| \mr r(u,v) \ml \ml \sup_{\mathcal{D}(0,\tu,\tv)}|\tmfa| \mr (v-u) + \ml \sup_{\mathcal{D}(0,\tu,\tv)}|\tp| \mr r(u,v) \mr \\
&\leq 24\pi\e \ml \mlm \phi_0 \mrm + \sup_{\mathcal{D}(0,\tu,\tv)}|\tp| \mr \ml \sup_{\mathcal{D}(0,\tu,\tv)}|\tmfa| \mr  r(u,v)^2.
\end{aligned}
\end{equation}

By utilizing (<ref>) and (<ref>), we further obtain

\begin{equation}\label{Area12'}
\begin{aligned}
|A_u|(u',v) &\leq \int^{v}_{u'} \frac{2|Q| (-\du r) \dv r}{r^2(1-\mu)} (u',v') \D v' \leq 24 \pi v_0 \e \ml \sup_{\mathcal{D}(0,\tu,\tv)}|\tmfa| \mr \ml \mlm \phi_0 \mrm + \sup_{\mathcal{D}(0,u',v)}|\tp| \mr.
\end{aligned}
\end{equation}

These two estimates will allow us to deduce that

\begin{equation}\label{Area9'}
\ml \sup_{\mathcal{D}(0,\tu,\tv)}|\tmfa| \mr \leq 3D \quad \mbox{ and } \quad \sup_{\mathcal{D}(0,\tu,\tv)}|\tp| \leq 3\ml \sup_{\mathcal{D}(0,\tu,\tv)}|\tmfa| \mr \leq 9D,
\end{equation}

which improves the bootstrap assumption in (<ref>). The desired estimates for $A$ and $P$ further imply the bound for $B$.

With these preparations, we start the proof of the theorem. Here, we generalize Christodoulou's ingenious arguments from [19] to the complex charged scenario. The proof in Section <ref> is relies on extensive calculations, and the estimates are intricately connected. Here, we provide an outline. For brevity, given a quantity $H$, we write $H \lesssim D$ to convey that there exists a non-negative constant $c$ (possibly depending on $\e$ and $|\phi_0|$) independent of $D$ such that $H \leq cD$. In view of Lemma 2 above, we then have

$$A \lesssim D, \quad B \lesssim D, \quad P \lesssim D.$$

Furthermore, $\dv r|_{\Gamma} = -\du r|_{\Gamma}$ and (<ref>) implies

$$\frac{5}{12} \leq \dv r(u,v) \leq \frac{1}{2}, \quad \text{ and } \quad \frac{1}{2} \leq (-\du r)(u,v) \leq \frac{2}{3}.$$

In addition, by

$$r \dv \tp = \dv \ml r \tp \mr - \tp \dv r, \quad r \du \tp = \du \ml r \tp \mr - \tp \du r, \quad \phi = \phi_0 + \tp,$$

together with estimates (<ref>) and (<ref>),
we deduce that

$$\sup_{\mathcal{D}(0,v_0)}\mlm r \dv \tp \mrm \lesssim D, \quad \sup_{\mathcal{D}(0,v_0)}\mlm r \du \tp \mrm \lesssim D, \quad \sup_{\mathcal{D}(0,v_0)}\mlm \phi \mrm \lesssim 1, \quad$$

$$|Q|(u,v) \lesssim D r^2(u,v), \quad |A_u|(u,v) \lesssim Dr(u,v), \quad |rD_u \phi|(u,v) \lesssim D.$$

We begin with deriving estimates for $\partial_v\tmfa$. This is done by computing $\du \dv \tmfa$ via using (<ref>) and we get

\begin{equation}\label{Area19'}
\du \ml \dv \tmfa \mr(u',v') - \du\ml \tp \dv \lambda \mr(u',v') = G_1(u',v')  + G_2(u',v')  + \frac{\p (\lambda, \tp)}{\p(u,v)}(u',v')
\end{equation}

G_1 &= -\ii \e \ml \phi_0 A_u + \frac{\du r }{1-\mu}\frac{Q\phi}{r} \mr \dv \lambda, \\
G_2 &=  -\ii \e \ml (\dv A_u) \tmfa + \phi_0 (\dv A_u)\dv r + A_u \dv \tmfa + \dv r \dv \ml \frac{\du r}{1-\mu}\frac{Q\phi}{r}\mr\mr.
\end{aligned}$$

For each $(u,v) \in \mathcal{D}(0,v_0)$, by integrating with respect to $u'$ from $0$ to $v$ and $v'$ from $u$ to $v_0$, we obtain

\int_u^{v_0} \mlm \dv \tmfa - \tp \dv \lambda \mrm (u,v') \D v' \leq \int^{v_0}_u \mlm \dv \tmfa \mrm(0,v') \D v' + \int_u^{v_0}\int^u_0 \ml |G_1| + |G_2| + \mlm\frac{\p(\lambda,\tp)}{\p(u,v)}\mrm \mr(u',v') \D u' \D v'.
\end{aligned}$$

Taking supremum over $u \in (0,v_0)$ and applying relevant estimates listed at the start of the proof, we derive:

\begin{equation}\label{Area13'}
\begin{aligned}
\sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\tmfa] \leq& \; D + P \sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\lambda] + A[\lambda,\tp] \\
&+ \sup_{u\in(0,v_0)}\int_u^{v_0}\int^u_0 |G_1| (u',v') \D u' \D v' + \sup_{u\in(0,v_0)}\int_u^{v_0}\int^u_0 |G_2| (u',v') \D u' \D v'.
\end{aligned}
\end{equation}

Furthermore, by applying the obtained estimates, we also show that

\begin{equation}\label{Area14'}
\begin{aligned}
\int_u^{v_0}\int^u_0 |G_1| (u',v') \D u' \D v' &\lesssim D \int_0^{u}\int^{v_0}_u |\dv \lambda| (u',v') \D v' \D u' \lesssim D \sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\lambda],
\end{aligned}
\end{equation}

\begin{equation}\label{Area15'}
\int_u^{v_0}\int^u_0 |G_2| (u',v') \D u' \D v' - \frac{1}{2}\sup_{u \in (0,v_0)}TV_{\{u\} \times (u,v_0) }[\tmfa] \lesssim  \int_u^{v_0}\int^u_0 D \D u' \D v' \lesssim D.
\end{equation}

These then simplify (<ref>) to

\begin{equation}\label{Area16'}
\begin{aligned}
\frac{1}{2}\sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\tmfa] \lesssim & \; D\ml 1 + \sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\lambda]\mr  + A[\lambda,\tp]. \\
\end{aligned}
\end{equation}

Moving forward, we proceed to derive estimates for $\ttmfb$. By (<ref>) and taking derivatives in $u$, $v$, we arrive at

$$\dv \du \ttmfb = \dv \ml \tp \du \nu \mr + G_3 + G_4 - \frac{\p(\nu,\tp)}{\p(u,v)}$$

G_3 = \ii \e \du \nu \ml \frac{\dv r}{1-\mu}\frac{Q\phi}{r}\mr \quad \text{ and } \quad G_4 = \ii \e \du r \du \ml \frac{\dv r}{1-\mu}\frac{Q\phi}{r} \mr.
\end{aligned}$$

Integrating along a future-directed outgoing null curve originating from $\Gamma$, we obtain

\begin{equation}\label{Area17'}
\begin{aligned}
\int^v_0 \mlm \du \ttmfb - \tp \du \nu\mrm(u',v) \D u' \leq&\; TV_{(0,v_0)}[\ttmfb|_{\Gamma}] + P \cdot TV_{(0,v_0)}[\nu|_{\Gamma}] + A[\nu,\tp] \\
&\; + \int^v_0 \int^v_{u'} |G_3|(u',v') \D v' \D u' + \int^v_0 \int^v_{u'} |G_4|(u',v') \D v' \D u'.
\end{aligned}
\end{equation}

By utilizing the obtained estimates, we deduce that

$$\int^v_0 \int^v_{u'} |G_3|(u',v') \D v' \D u' \lesssim D \cdot \sup_{v \in (0,v_0)}TV_{(0,v) \times \{v\}}[\nu] \quad \text{ and } \quad \int^v_0 \int^v_{u'} |G_4|(u',v') \D v' \D u' \lesssim D.$$

Furthermore, by employing relevant regularity conditions along $\Gamma$, we get

$$TV_{(0,v_0)}[\ttmfb|_\Gamma] = TV_{(0,v_0)}[\tmfa|_\Gamma], \quad TV_{(0,v_0)}[\nu|_\Gamma] = TV_{(0,v_0)}[\lambda|_\Gamma].$$

Combining these with (<ref>) then yields:

\begin{equation}\label{Area18'}
\begin{aligned}
\sup_{v \in (0,v_0)}TV_{(0,v) \times \{v\}}[\ttmfb] \lesssim & \; TV_{(0,v_0)}[\tmfa|_\Gamma] + D \ml 1 + TV_{(0,v_0)}[\lambda|_{\Gamma}] + \sup_{v \in (0,v_0)} TV_{(0,v) \times \{v\}}[\nu]\mr  + A[\nu,\tp].
\end{aligned}
\end{equation}

On the other hand, if we first integrate along a future-directed incoming null curve $v = u$ starting from $C_0^+$ to $(u,u) \in \Gamma$, and then integrate with respect to $u$ for $u \in (0,v_0)$, together with the boundary conditions along $\Gamma$, we will arrive at:

TV_{(0,v_0)}[\tmfa|_{\Gamma}] &\leq  P \cdot TV_{(0,v_0)}[\lambda|_{\Gamma}] + D  + A[\lambda,\tp] + \int_0^{v_0}\int^{v_0}_{u'} |G_1|(u',v)  + |G_2|(u',v) \D v \D u'.
\end{aligned}$$

Using a similar argument for the double integrations of $|G_1|$ and $|G_2|$ as in (<ref>) and (<ref>), we then deduce that

\begin{equation}\label{Area20'}
\lesssim D\ml 1 + TV_{(0,v_0)}[\lambda|_{\Gamma}] + \sup_{u\in(0,v_0)}TV_{\{u\}\times(u,v_0)}[\lambda]\mr + A[\lambda,\tp].
\end{equation}

Next, we will attempt to control $\sup_{u\in(0,v_0)}TV_{\{u\}\times(u,v_0)}[\lambda]$ appearing in (<ref>). Starting from (<ref>),  by applying (<ref>), (<ref>), and by carefully exploiting the relevant cancellations between the terms, we can show that

\begin{equation}\label{Area23'}
\begin{aligned}
\dv \du \log \lambda =&\; \du\ml \frac{\mu - Q^2/r^2}{1-\mu}\frac{\dv r}{r}\mr + \frac{4\pi}{r^2}\ml \frac{1-Q^2/r^2}{1-\mu}\mr\ml \frac{(r|\dv \phi|)^2(\du r)}{(\dv r)} - \frac{(r|D_u \phi|)^2(\dv r)}{\du r}\mr\\
&+ \ml \frac{8\pi \e Q}{1-\mu} \mr \ml \frac{1}{1-\mu}\mr\ml (\du r)  \im(\phi^{\dagger}\dv \phi ) + (\dv r)\im(\phi^{\dagger}D_u \phi)\mr
\end{aligned}
\end{equation}

\begin{equation}\label{Area24'}
\begin{aligned}
\du \dv \log(-\nu) =&\; \dv\ml \frac{\mu - Q^2/r^2}{1-\mu}\frac{\du r}{r}\mr - \frac{4\pi}{r^2}\ml \frac{1-Q^2/r^2}{1-\mu}\mr\ml \frac{(r|\dv \phi|)^2(\du r)}{(\dv r)} - \frac{(r|D_u \phi|)^2(\dv r)}{\du r}\mr\\
&- \ml \frac{8\pi \e Q}{1-\mu} \mr \ml \frac{1}{1-\mu}\mr\ml (\du r)  \im(\phi^{\dagger}\dv \phi ) + (\dv r)\im(\phi^{\dagger}D_u \phi)\mr.
\end{aligned}
\end{equation}

\begin{equation*}
\tf(u,v) := 4\pi\re\ml \frac{r (\dv \tp)^\dagger}{r}\ml \frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu}\mr\mr(u,v)
\end{equation*}

\begin{equation*}
\tfs(u,v) := 4\pi\re\ml \frac{r (\du \tp)^{\dagger}}{r}\ml \frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu}\mr \mr(u,v).
\end{equation*}

By performing laborious computations and utilizing the fact that $( \du \tp)^{\dagger}(\dv \tp) - (\du \tp)(\dv \tp)^\dagger$ is purely imaginary, along with the identity below,

$$|D_u \phi|^2 = (\du \phi + \ii \e A_u \phi)(\du \phi - \ii \e A_u \phi)^\dagger = |\du \tp|^2 - 2 \e A_u \im(\phi \du \phi^\dagger) + \e^2 A_u^2 |\phi|^2,$$

we derive

\begin{equation}\label{Area25'}
\begin{aligned}
\du\ml \dv \log \lambda \mr = \du \ml \frac{\mu - Q^2/r^2}{1-\mu}\frac{\dv r}{r} - \tf\mr + 4\pi \re \ml \frac{\p(\tp^\dagger,\ttmfb/\nu)}{\p(u,v)}\mr + E_5 + E_6
\end{aligned}
\end{equation}

\begin{equation}\label{Area26'}
\begin{aligned}
\dv\ml \du \log (-\nu) \mr = \dv \ml \frac{\mu - Q^2/r^2}{1-\mu}\frac{\du r}{r} + \tfs\mr - 4\pi \re \ml \frac{\p(\tp^\dagger,\tmfa/\lambda)}{\p(u,v)}\mr - E_5 + E_6.
\end{aligned}
\end{equation}

Here, $E_5$ and $E_6$ are terms satisfying $|E_5| \lesssim D^2$ and $|E_6| \lesssim D^2$. From (<ref>), for a fixed $(u,v) \in \mathcal{D}(0,v_0)$, by first integrating along a future-directed incoming null curve and further integrating for $v \in (u,v_0)$, we obtain

\begin{equation}\label{Area27'}
\begin{aligned}
\int^{v_0}_u \mlm \dv \log \lambda - \frac{\mu - Q^2/r^2}{1-\mu}\frac{\dv r}{r} + \tf\mrm(u,v') \D v'\lesssim \int^{v_0}_u \mlm \frac{\mu - Q^2/r^2}{1-\mu}\frac{\dv r}{r} - \tf\mrm(0,v') \D v' + A[\tp^\dagger,\ttmfb/\nu] + D^2.
\end{aligned}
\end{equation}

By employing relevant derived estimates and boundary conditions along $\Gamma$, we have

\begin{equation*}
\begin{aligned}
\int^{v_0}_u \frac{\mu - Q^2/r^2}{1-\mu}\frac{(\dv r)}{r}(u,v') \D v' \lesssim DX, \quad
&\int^{v_0}_u \frac{\mu - Q^2/r^2}{1-\mu}\frac{(\dv r)}{r}(0,v') \D v' \lesssim D^2, \\
\int^{v_0}_u |\tf|(u,v') \D v' \lesssim DX, \quad
&\int^{v_0}_u |\tf|(0,v') \D v' \lesssim D^2.
\end{aligned}
\end{equation*}

Substituting these back to (<ref>), we arrive at

\begin{equation}\label{Area28'}
\sup_{u \in (0,v_0)} TV_{ \{u\} \times (u,v_0) } [\log \lambda] \lesssim D^2 + DX + A[\tp^\dagger, \ttmfb/\nu].
\end{equation}

An analogous calculation for (<ref>) yields

\begin{equation}\label{Area29'}
\begin{aligned}
\int^v_0 \mlm \du \log(-\nu) - \frac{\mu - Q^2/r^2}{1-\mu}\frac{\du r}{r} - 4\pi \tfs \mrm(u',v) \D u' \lesssim  TV_{(0,v_0)}[\log(-\nu)|_{\Gamma}] +  A[\tp^\dagger,\tmfa/\lambda] + D^2.
\end{aligned}
\end{equation}

$$\int^v_0 \frac{\mu - Q^2/r^2}{1-\mu}\frac{(-\du r)}{r}(u',v) \D u' \lesssim DY + D^2 \quad \text{ and } \quad \int^v_0 |\tfs|(u',v) \D u' \lesssim DY,$$

inequality (<ref>) implies

\begin{equation}\label{Area30'}
\sup_{v\in(0,v_0)} TV_{(0,v) \times \{v\}}[\log |\nu|] \lesssim D^2 + DY + TV_{(0,v_0)}[\log\lambda|_{\Gamma}] + A[\tp^\dagger,\tmfa/\lambda].
\end{equation}

By a similar approach for $\log(\lambda|_{\Gamma})$, we also get

\begin{equation}\label{Area31'}
TV_{(0,v_0)}[\log(\lambda|_{\Gamma})] \lesssim D^2 + A[\tp^\dagger,\ttmfb/\nu].
\end{equation}

It now remains to establish relations between the different area terms that have appeared in the estimates above. Starting from $A[\tp,\tmfa/\lambda]$, we observe that

\begin{equation}\label{Area32'}
\frac{\p(\tp^\dagger,\tmfa/\lambda)}{\p(u,v)} = \frac{(\du \tp)^\dagger}{\lambda}(\dv \tmfa - \tp \dv \lambda) + \frac{(r \dv \tp)}{\lambda^2}\frac{\p(\lambda, \tp^\dagger)}{\p(u,v)}- \frac{(\dv \tp)^\dagger E_7}{\lambda}
\end{equation}

with $E_7$ satisfying $|E_7| \lesssim Dr$. Since $A[\lambda,\tp^\dagger] = A[\lambda,\tp]$ and

$$\int_{D(0,v_0)} \frac{|\du \tp|}{\lambda}|\dv \tmfa - \tp \dv \lambda|(u,v) \D u \D v \lesssim DY\ml 1 + \sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\lambda]\mr  + Y\cdot A[\lambda,\tp^\dagger],$$

identity (<ref>) then implies

\begin{equation}\label{Area33'}
A[\tp^\dagger, \tmfa/\lambda] \lesssim DY\ml 1 + \sup_{u \in (0,v_0)}TV_{\{u\}\times(u,v_0)}[\lambda]\mr + D^2 + (D+Y)\cdot A[\lambda,\tp^\dagger].
\end{equation}

A similar strategy also gives

\begin{equation}\label{Area34'}
\begin{aligned}
A[\tp^\dagger, \ttmfb/\nu] \lesssim&\; X \cdot TV_{(0,v_0)}[\ttmfb|_{\Gamma}] + DX\ml 1 +  TV_{(0,v_0)}[\lambda|_{\Gamma}] + \sup_{v \in (0,v_0)} TV_{(0,v) \times \{v\} }[\nu] \mr \\
&+ D^2 + (D + X)\cdot A[\nu,\tp^\dagger].
\end{aligned}
\end{equation}

In view of (<ref>) and (<ref>), we proceed to estimate $A[\lambda,\tp^\dagger]$ and $A[\nu,\tp^\dagger]$. Inspired by the definition of $\tf$ above, we further define

\begin{equation}\label{Area80'}
\tg := 4 \pi \im \ml \frac{\ttheta^\dagger}{r}\ml\frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu} \mr \mr
\end{equation}

so that

\begin{equation*}
\tf + \ii \tg = 4\pi \ml \frac{\ttheta^\dagger}{r} \mr\ml  \frac{\ttheta}{\lambda} - \frac{\tz}{\nu} + E_2\mr
\end{equation*}

with $E_2$ satisfying $|E_2| \lesssim Dr^2$. By computing $\frac{1}{\lambda}\frac{\p(\lambda,\tp)}{\p(u,v)}$, we deduce that

\begin{equation*}
\begin{aligned}
\frac{1}{\lambda}\frac{\p(\lambda,\tp)}{\p(u,v)}
=&\;\frac{1}{r^2}\ml \frac{ \nu \ttheta - \lambda \tz }{\lambda \nu}\mr\ml \frac{\mu - Q^2/r^2}{1-\mu}\lambda \nu + 4 \pi \ttheta^\dagger \ttz\mr - \frac{\tz}{r}\ml \dv \log \lambda - \frac{\mu - Q^2/r^2}{1-\mu}\frac{\lambda}{r} + \tf \mr \\
&- \ii \frac{\tz \tg}{r} + \frac{4\pi \tz \ttheta^\dagger}{r^2}E_2 - 4\pi \ii \e \frac{\nu \ttheta - \lambda \tz}{\lambda \nu}\frac{\ttheta^\dagger A_u \phi}{r}.  \\
\end{aligned}
\end{equation*}

Note that the introduction of $\tg$ and the derivation of its desired bound are novel and crucial for the charged scenario. Next, we define

$$\tr := r \ml \frac{\mu - Q^2/r^2}{1-\mu}\lambda \nu + 4\pi \ttheta^\dagger \ttz\mr.$$

By estimating $\dv \log \lambda - \frac{\mu - Q^2/r^2}{1-\mu}\frac{\lambda}{r} + \tf$ via (<ref>) above, we obtain

\begin{equation}\label{Area35'}
\begin{aligned}
\int_{\mathcal{D}(0,v_0)}\mlm \frac{1}{\lambda}\frac{\p(\lambda,\tp)}{\p(u,v)} \mrm(u,v) \D u \D v \lesssim&\;  D \int_{\mathcal{D}(0,v_0)}\frac{|\tr|}{r^3}(u,v) \D u \D v
+ D \int_{\mathcal{D}(0,v_0)}\frac{|r^2\tg|}{r^3}(u,v) \D u \D v \\ &+\; D^3 + D^2Y + Y\cdot A[\tp,\ttmfb/\nu].
\end{aligned}
\end{equation}

For the first integration on the right of (<ref>), by conducting integration by parts and using the fact that $\tr/r^2|_{\Gamma} = 0$, we derive

\begin{equation*}
\begin{aligned}
\int_{\mathcal{D}(0,v_0)} \frac{|\tr|}{r^3}(u,v) \D u \D v &=
\frac{1}{2}\int^{v_0}_0 \ml \int^v_0 |\tr/\nu| \du\ml \frac{1}{r^2}\mr(u,v) \D u \mr \D v \\
&= - \frac{1}{2}\int^{v_0}_0 \frac{|\tr/\nu|}{r^2}(0,v) \D v - \frac{1}{2}\int^{v_0}_0 \ml \int^v_0 \frac{1}{r^2}\du \mlm \frac{\tr}{\nu}\mrm(u,v) \D u\mr \D v \\
&\leq \frac{1}{2}\int_{\mathcal{D}(0,v_0)}\frac{1}{r^2}\mlm \du\ml \frac{\tr}{\nu}\mr \mrm (u,v) \D u \D v.
\end{aligned}
\end{equation*}

\begin{equation*}
\du \ml \frac{\ttmfb}{\nu}\mr = \du \ml \frac{\nu \tp + r D_u \phi}{\nu}\mr = \frac{\ttz}{r} - \ii \e A_u \tp + \du \ml \frac{\ttz}{\nu}\mr
\end{equation*}

\begin{equation*}
\begin{aligned}
\ttheta^\dagger \du \ml \frac{\ttmfb}{\nu}\mr &= r\ml \frac{\p(\ttmfb/\nu,\tp^\dagger)}{\p(u,v)}\mr - \frac{\lambda(\mu - Q^2/r^2)|\ttz|^2}{r\nu(1-\mu)} + E_{10} \quad \mbox{ with } \quad |E_{10}| \lesssim D^2r,
\end{aligned}
\end{equation*}

through detailed computations, we unravel key cancellations in the expressions. Upon using them, we will arrive at

$$\du \ml \frac{\tr}{\nu}\mr = 4\pi r^2 \frac{\p(\ttmfb/\nu,\tp^\dagger)}{\p(u,v)} + E_{11} \quad \mbox{ with } \quad |E_{11}| \lesssim D^2r^2.$$

Furthermore, we deduce that

\begin{equation}\label{Area36'}
\int_{\mathcal{D}(0,v_0)} \frac{|\tr|}{r^3}(u,v) \D u \D v \lesssim A[\tp^\dagger,\ttmfb/\nu] + D^2.
\end{equation}

We also employ integration by parts to treat the second integration on the right of (<ref>). Together with $\tg|_{\Gamma}=0$, we obtain

\begin{equation}\label{Area60'}
\begin{aligned}
\int_{\mathcal{D}(0,v_0)} \frac{|r^2 \tg|}{r^3}(u,v) \D u \D v \leq \frac{1}{2}\int_{\mathcal{D}(0,v_0)}\frac{1}{r^2}\mlm \du\ml \frac{r^2\tg}{\nu}\mr \mrm (u,v) \D u \D v.
\end{aligned}
\end{equation}

Since $\tg= 4 \pi \im \ml \frac{\ttheta^\dagger}{r}\ml\frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu} \mr \mr$,
to compute $\du\ml \frac{r^2 \tg}{\nu}\mr$, we first calculate $\du\ml \frac{\ttheta^\dagger r}{\nu} \ml \frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu}\mr\mr$.Through a rigorous calculation, along with several crucial cancellations,  we derive

\begin{equation*}
\begin{aligned}
\du\ml\frac{\ttheta^\dagger r}{\nu} \ml \frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu}\mr\mr
=&\; r^2 \frac{(\tz/\nu - E_2)}{\nu^2} \frac{\p(\nu,\tp^\dagger)}{\p(u,v)} + r^2\frac{1}{\nu}\frac{\p(\tp^\dagger,\ttmfb/\nu)}{\p(u,v)} + R_1 + E_{12},
\end{aligned}
\end{equation*}

where $R_1$ representing a term that is purely real (i.e., $\im(R_1) = 0$), and $E_{12}$ satisfying $|E_{12}|\lesssim D^2 r^2$.
This implies that

\begin{equation*}
\begin{aligned}
\du\ml \frac{r^2 \tg}{\nu}\mr &=   4\pi \im \ml \du \ml \frac{\ttheta^\dagger r}{\nu} \ml \frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu}\mr\mr \mr \\
&= 4\pi \im \ml r^2 \frac{(\tz/\nu - E_2)}{\nu^2} \frac{\p(\nu,\tp^\dagger)}{\p(u,v)} + r^2\frac{1}{\nu}\frac{\p(\tp^\dagger,\ttmfb/\nu)}{\p(u,v)}  + E_{12}\mr.
\end{aligned}
\end{equation*}

Going back to (<ref>), we then deduce

\begin{equation}\label{Area37'}
\int_{\mathcal{D}(0,v_0)} \frac{|r^2 \tg|}{r^3}(u,v) \D u \D v \lesssim D \cdot A[\nu,\tp^\dagger] + A[\tp^\dagger,\ttmfb/\nu] + D^2.
\end{equation}

Plugging (<ref>) and (<ref>) into (<ref>), we obtain

\begin{equation}\label{Area38'}
A[\lambda,\tp^\dagger] \lesssim (D+Y)A[\tp^\dagger,\ttmfb/\nu] + D^2\cdot A[\nu,\tp^\dagger] + D^2Y + D^3.
\end{equation}

We also introduce

$$\tgs := 4\pi \im\ml \frac{\tz^\dagger}{r}\ml \frac{\tmfa}{\lambda} - \frac{\ttmfb}{\nu}\mr\mr \quad \text{and} \quad \trs := r\ml \frac{\mu - Q^2/r^2}{1-\mu}\lambda \nu + 4\pi \ttheta \tz^\dagger\mr.$$

Here $\tgs$ is associated with the new complex charged scenario. Employing an analogous strategy as for $\tg$ and $\tr$, while giving additional attention to boundary terms, we arrive at

\begin{equation}\label{Area39'}
\begin{aligned}
A[\nu,\tp^\dagger] &\lesssim  X \cdot TV_{(0,v_0)}[\lambda|_{\Gamma}] + (X + D) \cdot A[\tp^\dagger,\tmfa/\lambda] + D^2 \cdot A[\lambda,\tp^\dagger] + D^2X + D^3.\\
\end{aligned}
\end{equation}

Combining (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>),  (<ref>), together with $D$ and $v_0$ being small, we conclude that $Z \lesssim D$, and inequalities in (<ref>) and (<ref>) all hold.

§.§.§ C1 Extension Theorem

For spherically symmetric Einstein-scalar field system, Christodoulou proved the following:

Theorem. [19] ($C^1$ Extension Theorem with Mass Ratio $\mu$.) Set $v_* > 0$ and assume that for every $\hat v \in (0,v_*)$ there is a $C^1$ solution on $\mathcal{D}(0,\hat v)$. Then, there exists a constant $\varepsilon' > 0$, such that if we have

$$ \lim_{u \rightarrow v_*} \sup_{\mathcal{D}(u,v_*)} \mu(u,v) < \varepsilon',$$

then we can find a $\tilde{v} > v_*$ and a $C^1$ solution in the extended region $\mathcal{D}(0,\tilde{v})$.

In Section <ref>, we prove two independent $C^1$ extension theorems. The first is novel, and is as follows:
Theorem. ($C^1$ Extension Theorem with Doppler Exponent $\gamma$.) Let $v_* > 0$ and assume for every $\hat{v} \in (0,v_*)$, a $C^1$ solution exists on $\mathcal{D}(0,\hat{v})$. If we have

\begin{equation}
\sup_{\mathcal{D}(0,v_*)} \gamma < + \infty,
\end{equation}

then a $C^1$ solution can be extended to $\mathcal{D}(0,\tilde{v})$ with $\tilde{v} > v_*$.

This is our main theorem in Section <ref> and is directly related to the later instability arguments. In this paper, we also generalize the aforementioned Christodoulou's $C^1$ extension theorem with mass-ratio $\mu$ to the charged case. It is stated and proved as Theorem <ref>. For both proofs, we employ the quantities:

\begin{equation*}
\alpha:=\partial_v(r\phi) \quad \mbox{ and } \quad \alpha' := \frac{1}{\dv r} \dv \left( \frac{\dv (r\phi)}{\dv r} \right) =\frac{\dv \alpha}{\dv r}.
\end{equation*}

We further define

\begin{equation}\label{GandBu introduction}
G:= \sup_{\mathcal{D}(0,v_*)} \gamma(u,v) \quad \text{ and } \quad B'(u):=\sup_{v\in[u, v_*]}|\alpha'|(u,v).
\end{equation}

With $\alpha'$, we then compute $\du \alpha'$ and obtain

\begin{equation}\label{a' intro1}
\begin{aligned}
\du \alpha'=& \; \frac{4 \pi r (Q^2/r^2-1)|\dv \phi|^2(\dv \phi)(\du r) }{(\dv r)^3 (1- \mu)} - \frac{2\du r}{r}\ml \frac{\mu - Q^2/r^2}{1-\mu}\mr \alpha' + \frac{\du r \dv \phi}{r \dv r}\ml \frac{5Q^2/r^2 -
3 \mu}{1-\mu}\mr\\
&\; -\frac{8 \pi \e Q (\im(\phi^\dagger \dv \phi))(\dv \phi)(\du r)}{(\dv r)^2(1-\mu)}   \\
&\;  -\ii \e \ml A_u \alpha' + \frac{3 Q \du r \dv \phi}{r(1-\mu)(\dv r)} + \frac{Q\phi \du r}{r^2 (1-\mu)} + \frac{4 \pi |\dv \phi|^2 Q\phi \du r}{(1-\mu)(\dv r)^2} - \frac{4  \pi \e \phi\im(\phi^\dagger \dv \phi) r\du r }{ \dv r (1-\mu)} \mr. \\
\end{aligned}
\end{equation}

Notice that, since  $Q=0$ and $\im{\phi}=0$ for the uncharged case, there are only three terms on the right of (<ref>). For the charged scenario, additional terms arise. Henceforth, we get

\begin{equation*}
\begin{aligned}
\alpha'(u,v) e^{\int^u_{0}\ii \e A_u (u',v) \mathrm{d}u'} &= \alpha'(0,v) +  \sum_{i=1}^{11} I_i(u,v),
\end{aligned}
\end{equation*}

with $I_i$ given in (<ref>) via (<ref>).
This implies

\begin{equation}\label{alpha' Ii introduction}
|\alpha'|(u,v)\leq \sup_{\{0\}\times [0, v_*]}|\alpha'|+\sum_{i=1}^{11}|I_i|(u', v)du'.
\end{equation}

To illustrate our proof strategy, here we will focus on estimating $I_3$ and $I_{10}$. For $I_3$ we have

\begin{equation}\label{I_3' introduction}
|I_3|(u,v) \leq  4\pi \int^{u}_{0} (-\du r)\ml \frac{1}{r^2}\mr\ml \frac{1}{1-\mu}\mr\ml \frac{r|\dv \phi|}{\dv r}\mr^3 (u',v) \mathrm{d}u'.
\end{equation}

Observe that, since

\begin{equation*}
\begin{aligned}
\dv  \ml \frac{r \dv \phi}{\dv r}\mr^3 &= \dv \ml \frac{\dv (r\phi)}{\dv r} - \phi \mr^3 = 3 \alpha' \frac{r^2(\dv \phi)^2}{\dv r} - \frac{3 \dv r}{r}\ml \frac{r \dv \phi}{\dv r}\mr^3,
\end{aligned}
\end{equation*}

we then obtain

\begin{equation*}
\begin{aligned}
\dv  \ml r^3\ml \frac{r \dv \phi}{\dv r}\mr^3 \mr &= 3 \alpha' r^3 \ml \frac{r^2(\dv \phi)^2}{\dv r}\mr.
\end{aligned}
\end{equation*}

Together with $r^3\ml \frac{r \dv \phi}{\dv r}\mr^3 |_{\Gamma} = 0$, integrating the above equation with respect to $v$, we then derive

\begin{equation}\label{partial phi 3}
\begin{aligned}
r^3\ml \frac{r|\dv \phi|}{\dv r} \mr^3 (u,v)
&\leq 3B'(u)r^3(u,v)\int_u^v \frac{r^2|\dv \phi|^2}{\dv r}(u,v') \D v' \leq \frac{3}{2\pi}B'(u)r^3(u,v) \int^v_u \frac{\dv m}{1-\mu}(u,v') \D v' \\
&\leq \frac{3e^G}{2\pi(1-\mu_*)} B'(u)r^3(u,v) \int^v_u \dv m(u,v') \D v' \leq \frac{3e^G}{\pi(1-\mu_*)} B'(u)r^4(u,v) \mu(u,v).
\end{aligned}
\end{equation}

To prove the third inequality, we employ the equation

$$\du\ml \frac{1}{1-\mu(u,v)}\mr  = \frac{1}{1-\mu(u,v)} \ml -\frac{4\pi r|D_u \phi|^2}{(-\du r)} + \frac{(-\du r)}{r}\frac{\mu - Q^2/r^2}{1-\mu}\mr$$

and its implication

\begin{equation*}
\begin{aligned}
\frac{1}{1-\mu(u,v)} &= \frac{1}{1-\mu(0,v)}\exp\ml {\int^u_{0} \ml -\frac{4\pi r|D_u \phi|^2}{(-\du r)} + \frac{(-\du r)}{r}\frac{\mu - Q^2/r^2}{1-\mu}\mr (u',v) \mathrm{d}u'} \mr\\
&\leq \frac{1}{1-\mu(0,v)}\exp\ml {\int^u_{0} \ml  \frac{(-\du r)}{r}\frac{\mu - Q^2/r^2}{1-\mu}\mr (u',v) \mathrm{d}u'} \mr\leq \frac{1}{1-\mu_*}e^G\\
\end{aligned}
\end{equation*}

with $G$ defined in (<ref>).
Plugging (<ref>) back to (<ref>), we then obtain

\begin{equation*}
|I_3|(u,v) \leq \frac{12 e^G}{1-\mu_*} \int_0^u \frac{- \du r}{r} \frac{\mu}{1-\mu}(u',v) B(u') \D u'.
\end{equation*}

For $|I_{10}|$, with our improved estimate for $Q$, we deduce

\begin{equation*}
\begin{split}
|I_{10}|(u,v) \lesssim& \; \e  \int^{u}_{0} (-\du r) \ml \frac{|Q|}{r^2}\mr  \ml \frac{1}{1-\mu}\mr\ml \frac{r|\dv \phi|}{\dv r}\mr (u',v) \mathrm{d}u'\lesssim\frac{\e e^G r(0,v_*)^\oh}{1-\mu_*} \int_0^u \frac{- \du r}{r^{\oh}}(u',v) B(u') \D u'.
\end{split}
\end{equation*}

Together with our treatment for all other terms from $I_1$ to $I_{10}$, in Section <ref>, we arrive at

\begin{equation*}
B(u) \leq \sup_{\{0\}\times [0, v_*]}|\alpha'| + C_{11} + \int_0^u \ml C_9\frac{- \du r}{r} \frac{\mu}{1-\mu} + C_{10} \frac{- \du r}{r^\oh}\mr(u',\tilde{v}) B(u') \D u'.
\end{equation*}

Note that the exact expressions for constants $C_9, C_{10},$ and $C_{11}$ are provided in Section <ref>.
By Grönwall's inequality, we then obtain

\begin{equation}\label{ExtensionGronwall}
\begin{aligned}
B(u) &\leq \ml \sup_{\{0\}\times [0, v_*]}|\alpha'|+C_{11} \mr \exp\ml \int_0^u \ml C_9\frac{- \du r}{r} \frac{\mu}{1-\mu} + C_{10} \frac{- \du r}{r^\oh}\mr(u',\tilde{v}) \D u' \mr \\
&\leq \ml \sup_{\{0\}\times [0, v_*]}|\alpha'|+C_{11} \mr \exp\ml C_9 \gamma(u,\tilde{v}) + 2C_{10}r(0,\tilde{v})^{\oh} \mr \\
&\leq \ml \sup_{\{0\}\times [0, v_*]}|\alpha'|+C_{11} \mr \exp\ml C_9 G + 2C_{10}r(0,v_*)^{\oh} \mr<\infty.\\
\end{aligned}
\end{equation}

We proceed to extend the $C^1$ solution beyond the triangular boundary. We consider the rectangular strip $U_{\xi}(u_1) = [0,u_1) \times [v_*,v_*+\xi]$.
We also define

$$\overline{B'_{\xi}}(u_1) := \sup_{U_{\xi}(u_1)} (|\alpha'|)\quad \quad \mbox{ and } \quad \quad \mathcal{D}(0,u_1,v_* + \xi) := \mathcal{D}(0,u_1,v_*) \cup U_{\xi}(u_1).$$

Now, fix some $\hat{\xi}>0$. Repeating the proof outlined as above, we obtain

Lemma. For any $u_1 \in (0,v_*)$ and $\xi \in (0,\hat{\xi}]$, if $\gamma$ satisfies

\begin{equation}
\sup_{\mathcal{D}(0,u_1,v_* + \xi)} \gamma \leq 3G,
\end{equation}

then we have

\begin{equation}
\overline{B'_{\xi}}(u_1) \leq \max\{  C_{12}(G,v_*), C_{12}(3G,v_*+\hat{\xi}) \} =: B_{2}.
\end{equation}

With the above lemma, we then define

\begin{equation*}
u_2 = \sup\left\{ u_1 \in [0,v_*): \sup_{\mathcal{D}(0,u_1,v_*+\xi)} \gamma \leq 3G\right\}.
\end{equation*}

Subsequently, we prove that for $\xi$ sufficiently small, $u_2$ has to be $v_*$ and this will give uniform bound of $|\alpha'|$ in $\mathcal{D}(0,v_*,v_* + \xi)$. To achieve this, we consider the equation

\begin{equation*}
\dv \gamma(u,v) = \int_0^u (T_1 + T_2 + T_3)(u',v) \D u'.
\end{equation*}

\begin{equation*}
T_1 := \frac{-\du r}{r}\ml \frac{8 \pi \e Q \im(\phi^\dagger \dv \phi)}{1-\mu}\mr, \; T_2 := \frac{-\du r}{r}\ml \frac{4\pi r(1-Q^2/r^2)|\dv \phi|^2}{\dv r(1-\mu)}\mr, \;
T_3 := \frac{2 \du r \dv r}{r^2}\ml \frac{\mu - 2Q^2/r^2}{1-\mu}\mr.
\end{equation*}

Employing estimates previously derived, we get

\begin{equation*}
|\dv \gamma|(u,v) \leq C_{14}
\end{equation*}

\begin{equation*}
C_{14} := \ml \frac{4\pi \e e^{4G} C_8' C_{13} B_2 }{6\sqrt{2}(1-\mu_*)}(v_* + \hat \xi)^{\frac{1}{2}} + \frac{\pi e^{4G}B_2^2}{2(1-\mu_*)}\ml 1+\frac{C_{13}^2(v_* + \hat \xi)^2}{4}\mr + \frac{e^{4G}}{2(1-\mu_*)}\ml \frac{\pi B_2^2}{12} + \frac{25 C_{13}^2}{12}\mr\mr (v_* + \hat \xi).
\end{equation*}

For $(u,v) \in U_\xi(u_2)$, we then derive

\begin{equation*}
\begin{aligned}
\gamma(u,v) &\leq \gamma(u,v_*) + \int_v^{v_*}|\dv \gamma|(u,v') \D v' \leq G + C_{14}\xi \leq 2G,
\end{aligned}
\end{equation*}

if we pick $\xi := \min \left\{  \hat{\xi}, \frac{G}{C_{14}}\right\}$. This concludes the proof of Theorem <ref>.

To prove Theorem <ref>, with $0<\delta<1$, we set

$$A'(u_1):=\sup_{\mathcal{D}(u_0, u_1, v_*)}(r^{\delta}|\alpha'|).$$
Using $\mu(u,v)$ being small in a region $\mathcal{D}(u_0, u_1, v_*)$ close to $O'$, via detailed estimate for each $I_i$ in (<ref>), after conducting extensive estimates involving $Q$ and $\phi$, we prove that $A'(u_1)\leq C_{13}$ with $C_{13}$ given in (<ref>).

For all $(u,v) \in \mathcal{D}(u_0, u_1, v_*)$, using the definition of $A'(u_1)$, we then acquire

\begin{equation}\label{dv phi intro1}
\begin{aligned}
\frac{r |\dv \phi|}{\dv r}(u,v) &\leq C_{14}r^{1-\delta}(u,v), \quad  \mbox{ and } \quad C_{14}=\frac{C_{13}}{1-\delta}.
\end{aligned}
\end{equation}

Applying (<ref>), together with $m|_{\Gamma} = 0$, we deduce

\begin{equation}\label{FET108'}
m(u,v) \leq \int^v_u \ml \frac{2\pi r^2(1-\mu)|\dv \phi|^2}{\dv r} + \frac{Q^2 \dv r}{2r^2} \mr(u,v') \D v'.
\end{equation}

Using (<ref>), we hence obtain

\begin{equation*}
\begin{aligned}
\int^v_u \frac{2\pi r^2(1-\mu)|\dv \phi|^2}{\dv r}(u,v') \D v'&\leq 2\pi \int^v_u \frac{r^2|\dv \phi|^2}{(\dv r)^2}(\dv r)(u,v') \D v'\\
&\leq 2\pi \int^v_u C_{14}^2 r^{2-2\delta}(\dv r)(u,v') \D v' = \frac{2\pi C_{14}^2}{3-2\delta}r^{3-2\delta}(u,v).
\end{aligned}
\end{equation*}

At the same time, when $0<\chi<\delta<1$, the term involving $Q$ in (<ref>) also obeys

\begin{equation*}
\begin{aligned}
\int^v_u \frac{Q^2 \dv r}{2r^2}(u,v') \D v' &\leq \frac{C_{6}^2}{2}\int^v_u r^{1-\chi}(\dv r)(u,v') \D v' = \frac{C_{6}^2}{2(2-\chi)}r^{2-\chi}(u,v)
\end{aligned}
\end{equation*}

with $C_6$ given in (<ref>).  If we further restrict our choice of $\delta$ to $\left( 0, \frac{1}{2} \right]$, we then get

\begin{equation*}
m(u,v) \leq \frac{C_{15}}{2} r^{2-\chi}(u,v) \mbox{ with } C_{15}(\delta,\chi) := \frac{4\pi C_{14}^2}{3-2\delta}r^{1-2\delta+\chi}(0,v_*) + \frac{C_6^2}{(2-\chi)}.
\end{equation*}

Consequently, this implies the following improved estimate for $\mu$:

\begin{equation*}
\mu(u,v) \leq C_{15}r^{1-\chi}(u,v).
\end{equation*}

With these enhanced bounds, we then employ the estimates for $I_i$ in

\begin{equation*}
|\alpha'(u,v)| \leq \sup_{\{u_0\} \times [u_0,v_*]} (|\alpha'|) + \sum_{i=1}^{11}|I_i|(u,u_0,v)
\end{equation*}

and obtain a uniform bound for $|\alpha'(u,v)|$. This in turn implies that the solution extends to $\mathcal{D}(0,v_*).$

In a similar fashion, we can extend the $C^1$ solution beyond the triangular boundary as follows. Consider a thin rectangle region

\begin{equation*}
U_{\xi}(u_1) = [u_0,u_1) \times [v_*,v_*+\xi].
\end{equation*}

We then derive uniform bounds for quantities

$$\overline{A'_{\xi}}(u_1) := \sup_{U_{\xi}(u_1)} (r^{\delta}|\alpha'|) \quad
\mbox{ and }
\quad A'_* := \sup_{u_1\in(u_0,v_*)}A'(u_1).$$

With $0<\delta<1$, we arrive at

$$\sup_{\mathcal{D}(u_0,u_1,v_*+\xi)}(r^{\delta}|\alpha'|) \leq \max\{ A'(u_1), \overline{A'_\xi}(u_1)\} \leq \max\{A'_*, \overline{A'_\xi}(u_1)\}.$$

By replacing $A'(u_1)$ with $\max\{A'_*, \overline{A'_\xi}(u_1)\}$, we also obtain an analogous bound for $\overline{A'_{\xi}}(u_1)$. As explained before, we then employ improved estimates for $m(u,v)$ and $\mu(u,v)$, leading to the derivation of the uniform bound for $|\alpha'(u,v)|$ in the rectangular region $U_{\xi}(u_1)$. The desired conclusion then follows.

§.§.§ First Instability Theorem

In this paper, to explore the blueshift effect, we define the quantities:

\begin{equation}
\begin{split}
\gamma(u,v) :=& \int^u_{0} \frac{(- \du r)}{r} \frac{\mu - Q^2/r^2}{1 - \mu} (u',v) \mathrm{d}u',\quad \quad \gamma_0(u) := \int^u_{0} \frac{(- \du r)}{r} \frac{\mu - Q^2/r^2}{1 - \mu} (u',v_0) \mathrm{d}u',\\
\gamma_s(u,v) :=& \int^u_{0} \frac{(- \du r)}{r} \frac{\mu}{1 - \mu} (u',v) \mathrm{d}u', \quad \quad \gamma_{s, 0}(u) :=\int^u_{0} \frac{(- \du r)}{r} \frac{\mu}{1 - \mu} (u',v_0) \mathrm{d}u'.
\end{split}
\end{equation}

Note that $\gamma_s$ was introduced by Christodoulou in [21]. For our use, with $\overline{u} \in [0,v_0)$ we also set

\begin{equation}
\gamma(u,v;\overline{u}) = \tgam(u,v) := \int^u_{\overline{u}} \frac{(- \du r)}{r} \frac{\mu - Q^2/r^2}{1 - \mu} (u',v) \mathrm{d}u'.
\end{equation}

The following two properties reveal that the quantity $\gamma$ is the source of the blueshift effect:

$$(\dv r) (u,v) = (\dv r)(0,v) e^{-\gamma(u,v)},$$

$$ \frac{1}{1-\mu(u,v)} \leq \frac{1}{1-\mu(0,v)}e^{\gamma(u,v)}.$$

Hence, if $\gamma(u,v)$ is large, then $\partial_v r(u,v)$ would be close to $0$.

The first property above is obtained from integrating

\begin{equation*}
\du \log \ml \dv r\mr = \frac{\du r}{r}\ml \frac{\mu-Q^2/r^2}{1-\mu}\mr.
\end{equation*}

The second property is a result of  considering

\begin{equation*}
\begin{aligned}
\du\ml \frac{1}{1-\mu(u,v)}\mr=& \frac{1}{1-\mu(u,v)} \ml -\frac{4\pi r|D_u \phi|^2}{(-\du r)} + \frac{(-\du r)}{r}\frac{\mu - Q^2/r^2}{1-\mu}\mr,
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\frac{1}{1-\mu(u,v)} =& \frac{1}{1-\mu(0,v)}\exp\ml {\int^u_{0} \ml -\frac{4\pi r|D_u \phi|^2}{(-\du r)} + \frac{(-\du r)}{r}\frac{\mu - Q^2/r^2}{1-\mu}\mr (u',v) \mathrm{d}u'} \mr\\
\leq& \frac{1}{1-\mu(0,v)}\exp\ml {\int^u_{0} \ml  \frac{(-\du r)}{r}\frac{\mu - Q^2/r^2}{1-\mu}\mr (u',v) \mathrm{d}u'} \mr.
\end{aligned}
\end{equation*}

In the subsequent portions of that section, we will select a sequence of $u_n \in [u_2,v_0)$ and determine corresponding values of $v(u_n) \in [v_0,v_0 + \te]$, where $0<\te<1$. This selection ensures that for all $u \in [u_2,u_n]$, the following inequality holds:

\begin{equation*}
\delta(u,v(u_n)) \log \ml \frac{1}{\delta(u,v(u_n))}\mr \leq C_{33}e^{-2\gamma_0(u)}
\end{equation*}
with $C_{33}$ given in (<ref>) and $\delta(u,v(u_n)) \leq 1/2$. Then, within the considered regions, we proceed to obtain

\begin{equation*}
\begin{aligned}
\frac{1}{1-\mu(u,v)} &\leq \frac{8}{7}\frac{1}{1-\mu(u, v_0)}.
\end{aligned}
\end{equation*}

We further state an important property for the function $\delta(u,v)$ in double-null coordinates.

Lemma.  For a sequence of $u_n$ satisfying $u_n \in [u_2,v_0)$, suppose that there exist corresponding values of $v(u_n) \in [v_0,v_0 + \te]$ such that, for all $u \in [u_2,u_n]$, the following inequalities hold:
\begin{equation}\label{FIT35c'}
\delta(u,v(u_n)) \leq \min\left\{ \frac{1}{2},\frac{2(1 - \mu_0(0))e^{-\gamma_0(u)}}{3}, \frac{e^{-\frac{3}{2}(1+2\xi)\gamma_0(u)}}{2 C_{34}} \right\}
\end{equation}

\begin{equation}\label{FIT35c''}
\delta(u,v(u_n))\log \ml \frac{1}{\delta(u,v(u_n))} \mr \leq C_{33}e^{-2\gamma_0(u)}
\end{equation}

with $C_{34}$, $C_{33}$ given in (<ref>), (<ref>) and $\xi\in(0, \frac12)$. Then, for a fixed $v \in (v_0,v(u_n)]$ and for any $u_3,u_4 \in [u_2,u_n]$ with $u_3 < u_4$, we have

\begin{equation}\label{FIT83'}
\delta(u_3,v) \leq 4 e^{-\frac{1}{2}\log\ml \frac{r_0(u_3)}{r_0(u_4)}\mr + \frac{3}{2}(1+2\xi)\gamma_{0}(u_4)}\delta(u_4,v).
\end{equation}

The conclusion in (<ref>) means, along the null curve with the same $v$, the outer value of $\delta(u,v)$ can be controlled by its inner value, since $(u_4, v)$ is closer to the center compared with $(u_3, v)$. In double-null coordinates, this is new, constituting a non-trivial feature of the geometry that is crucially used in later arguments.

To prove this lemma, we first the following employ inequality:

\begin{equation*}
\frac{1}{2}\delta(u,v) \leq \log \ml \frac{r(u,v)}{r_0(u)}\mr \leq \delta(u,v)
\end{equation*}

with $u \in [u_3,u_4]$, $v \in (v_0,v(u_n)]$ and $\delta(u, v(u_n))\leq 1/2$. We then apply

\begin{equation*}
\begin{aligned}
\dv \ml \log \ml \frac{-\du r}{r}\mr\mr =& \; \frac{\dv r}{r}\ml \frac{\mu - Q^2/r^2}{1-\mu} - 1\mr\\
\leq& \;\frac{\dv r}{r} \ml \frac{\mu}{1-\mu} - 1\mr
= \frac{\dv r}{r} \ml \frac{1}{1-\mu} - 2\mr \leq \frac{\dv r}{r}\ml \frac{3}{2}\frac{1}{1-\mu_0}-2\mr.
\end{aligned}
\end{equation*}

Integrating from $v_0$ to $v$ implies that

\begin{equation*}
\begin{aligned}
\log \ml \frac{-\frac{\du r}{r}(u,v)}{\ml -\frac{\du r}{r} \mr_0(u)}\mr &\leq \ml \frac{3}{2}\frac{1}{1-\mu_0(u)} - 2\mr \log \ml \frac{r(u,v)}{r_0(u)}\mr,
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\ml - \frac{\du r}{r}\mr(u,v) &\leq \ml -\frac{\du r}{r}\mr_0(u) e^{\ml \frac{3}{2}\frac{1}{1-\mu_0(u)} - 2\mr \log \ml \frac{r(u,v)}{r_0(u)}\mr}.
\end{aligned}
\end{equation*}

This further yields

\begin{equation*}
\ml \frac{- \du r}{r}\mr (u,v) - \ml \frac{- \du r}{r}\mr_0(u) \leq \ml e^{\ml \frac{3}{2}\frac{1}{1-\mu_0(u)} - 2\mr \log \ml \frac{r(u,v)}{r_0(u)}\mr} - 1 \mr \ml \frac{-\du r}{r}\mr_0.
\end{equation*}

With these, we derive

\begin{equation*}
\begin{aligned}
-\du \log\ml \frac{r(u,v)}{r_0(u)}\mr  &= -\frac{\frac{\du r}{r_0}(u,v) - \frac{r(u,v)}{r_0(u)^2}(\du r)_0(u)}{\frac{r(u,v)}{r_0(u)}} =\frac{- \du r}{r}(u,v) - \ml\frac{- \du r}{r}\mr_0(u) \\
&\leq \ml e^{\ml \frac{3}{2}\frac{1}{1-\mu_0(u)} - 2\mr \log \ml \frac{r(u,v)}{r_0(u)}\mr} - 1 \mr \ml \frac{-\du r}{r}\mr_0.
\end{aligned}
\end{equation*}

Inspired by [21], we utilize the inequality $e^x - 1 \leq x + (e-2)x^2$ for $ x \leq 1$. Denoting

\begin{equation*}
\td(u,v) := \log \ml \frac{r(u,v)}{r_0(u)}\mr,
\end{equation*}

for all $(u,v) \in [u_2,u_n] \times (v_0,v(u_n)]$, we then obtain

\begin{equation*}
\begin{aligned}
-\du \td &\leq \ml \ml \frac{3}{2(1-\mu_0(u))} - 2 \mr \td + (e-2)\ml \frac{3}{2(1-\mu_0(u))} - 2 \mr^2\td^2 \mr \ml \frac{-\du r}{r}\mr_0
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\du \ml \frac{1}{\td}\mr &\leq \ml \ml \frac{3}{2(1-\mu_0(u))} - 2 \mr\ml \frac{1}{\td}\mr + (e-2)\ml \frac{3}{2(1-\mu_0(u))} - 2 \mr^2 \mr\ml \frac{-\du r}{r}\mr_0.
\end{aligned}
\end{equation*}

By employing a Grönwall-like argument, we arrive at

\begin{equation*}
\begin{aligned}
\frac{e^{- \frac{3}{2}\gamma_{s,0}(u_4) + \frac{1}{2}\log\ml\frac{r_0(0)}{r_0(u_4)}\mr}}{\td(u_4,v)} - \frac{e^{- \frac{3}{2}\gamma_{s,0}(u_3)+ \frac{1}{2}\log\ml\frac{r_0(0)}{r_0(u_3)}\mr}}{\td(u_3,v)} &\leq \frac{e^{-\frac{3}{2}\gamma_{s,0}(u_4) + \frac{1}{2}\log\ml\frac{r_0(0)}{r_0(u_4)}\mr}}{2\td(u_4,v)},
\end{aligned}
\end{equation*}

which implies the above-stated lemma.

With the above preparations, we now delve into the physical arguments as outlined in [21]. Employing

\begin{equation*}
\begin{aligned}
\du \ml \frac{r \dv \phi}{\dv r} \mr
= \ml -D_u \phi - \ii \e \frac{Q \phi (\du r)}{r(1-\mu)}\mr + \ml \frac{r \dv \phi}{\dv r}\mr \ml -\ii \e A_u + \frac{(-\du r)}{r} \frac{\mu - Q^2/r^2}{1 - \mu} \mr,
\end{aligned}
\end{equation*}

we deduce that

\begin{equation*}
\ml \frac{r |\dv \phi|}{|\dv r|} \mr(u,v)  = \mlm \ml \frac{r \dv \phi}{\dv r} \mr(0,v) - I(u)\mrm e^{\gamma(u,v)}
\end{equation*}

\begin{equation*}
I(u)=\int^u_{0} g(u',v)e^{-f(u',v)}\mathrm{d}u'
\end{equation*}

\begin{equation*}
\begin{split}
g(u,v) =& D_u \phi(u,v) +  \ii \e \frac{Q \phi (\du r)}{r(1-\mu)}(u,v),\\
f(u,v) =& \int^u_{0} -\ii \e A_u(u',v) + \frac{(-\du r)}{r} \frac{\mu - Q^2/r^2}{1 - \mu}(u',v) \mathrm{d}u'.
\end{split}
\end{equation*}

Our to-be-established first instability theorem considers the following case:

\begin{equation}\label{first instability condition}
\ml \frac{r \dv \phi}{\dv r} \mr_0 (0):=\ml \frac{r \dv \phi}{\dv r} \mr(0,0) \neq \lim_{u \rightarrow v_0^-} I(u).
\end{equation}

For this scenario, as $u\rightarrow v_0^-$, there are various cases for the behaviors of the complex-valued $I(u)$. Since $I(u)$ is complex-valued, we rewrite it as

\begin{equation*}
I(u) = X(u) + \ii Y(u)
\end{equation*}

with $X(u) = \re{(I(u))}$ and $Y(u) = \im{(I(u))}$. Let us denote the following quantities:

\begin{equation}
\begin{aligned}
X_- := \liminf_{u \rightarrow v_0^-} X(u), \qquad & \qquad X_+ := \limsup_{u \rightarrow v_0^-} X(u), \\
Y_- := \liminf_{u \rightarrow v_0^-} Y(u), \qquad & \qquad Y_+ := \limsup_{u \rightarrow v_0^-} Y(u). \\
\end{aligned}
\end{equation}

A priori, as the asymptotic property of $I$ is unknown, we have to consider all possible behaviors of $X$ and $Y$ below. For notational convenience, by using $Z$ as a placeholder for either $X$ or $Y$, we characterize their behavior as follows:

* $-\infty < Z_- = Z_+ < \infty$ (limit exists and it is finite),

* $-\infty = Z_- = Z_+$ (limit exists and it is equals to $- \infty$),

* $Z_- = Z_+ = \infty$ (limit exists and it is equals to $+ \infty$),

* $-\infty < Z_- < Z_+ < \infty$ (limit does not exist but $Z_-$ and $Z_+$ are bounded),

* $-\infty < Z_- < Z_+ = \infty$ (limit does not exist),

* $-\infty = Z_- < Z_+ < \infty$ (limit does not exist),

* $-\infty = Z_- < Z_+ = \infty$ (limit does not exist).

With respect to $X$ and $Y$, there are in total $7\times 7=49$ cases, and we denote them as ordered integer pairs $(a,b)$ with $1\leq a, b \leq 7$. For simplicity of demonstration, we focus on the $(1,1)$ case here. Since both limits for $X$ and $Y$ exist, the limit for the complex-valued $I$ does exist. By the hypothesis stated in (<ref>), we set

\begin{equation}
h := \mlm \ml \frac{r\dv \phi}{\dv r}\mr_0(0) - l\mrm \neq 0 \quad \mbox{ with } \quad l = \lim_{u \rightarrow v_0^-} I(u).
\end{equation}

For $u_*$ sufficiently close to $v_0$ and for all $u \in [u_*,v_0)$, we then have

\begin{equation*}
|I(u)-l| \leq \frac{2h}{3}.
\end{equation*}

This implies that

\begin{equation*}
\begin{aligned}
\ml \frac{r |\dv \phi|}{\dv r} \mr_0(u)
&= \mlm \ml\frac{r \dv \phi}{\dv r} \mr_0(0) - l + l - I(u) \mrm e^{\gamma_0(u)} \geq \mlm \mlm \ml\frac{r \dv \phi}{\dv r} \mr_0(0) - l\mrm - |l - I(u)|  \mrm e^{\gamma_0(u)} \geq \frac{h}{3}e^{\gamma_0(u)}.
\end{aligned}
\end{equation*}

Let us define the following quantities:

\begin{equation*}
\Upxi(u,v) := \frac{r(u,v)}{r_0(u)}\ml \frac{r \dv \phi}{\dv r}\mr(u,v) - \ml \frac{r\dv \phi}{\dv r
}\mr_0 (u)
\end{equation*}

\begin{equation*}
\psi(u,v) := e^{-\gamma_0(u)}\Upxi(u,v).
\end{equation*}

We first assume that, with $n \in \mathbb{N}$ being sufficiently large, we can work under the condition

\begin{equation}\label{first instability condition 2}
\sup_{v' \in [v_0,v(u_n)]}|\psi(u_n,v')| \leq \frac{h}{6}.
\end{equation}

Under this assumption, for each $v' \in [v_0,v(u_n)]$, we have

\begin{equation*}
\begin{aligned}
\frac{r(u_n,v')}{r_0(u_n)}\frac{r|\dv \phi|}{\dv r}(u_n,v') &= \mlm \frac{r\dv \phi}{\dv r}(u_n,v')\ml \frac{r(u_n,v')}{r_0(u_n)}\mr - \ml \frac{r\dv \phi}{\dv r}\mr_0(u_n) + \ml \frac{r \dv \phi}{\dv r}\mr_0(u_n) \mrm  \\
&\geq \mlm \mlm \frac{r\dv \phi}{\dv r}(u_n,v')\ml \frac{r(u_n,v')}{r_0(u_n)}\mr - \ml \frac{r\dv \phi}{\dv r}\mr_0(u_n) \mrm - \mlm \ml \frac{r \dv \phi}{\dv r}\mr_0(u_n)\mrm\mrm \\
&= \mlm \mlm \ml \frac{r \dv \phi}{\dv r}\mr_0(u_n)\mrm - |\psi|(u_n,v')e^{\gamma_0(u_n)} \mrm \\
&\geq \mlm \frac{h}{3} - \frac{h}{6}\mrm e^{\gamma_0(u_n)}= \frac{h}{6}e^{\gamma_0(u_n)}.
\end{aligned}
\end{equation*}

Recalling the definition of $\eta^*(u,v)$, this implies:

\begin{equation*}
\begin{aligned}
\eta^*(u_n,v) &= \frac{2\int^v_{v_0} \dv m(u_n,v') \mathrm{d}v'}{r(u_n,v)} - \frac{2}{r(u_n,v)} \int^v_{v_0} \frac{Q^2(\dv r)}{2r^2} (u_n,v') \mathrm{d}v' \\
&=  \frac{2}{r(u_n,v)}\ml \int^v_{v_0} \frac{2\pi r^2(1-\mu)|\dv \phi|^2}{\dv r}(u_n,v') \mathrm{d}v' \mr  \\
&\geq \frac{2\pi(1-\mu_0(0))h^2 }{27r(u_n,v)}e^{\gamma_0(u_n)}r^2_0(u_n)\ml \int^v_{v_0} \frac{(\dv r)(u_n,v')}{r^2(u_n,v')} \mathrm{d}v' \mr  \\
&= \frac{2\pi(1-\mu_0(0))h^2 }{27}e^{\gamma_0(u_n)}\ml \frac{1}{1+\delta(u_n,v)}\mr^2\ml \frac{r(u_n,v) - r_0(u_n)}{r_0(u_n)}\mr  \\
&\geq C_{37} \delta(u_n,v)e^{\gamma_0(u_n)} \quad \mbox{ with } \quad C_{37} := \frac{8\pi(1-\mu_0(0))h^2 }{243}.  \\
\end{aligned}
\end{equation*}

On the other hand, assuming that there is no trapped surface formation, from (<ref>) (and its more detailed version (<ref>)), we obtain

\begin{equation*}
\begin{aligned}
\eta^*(u_n,v) &\leq C_{24} \frac{\delta(u_n,v)}{1+\delta(u_n,v)}\log\ml \frac{1}{\delta(u_n,v)}\mr + C_{23}\frac{\delta(u_n,v)r_0(u_n)}{1-\sup_{{v \in [v_0,v_0+\varepsilon]}}\mu(u_n,v)} \\
&\leq C_{24} \delta(u_n,v) \log\ml \frac{1}{\delta(u_n,v)}\mr + \frac{3 C_{23}}{2(1-\mu_0(0))} \delta(u_n,v) e^{\gamma_0(u_n)}r_0(u_n) \\
&\leq C_{36}\delta(u_n,v) \ml \log \ml \frac{1}{\delta(u_n,v)}\mr + e^{\gamma_0(u_n)} r_0(u_n) \mr
\end{aligned}
\end{equation*}

\begin{equation*}
C_{36} = C_{24} + \frac{3 C_{23}}{2(1-\mu_0(0))}.
\end{equation*}

By comparing the above two inequalities for $\eta^*(u_n,v)$ at $v = v(u_n)$, a contradiction arises if

\begin{equation}\label{first instability condition 3}
\log \ml \frac{1}{\delta(u_n,v(u_n))}\mr e^{-\gamma_0(u_n)} + r_0(u_n)  < \frac{C_{37}}{C_{36}}.
\end{equation}

This means that if (<ref>) is true, the assumption of no trapped surface is violated, and the formation of trapped surface is guaranteed.

It is crucial to have the additional $r_0(u_n)$ as in (<ref>) in the expression of the trapped surface formation criterion. By picking $n$ sufficiently large, we can set

\begin{equation*}
r_0(u_n) \leq \frac{C_{37}}{4C_{36}}.
\end{equation*}

At the same time, for a sufficiently large $n$, we choose $v(u_n)$ such that

\begin{equation*}
\delta(u_n,v(u_n)) = e^{-\frac{C_{37}}{4C_{36}}e^{\gamma_0(u_n)}},
\end{equation*}

hence verifying (<ref>),  ultimately concluding the proof of the first instability theorem. If we cannot obtain $r_0(u_n)$ in (<ref>) and relax the factor to $1$, we would enter the realm of comparing constant with both $C_{37}$ and $C_{36}$ being of size 1. In this case, the validation of (<ref>) remains open, and we would fail to provide the integrated proof as stated above.

Recall the assumption (<ref>) we imposed. The next lemma verifies it.

Lemma. Suppose that on top of having (<ref>) and (<ref>), we demand that

\begin{equation*}
\delta(u,v(u_n)) \leq \min \left\{ \ml \frac{\log(3)}{8C_{48}}\mr^2e^{-(9+6\xi)\gamma_0(u)}, \ml \frac{h}{1296C_{45}}\mr^2 e^{-\ml \frac{15}{2} + 3\xi\mr\gamma_0(u)}, \ml \frac{h}{1296 b C_{49}}\mr^2 e^{- \frac{19}{2}\gamma_0(u)} \right\}
\end{equation*}

also holds for all $u \in [u_2,u_n]$. Then, the following requirement for initial data:

\begin{equation}\label{first instability condition 4}
\sup_{v \in [v_0,v(u_n)]}|\psi|(u_2,v) \leq \frac{h}{54}
\end{equation}

implies that

\begin{equation*}
\sup_{v' \in [v_0,v(u_n)]}|\psi|(u_n,v') \leq \frac{h}{6}.
\end{equation*}

To prove this claim, we first compute

\begin{equation}\label{first instability condition 5}
\begin{aligned}
\du \psi &= e^{-\gamma_0} J_4 + \psi(J_5 - \ii \e A_u) + e^{-\gamma_0}\ml \frac{r \dv \phi}{\dv r}\mr_0 J_6.
\end{aligned}
\end{equation}

Here, we have

\begin{equation*}
\begin{aligned}
J_4 &=  \frac{r}{r_0}\ml - D_u \phi - \ii \e \frac{Q \phi (\du r)}{r(1-\mu)}\mr + \ml D_u \phi + \ii \e \frac{Q \phi (\du r)}{r(1-\mu)}\mr_0 , \\
J_5 &=  \du \gamma - \du \gamma_0 + \frac{\du r}{r} - \ml \frac{\du r}{r}\mr_0,  \\
J_6 &= J_5 - \ii \e (A_u - (A_u)_0).
\end{aligned}
\end{equation*}

To control $J_4$, we employ the following equation:

\begin{equation*}
\begin{aligned}
& \dv \ml \frac{r}{r_0} \ml D_u \phi + \ii \e \frac{Q \phi (\du r)}{r(1-\mu)} \mr \mr \\
=\; & \frac{1}{r_0}\ml (-\du r) \dv \phi + \ii \e \frac{Q (\du r)(\dv r) \phi}{r(1-\mu)} +  (4 \pi \ii \e^2)\frac{\phi(-\du r)r^2 \im{(\phi^{\dagger} \dv \phi)}}{1-\mu} +\ii \e \frac{Q (\dv \phi)(\du r)}{1-\mu} +  4 \pi \ii \e \frac{Q r |\dv \phi|^2(\du r) \phi}{\dv r(1-\mu)}\mr.
\end{aligned}
\end{equation*}

In the subsequent detailed proofs, we bound each term on the right. In one of the upcoming steps, we employ the following estimate:

\begin{equation*}
\begin{aligned}
\int^v_{v_0} r^2|\phi|^4 \frac{\dv r}{1-\mu}(u,v') \D v' \leq \; & 8C_{35}^4\int^v_{v_0} r^2 \frac{\dv r}{1-\mu}(u,v') \D v' \\
&+  \frac{1}{2\pi^2}\int^v_{v_0} r^2 \frac{\dv r}{1-\mu}(u,v') \ml  \log \ml \frac{\frac{1 - \mu}{ \dv r}(u,v)}{\frac{1 - \mu}{ \dv r}(u_2,v)}\mr  \mr^{2}\ml \log \ml \frac{r(u_2,v)}{r(u,v)}\mr \mr^{2} \D v'. \\
\end{aligned}
\end{equation*}

We then utilize $(\frac{\dv r}{1-\mu})^{\frac12}$ to balance the term in the logarithm and relate the other $(\frac{\dv r}{1-\mu})^{\frac12}$ to $e^{\frac{1}{2}\gamma_0(u)}$. Henceforth, we obtain

\begin{equation*}
\begin{aligned}
& \int^v_{v_0} r^2 \frac{\dv r}{1-\mu}(u,v') \ml  \log \ml \frac{\frac{1 - \mu}{ \dv r}(u,v)}{\frac{1 - \mu}{ \dv r}(u_2,v)}\mr  \mr^{2}\ml \log \ml \frac{r(u_2,v)}{r(u,v)}\mr \mr^{2} \D v' \\
\lesssim \; & \int^v_{v_0} \ml \frac{\dv r}{1-\mu}\mr^{\frac{1}{2}}(u,v')  \D v' \lesssim \;   e^{\frac{1}{2}\gamma_0(u)} \ml \int^v_{v_0} \dv r(u,v')  \D v' \mr^{\frac{1}{2}} \lesssim \;  e^{\frac{1}{2}\gamma_0(u)} \delta(u,v)^{\frac{1}{2}}.
\end{aligned}
\end{equation*}

This leads to

\begin{equation*}
\int^v_{v_0} r^2|\phi|^4 \frac{\dv r}{1-\mu}(u,v') \D v' \leq C_{41}e^{\frac{1}{2}\gamma_0(u)}\delta(u,v)^{\frac{1}{2}} \quad \mbox{ with } \quad  C_{41} \mbox{ given in } \eqref{FIT247}.
\end{equation*}

To bound the contribution from $J_5$, we first rewrite $J_5$ as

\begin{equation*}
J_5(u,v) = \int^v_{v_0} \dv \ml \frac{\du r}{r} + \du \gamma \mr(u,v') \, \D v'.
\end{equation*}

Furthermore, we can show that

\begin{equation*}
\begin{aligned}
|J_5|(u,v) &\leq C_{48} \ml \frac{-\du r}{r}\mr_0(u) e^{3 \gamma_0(u)} \delta(u,v)^{\frac{1}{2}}
\end{aligned}
\end{equation*}

with $C_{48}$ given in (<ref>). This further implies

\begin{equation*}
\begin{aligned}
\int^u_{u_2} |J_5|(u',v) \D v' &\leq C_{48} \int^u_{u_2} \ml \frac{-\du r}{r}\mr_0(u') e^{3\gamma_0(u')}\delta(u',v)^{\frac{1}{2}}\D u' \\
&\leq 2C_{48} e^{\ml \frac{15}{4} + \frac{3}{2}\xi \mr\gamma_0(u)}\delta(u,v)^{\frac{1}{2}} \int^u_{u_2} \ml \frac{-\du r}{r}\mr_0(u') e^{-\frac{1}{4}\log\ml \frac{r_0(u')}{r_0(u)}\mr} \D u' \\
&\leq 8C_{48} e^{\ml \frac{15}{4} + \frac{3}{2}\xi \mr\gamma_0(u)}\delta(u,v)^{\frac{1}{2}}.
\end{aligned}
\end{equation*}

Note that the above lemma for $\delta(u,v)$ is critically employed in the second inequality.

Back to the equation (<ref>) for $\du \psi$, for each $v \in [v_0,v(u_n)]$, we then arrive at

\begin{equation*}
|\psi|(u_n,v) \leq 3 \ml \frac{h}{54} \times 3\mr = \frac{h}{6}.
\end{equation*}

We are left with the task of verifying all the assumptions for $\delta$, as summarized and listed in (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), along with the requirement on the initial data in (<ref>). Our complete proof is presented in a later section. For instance, the verification of (<ref>) for $\delta$ can be carried out as follows:

\begin{equation*}
\begin{aligned}
\delta(u_*,v(u_n))\log \ml \frac{1}{\delta(u_*,v(u_n))}\mr &\leq \frac{2}{e}\delta(u_*,v(u_n))^{\frac{1}{2}}
\\
&\leq \frac{4}{e}\delta(u_n,v(u_n))^{\frac{1}{2}} e^{-\frac{1}{4}\log \ml \frac{r_0(u_*)}{r_0(u_n)}\mr + \frac{3}{4}\ml 1 + 2\xi \mr\gamma_0(u_n)} \\
&= 4e^{-\frac{C_{37}}{8C_{36}}e^{-\gamma_0(u_n)} + \frac{3}{4}\ml 1 + 2\xi \mr\gamma_0(u_n)-1}.
\end{aligned}
\end{equation*}

By continuity of $\delta(u_2,\cdot)$ and (<ref>), we can further verify the requirement (<ref>):

\begin{equation*}
\sup_{v \in [v_0,v(u_n)]}|\psi|(u_2,v) \leq \frac{h}{54}.
\end{equation*}

The above outlines the proof for the case $(1,1)$. Analogous arguments are conducted for case $(1,4), (4,1),$ and $(4,4)$.

For the remaining 45 cases, since $\limsup_{u \rightarrow v_0^-}|I(u)| = +\infty$, we define $b:[0,v_0) \rightarrow \mathbb{R}$ via

$$b(u) := \sup_{u'\in [0,u]} \ml \frac{r|\dv \phi|}{\dv r}\mr_0(u')e^{-\gamma_0(u')}$$

and deduce that there exists an increasing sequence $u_n \rightarrow v_0^-$ such that

$$b(u_n) =\ml \frac{r|\dv \phi|}{\dv r}\mr_0(u')e^{-\gamma_0(u')}$$

with $b(u_n) \rightarrow \infty$. Repeating the above arguments, similar to (<ref>), we can deduce that a contradiction arises if

\begin{equation}
\log\ml \frac{1}{\delta(u_n,v(u_n))}\mr e^{-\gamma_0(u_n)} + r_0(u_n) < \frac{C_{50}}{C_{36}}b(u_n)^2.
\end{equation}

In these cases, we instead pick $v(u_n)$ such that

$$\delta(u_n,v(u_n)) = e^{-b(u_n)^2\frac{C_{50}}{4C_{36}}e^{\gamma_0(u_n)}}.$$

Correspondingly, some of the required assumptions would have additional linear factors in $b(u_n)$ in their exponents. Henceforth, despite the stronger requirements, these are met through picking a value of $v(u_n)$ that is even closer to $v_0$ as described above.

§.§.§ Second Instability Theorem

We now investigate the scenario where

\begin{equation*}
\ml \frac{r \dv \phi}{\dv r} \mr_0 (0) = \lim_{u \rightarrow v_0^-} I(u).
\end{equation*}

For the Einstein-scalar field system, in [21], Christodoulou remarked that the proof for the first instability theorem still holds (and, therefore, a similar conclusion is valid) if there exists a positive constant $p > 1$ such that

\begin{equation}\label{SIT2'}
\limsup_{u \rightarrow v_0^-}\left\{ \mlm \ml \frac{r \dv \phi}{\dv r}\mr_0(0) - I(u)\mrm e^{\frac{1}{2}p\gamma_0(u)} \right\} \neq 0.
\end{equation}

For the uncharged situation, assumption (<ref>) can be satisfied. However, for the charged case, it is not clear whether (<ref>) can still be imposed and verified. Here we take a different path and avoid requiring (<ref>).

We first define

\begin{equation*}
V(u) := \mlm \ml \frac{r \dv \phi}{\dv r}\mr_0(0) - I(u)\mrm, \quad v(u) := v_0 + e^{-14\gamma_0(u) - \log \ml \frac{r_0(0)}{r_0(u)}\mr}.
\end{equation*}

\begin{equation*}
g(v(u)) := \sqrt{\max\left\{V^2(u),e^{-\frac{1}{2}\gamma_0(u)},e^{-\log\ml\frac{r_0(0)}{r_0(u)}\mr} \right\}}.
\end{equation*}

We also set

\begin{equation*}
h^2(v(u)) := \frac{1}{v(u) - v_0}\int_{v_0}^{v(u)} |\psi|^2(0,v') \D v'.
\end{equation*}

Subsequently, we will observe that the second instability theorem still holds if

\begin{equation*}
\limsup_{u \rightarrow v_0^-}\frac{h(v(u))}{g(v(u))} = \infty,
\end{equation*}

indicating that an additional instability mechanism has been triggered. In particular, we allow
$$\lim_{u\rightarrow v_0^-}h(v(u))=0.$$

We first state a useful lemma below.

Lemma. Let $\tu \in [u_2,v_0)$ be given. Suppose there exists a corresponding value $v(\tu) \in [v_0,v_0 + \te]$ with $0<\te<1$ such that, for all $u \in [u_2,\tu]$, we have

\begin{equation*}
\delta(u,v(\tu)) \leq \frac{e^{-\ml \frac{15}{2} + 3 \xi \mr\gamma_0(u)}}{64C_{53}^2}
\end{equation*}

with $C_{53}$ defined in (<ref>).
Then, for each $u \in [u_2,\tu]$ and $v' \in [v_0,v(\tu)]$, we have

\begin{equation*}
|\tgam(u,v) - \tgam_0(u)| \leq 1,
\end{equation*}

$$\tgam(u,v):=\gamma(u,v;\bar{u})=\int_{\bar{u}}^u\frac{(-\partial_u r)}{r}\frac{\mu-Q^2/r^2}{1-\mu}(u', v)du' \quad \mbox{ and } \quad \tgam_0(u):=\tgam(u, v_0).$$

To prove this lemma, we compute $\dv \gamma$ and obtain

\begin{equation}
|\tgam(u,v) - \tgam_0(u)| \leq \int^u_{u_2} \int^v_{v_0} |J_9|(u',v') + |J_{10}|(u',v') + |J_{11}|(u',v') \D v' \D u'
\end{equation}

with the expressions for $J_8, J_{9}, J_{10}$ available in (<ref>). By carefully estimating each of these integrals, we deduce that

\begin{equation}
\int^v_{v_0} |J_8|(u',v') + |J_{9}|(u',v') + |J_{10}|(u',v') \D v' \leq C_{53}\ml \frac{-\du r}{r} \mr_0(u')e^{3\gamma_0(u')}\delta(u',v)^{\frac{1}{2}}.
\end{equation}

This gives

\begin{equation*}
\begin{aligned}
|\tgam(u,v) - \tgam_0(u)| &\leq C_{53} e^{3\gamma_0(u)}\int^u_{u_2} \ml \frac{-\du r}{r}\mr_0(u') \delta(u',v)^{\frac{1}{2}}\D u' \\
&\leq 2C_{53} e^{ \ml \frac{15}{4} + \frac{3 \xi}{2} \mr \gamma_0(u)}\delta(u,v)^{\frac{1}{2}} \int^u_{u_2} \ml \frac{-\du r}{r}\mr_0(u') e^{-\frac{1}{4}\log \ml \frac{r_0(u_2)}{r_0(u')}\mr}\D u' \\
&\leq 8C_{53} e^{\ml \frac{15}{4} + \frac{3 \xi}{2} \mr\gamma_0(u)}\delta(u,v(\tu))^{\frac{1}{2}} \leq 1.
\end{aligned}
\end{equation*}

Applying the above lemma, we obtain

\begin{equation*}
\begin{aligned}
\delta(\tu,v(\tu)) &= \frac{1}{r_0(\tu)}\int^{v(\tu)}_{v_0} (\dv r)(\tu,v')\D v' = \frac{1}{r_0(\tu)} \int^{v(\tu)}_{v_0} (\dv r)(0,v')e^{-\gamma_0(\tu) + \ml \gamma_0(\tu) -\tgam(\tu,v') \mr - \gamma(u_2,v')} \D v' \\
&\quad\quad\quad\leq \frac{e C_{56}}{2r_0(\tu)} \int^{v(\tu)}_{v_0} e^{-\gamma_0(\tu) } \D v' \leq \frac{e C_{56} e^{-\gamma_0(\tu)}}{2r_0(\tu)}(v(\tu) - v_0)
\end{aligned}
\end{equation*}

\begin{equation*}
C_{56}(u_2) := \max_{v'\in[v_0,v_0 + \te]}e^{|\gamma|(u_2,v')} < + \infty.
\end{equation*}

Note that the direction of the above inequality for $\delta$ is reversed as compared to (<ref>). In (<ref>) the outer $\delta(u_3, v')$ is bounded by the inner $\delta(u_4, v')$ with inflation factors.

The bulk of the arguments for that section relies on a simple inequality mentioned in Lemma <ref>, which states that for any arbitrary measurable subsets in $\mathbb{R}$,

\begin{equation}\label{second1}
\int_X f|g|^2 \geq 4 \int_X f|h|^2 \implies \int_X f|g+h|^2 \geq \frac{1}{4}\int_X f|g|^2 \geq \int_X f|h|^2.
\end{equation}

To proceed, we first deduce:

\begin{equation}\label{second2}
4 \int_{v_0}^{v}  \mlm \frac{r \dv \phi}{\dv r} \mrm_0^2  (\tu,v')  \D v' \leq 4( v(\tu) - v_0 ) V^2(\tu)e^{2\gamma_0(\tu)}.
\end{equation}

Subsequently, we will attempt to show that

\begin{equation}\label{V(u) and Theta}
4( v(\tu) - v_0 ) V^2(\tu)e^{2\gamma_0(\tu)}\leq \int_{v_0}^{v(\tu)} \mlm \Upxi \mrm^2 (\tu,v')  \D v' .
\end{equation}

To prove (<ref>), we employ

\begin{equation}\label{second3}
\int_{v_0}^{v(\tu)} \mlm \Upxi \mrm^2 (\tu,v')  \D v'
\geq \frac{e^{-2C_{55}}}{9}e^{2\gamma_0(\tu)}\int^{v(\tu)}_{v_0} |\psi(0,v') + J_{13}(\tu,v')|^2  \D v'.
\end{equation}

In the later sections, we will verify that

\begin{equation}\label{second4}
\int_{v_0}^{v(\tu)} |\psi|(0,v') \D v' \geq 4 \int_{v_0}^{v(\tu)} |J_{13}|(\tu,v') \D v' .
\end{equation}

Next, we define

$$h^2(v):=\frac{1}{v-v_0}\int_{v_0}^v |\psi|^2(0, v')dv'$$

and choose

\begin{equation*}
h^2(v(\tu)) \geq 144 e^{2C_{55}} V^2(\tu).
\end{equation*}

These imply that

\begin{equation}\label{second5}
4(v(\tu)-v_0)V^2(\tu)e^{2\gamma_0(\tu)} \leq \frac{e^{-2C_{55}}}{36}e^{2\gamma_0(\tu)}\int^{v(\tu)}_{v_0} |\psi|^2(0,v') \D v'.
\end{equation}

Stitching (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) together, we obtain the inequality:

\begin{equation*}
\int_{v_0}^{v(\tu)}  \mlm \ml \frac{r \dv \phi}{\dv r}\mr_0 + \Upxi \mrm^2 (\tu,v') \D v' \geq \frac{e^{-2C_{55}}}{144}e^{2\gamma_0(\tu)}\int^{v(\tu)}_{v_0}|\psi|^2(0,v')\D v'.
\end{equation*}

Plugging this back to the expression for $\eta^*(\tu,v(\tu))$, we thus obtain

\begin{equation*}
\begin{aligned}
\eta^*(\tu,v(\tu)) &= \frac{4\pi}{r(\tu,v(\tu))}\int^{v(\tu)}_{v_0} (1-\mu) \ml \frac{r^2|\dv \phi|^2}{\dv r}\mr(\tu,v') \D v' \\
&\geq \frac{16\pi C_{54}(1-\mu_0(0))}{27e r(\tu,v(\tu))}e^{-2\gamma_0(\tu)}\int_{v_0}^{v(\tu)}  \mlm \ml \frac{r \dv \phi}{\dv r}\mr_0 + \Upxi \mrm^2 (\tu,v') \D v' \\
&\geq \frac{2\pi C_{54}e^{-2C_{55}}(1-\mu_0(0))}{729 e r_0(\tu)} \int^{v(\tu)}_{v_0}|\psi|^2(0,v')\D v'.
\end{aligned}
\end{equation*}

This would contradict the no trapped-surface-formation condition (<ref>) if we choose

\begin{equation*}
\begin{aligned}
& \frac{eC_{36}C_{56}}{2r_0(\tu)}(v(\tu) - v_0)\ml r_0(\tu) + e^{-\gamma_0(\tu)}\ml \log \ml \frac{1}{v(\tu) - v_0}\mr + \log \ml \frac{2r_0(0)}{eC_{56}}\mr  +
\log \ml\frac{r_0(\tu)}{r_0(0)} \mr + \gamma_0(\tu)\mr\mr \\
\leq & \; \frac{\pi C_{54}e^{-2C_{55}}(1-\mu_0(0))}{729e r_0(\tu)} \int^{v(\tu)}_{v_0}|\psi|^2(0,v')\D v'.
\end{aligned}
\end{equation*}

Therefore, to trigger the second instability mechanism, it suffices to require

\begin{equation*}
\begin{aligned}
h^2(v(\tu)) \geq  C_{61}\ml r_0(\tu) + e^{-\gamma_0(\tu)}\ml \log \ml \frac{1}{v(\tu) - v_0}\mr + \log \ml \frac{2r_0(0)}{eC_{56}}\mr  +
\log \ml\frac{r_0(\tu)}{r_0(0)} \mr + \gamma_0(\tu)\mr \mr
\end{aligned}
\end{equation*}

\begin{equation*}
C_{61}(u_2) := \frac{729 e^2 C_{36} C_{56}}{2\pi C_{54}e^{-2C_{55}}(1-\mu_0(0))}.
\end{equation*}

For the inequality

$$h^2(v(\tu)) \geq 144 e^{2C_{55}} V^2(\tu)$$

as mentioned above, we carefully choose

\begin{equation*}
v(\tu) - v_0 = e^{-A\gamma_0(\tu) - B \log \ml \frac{r_0(0)}{r_0(\tu)}\mr} \quad \mbox{ with } \quad A=14 \mbox{ and } B=1.
\end{equation*}

By employing a small bootstrap argument involving the use of $\du \log \delta(u,v)$,
we ultimately satisfy all the imposed requirements and establish the second instability theorem. It is noteworthy that setting $B = 1$ is necessary, while $A$ can be chosen to be as large as required.

§.§ Related Works

In this section, we list some references related to our approach. In [15, 16, 17], Christodoulou initiated the exploration of the gravitational collapse for the spherically symmetric Einstein-scalar field system. In a series of breakthrough works spanning four papers [18]-[21], Christodoulou proved the weak cosmic censorship conjecture for this system. Subsequently, the proofs and the conclusions in these four papers have been revisited as follows:

- The method used to prove trapped surface formation as in [18] is subsequently applied to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in [24] by Costa, and to the Einstein-Maxwell-charged scalar field system within spherical symmetry in [6] by the first author and Lim.

- Within spherical symmetry and self-similar ansatz, the construction of naked singular solutions in [20] for the Einstein-scalar field system later inspires its extension to higher dimensions by the first author and Zhang in [3], and by Cicortas in [25]. A recent revisit of [20] by Singh is presented in [48]. Additionally, for the Einstein-Euler system, under the symmetry assumption, naked singularities are successfully constructed by Guo-Hadzic-Jang in [30].

- In [21], Christodoulou presented two instability theorems, consistent with his earlier established sharp trapped surface formation criterion. Building on BV regularity estimates from [19] and the blueshift effect discussed in [21], Li-Liu [36] gave a different argument to obtain instability results with a priori estimates. Note that their instability theorem diverges from Christodoulou's two instability theorems in [21].

Within spherical symmetry, based on the aforementioned Christodoulou's works, various investigations have been unfolded regarding gravitational collapse. For insights into spacelike singularities in the Einstein-scalar field system, refer to [5] by the first author and Zhang, [9] by the first author and Gajic, and [38] by Li. For the study of weak null singularity, related to the proof presented in this paper for the charged scenario, see [26] by Dafermos, [40] by Luk-Oh, [41], [42] by Van de Moortel, [39] by Li-Van de Moortel, [14] by Cardoso-Costa-Destounis-Hintz-Jansen, [46] by Rossetti, and to the references therein. We also note a recent result [31] by Kehle and Unger investigating extremal critical collapse for the Einstein-Maxwell-Vlasov system. The approach of this current paper with double-null foliations is quite robust. The first author and collaborators are extending this approach to cosmological spacetimes and other matter models.

Beyond spherical symmetry, in a monumental work [23] Christodoulou invented the short-pulse method and proved the first trapped-surface-formation result for the Einstein vacuum equations. Subsequently, Klainerman-Rodnianski[32] simplified and expanded this work by introducing the signature for the short pulse. In [33], Klainerman-Luk-Rodnianski further relaxed the initial requirements and provided a fully anisotropic mechanism for the formation of trapped surfaces. The first scale-critical result on trapped surface formation was obtained by the first author and Luk in [2], opening a door to study the region near the central singularity. In [4] by the first author and in [10] by the first author and Han, emergences of (isotropic and anisotropic) apparent horizons from the center of gravitational collapse are further established. Dynamics of apparent horizon is also studied by the first author and He in [12]. In [1], the first author introduced the signature for decay rates. Utilizing only this signature, a new concise proof for the (scale-critical) trapped surface formation criterion is provided by the first author in [7] . Later, this approach is generalized to the Einstein-Maxwell system by the first author and Athanasiou in [8]. For the Einstein-scalar field system, with singular initial data, Li-Liu proved an almost scale-critical trapped surface formation criterion in [37]. It is worth noting that the construction of the naked-singularity solutions for the Einstein vacuum equations was recently accomplished by Rodnianski-Shlapentokh-Rothman [45] and by Shlapentokh-Rothamn [47]. Based on Christodoulou's naked singularity constructed in [20], for the Einstein-scalar field system with no symmetric assumption, the proof of naked-singularity censoring with anisotropic apparent horizon was also recently established by the first author in [11].

§.§ Acknowledgement
Both authors would like to thank Demetrios Christodoulou for insightful correspondence. XA extends his thanks to Jonathan Luk for a valuable discussion of Christodoulou's work back to Princeton. Additionally, both authors would like to thank Haoyang Chen and Kai Dong for the beneficial discussions and for their careful reading of this manuscript.
XA is supported by MOE Tier 1 grants A-0004287-00-00, A-0008492-00-00 and MOE Tier 2 grant A-8000977-00-00. HKT is partially supported under the MOE Tier 1 grant R-146-000-269-133 during his time at the National University of Singapore.

§ PRELIMINARIES

In this study, we utilize the double-null coordinate system to re-express the Einstein-Maxwell-charged-scalar-field system as presented in (<ref>).

We say that a spacetime $(\mathcal{M},g)$ is spherically symmetric if $SO(3)$ acts on it by isometry, and the orbits of the group are (topological) $2$-dimensional spheres $\mathcal{S}$. Our spacetime $(\mathcal{M},g)$ can be represented with a $2$-dimensional Penrose's diagram, by considering the quotient manifold  $\mathcal{Q} = \mathcal{M}/\mathcal{S}$. Under spherical symmetry, we employ the following ansatz for the Lorentzian metric:

\begin{equation}\label{Setup4}
g = - \Omega^2(u,v) \; \mathrm{d}u\mathrm{d}v + r^{2}(u,v) (\mathrm{d}\theta^2 + \sin^2 \theta \mathrm{d} \phi^2),
\end{equation}

where $\Omega(u,v)$ and $r(u,v)$ are positive functions on $\mathcal{Q}$ representing lapse and area radius respectively. Here, $u$ and $v$ are optical functions satisfying $g^{\alpha\beta}\partial_{\alpha} u \partial_{\beta} u = 0$ and $g^{\alpha\beta}\partial_{\alpha} v \partial_{\beta} v = 0$, with their level sets corresponding to outgoing and incoming null hypersurfaces respectively. In Penrose's diagram, these null geodesics are $45$-degree slanted. We further denote the axis of symmetry to be $\Gamma$, where the area radius function vanishes, and we represent $\Gamma$ as the vertical axis. Mathematically, we express this as

\begin{equation}\label{Setup1}
\Gamma := \{ (u,v) \in \mathcal{Q} \hspace{3pt}|\hspace{3pt} r(u,v) = 0\}.
\end{equation}

As shown in Figure <ref>, we proceed to set up the coordinate system in $\mathcal{Q}$ as follows.

[->] (0,0) node[anchor=north]$\mathcal{O} = (0,0)$ –  (0,6) node[anchor = east]$\Gamma$;
[->] (0,0) – (5,5) node[anchor = west]$C_0^+$;
(3,3) – (1,5)node[anchor = south]$p$;
(1,5) – (0,4)node[anchor = east]$p_*$;
(0,3) – (1,2);
(0,1) – (1,2);
(1,2) – (1.5,1.5);
[] at (3.25,2.9) $p_0$;
[] at (1.15,2.15) $q$;
[] at (0.4,2) $\mathcal{D}(q)$;
[] at (-0.7,5) $u = v$;
[rotate=45] at (0.6,1.2) $\mathcal{J}^-(q)$;

The coordinate system in $\mathcal{Q}$

* Let $\Gamma$ denote the axis of symmetry as described in (<ref>).

* For any point $p$ in the causal future of $C_0^+$, there is a past-directed incoming null curve emanating from it and intersecting $C_0^+$ at $p_0$. We then assign the $v$-coordinate of $p$ to be the same as that of $p_0$.

* For the points along $\Gamma$, by the aforementioned instruction, each point about $\mathcal{O}$ is associated with a $v$-coordinate. Furthermore, we choose its $u$-coordinate along $\Gamma$ to satisfy

\begin{equation}\label{Setup2}
\Gamma:\; u = v.
\end{equation}

* For an outgoing null curve emitting from a point $\mathcal{O}\in \Gamma$, we denote it $C_0^+$. We fix $u=0$ and prescribe the coordinate parameter $v$ along it. We require the point $\mathcal{O}$ to have the coordinate $(0,0)$.

* Back to the point $p$, there is a past-directed outgoing null curve emanating from $p$ and intersecting $\Gamma$ at $p_*$. We proceed to assign the $u$-coordinate of $p$ to be equal to that of $p_*$.

With the above-chosen coordinate system, we define some commonly used terminologies below.

* With $q = (u',v')$, we denote $\mathcal{D}(u',v')$ to be its domain of dependence determined by the point $q=(u', v')$ and we have

\begin{equation}\label{Setup3}
\mathcal{D}(u',v') = \{(u,v) \in \mathcal{Q}: u \in [u',v'), v \in (u,v']\}.
\end{equation}

* With $q = (u',v')$, we set $\mathcal{J}^-(u',v')$ to be its causal past and the following holds:

\begin{equation}\label{Setup28}
\mathcal{J}^-(u',v') = \{(u,v) \in \mathcal{Q}: u \in [0,u'], v \in (u,v')\}.
\end{equation}

We further eliminate the gauge freedom of $v$ for our double-null coordinates. Along $C_0^+$, we require our coordinate parameter $v$ to satisfy

\begin{equation}\label{Setup33}
C_0^+:\; v = 2r
\end{equation}

and we also impose

\begin{equation}\label{Setup34}
\Omega(0,0) = 1.
\end{equation}

Next, we remark that the system considered above is subjected to the following boundary conditions along $\Gamma$:

* For every $(u,u) \in \Gamma$ with $u \geq 0$ and $F \in C^1(\mathcal{Q})$, we first define

\begin{equation}\label{Setup74}
F|_{\Gamma}(u) := \lim_{v \rightarrow u^+}F(u,v).
\end{equation}

Under spherical symmetry, when we consider points infinitesimally close to the center, we note that the incoming null geodesics would become outgoing in the opposite direction (and vice versa) after bypassing $\Gamma$. Mathematically, for $u \geq 0$ and $F \in C^1(\mathcal{Q})$, we hence have

\begin{equation}\label{Setup75}
F|_{\Gamma}(u) = \lim_{\tilde{u} \rightarrow u^-}F(\tilde{u},u) \quad \mbox{ and } \quad \dv F|_{\Gamma}(u) = - \du F|_{\Gamma}(u).
\end{equation}

* As employed in [19], at regular points along $\Gamma$, we impose the following boundary conditions:

\begin{equation}\label{Setup35}
|g_{\mathcal{Q}}(\nabla r,\nabla r)| < +\infty, \quad |\phi|_\Gamma < +\infty, \quad |F_{uv}|_{\Gamma} < + \infty.
\end{equation}

These imply

\begin{equation}\label{Setup36}
m|_{\Gamma} = 0, \quad r\phi|_{\Gamma} = 0, \quad Q|_{\Gamma} = 0.
\end{equation}

An alternative approach to observe that (<ref>) must hold at regular points along $\Gamma$ is to consider the Kretschmann scalar of this system. To avoid being a singular point, (<ref>) must hold.

* In addition, it is worth noting that the electromagnetic potential $A_u$ enjoys a gauge freedom as described in (<ref>). Considering (<ref>), without loss of generality, we set

\begin{equation}\label{Setup37}
A_u|_{\Gamma} = 0.
\end{equation}

*  By the spherical symmetry assumption, for points infinitesimally close to the center, its incoming null geodesics essentially become outgoing in the opposite direction. Mathematically, for all $u \geq 0$, we have

\begin{equation}\label{Setup38}
\dv r|_{\Gamma} = -\du r|_{\Gamma}.
\end{equation}

For the study of gravitational collapse, it is natural to assume that there is no anti-trapped surface along $C_0^+$, that is, $\du r(0,v) < 0$ for all $v > 0$. By (<ref>), this in turn implies that

\begin{equation}\label{Setup40}
\du r(u,v) < 0
\end{equation}

for all $(u,v) \in \mathcal{Q}$.

Next, we define the regular region $\mathcal{R}$, the apparent horizon $\mathcal{A}$, and the trapped region $\mathcal{T}$ below.

* The regular region $\mathcal{R}$ refers to the set of points on $\mathcal{Q}$ in which $\dv r > 0$;

* The apparent horizon $\mathcal{A}$ refers to the set of points  on $\mathcal{Q}$, where $\dv r = 0$.

* The trapped region $\mathcal{T}$ refers to the set of points  on $\mathcal{Q}$ in which $\dv r < 0$.

A $2$-sphere is termed a trapped surface if $\dv r(u,v) < 0$. If $\dv r(u,v) = 0$, we labelthe $2$-sphere at $(u,v)$ as a marginally outer trapped surface (MOTS).

In view of the definitions above, using (<ref>), we have an equivalent characterization as shown below:

Regions     Sign of $\dv r$     Sign of $1- \mu$     Value of $\mu$

$(u,v) \in \mathcal{R}$     $\dv r > 0$     $1 - \mu > 0$     $\mu < 1$

$(u,v) \in \mathcal{A}$     $\dv r = 0$     $1 - \mu = 0$     $\mu = 1$

$(u,v) \in \mathcal{T}$     $\dv r < 0$     $1 - \mu < 0$     $\mu > 1$

Related to the gauge fixing along $C_0^+$, since we set $v=2r$ along $C_0^+$, we have $\partial_v r|_{C_0^+}=1/2$. Referring to the table above, we note that neither the trapped surface nor the MOTS appears along $C_0^+$. In addition, we introduce the term

\begin{equation}\label{Setup42}
\mu_* := \sup_{v > 0} \mu(0,v)
\end{equation}

which lies in the range $[0,1)$.

We further record a proposition that will be commonly referenced in subsequent sections.

Along any future-directed outgoing null curve with $u > 0$, if $(u,v_1) \in \mathcal{A}$, then for all $(u,v)$ with $v \geq v_1$, we have $(u,v) \in \mathcal{A} \cup \mathcal{T}$. Moreover, if $(u,v_1) \in \mathcal{T}$, then for all $(u,v)$ with $v \geq v_1$, we have $(u,v) \in \mathcal{T}$.

From (<ref>), we see that $\dv(\Omega^{-2}\dv r) \leq 0$. For all $v \geq v_1$, this implies

\begin{equation}\label{Setup44}
\Omega^{-2}\dv r(u,v) \leq \Omega^{-2}\dv r(u,v_1).
\end{equation}

Thus, if $(u,v_1) \in \mathcal{A}$ with $\dv r(u,v_1) = 0$, the above yields that $\Omega^{-2}\dv r(u,v) \leq 0$. In particular, we have $\dv r(u,v) \leq 0$ since $\Omega^2 > 0$ for all $(u,v) \in \mathcal{Q}$, and thus $(u,v) \in \mathcal{A} \cup \mathcal{T}$. Similarly, if $(u,v_1) \in \mathcal{T}$ with $\dv r(u,v_1) < 0$, for $v\geq v_1$, we then conclude $\dv r(u,v) < 0$ and thus $(u,v) \in \mathcal{T}$.

Intuitively, this implies that once we encounter a MOTS along a fixed future-directed outgoing null curve at some $(u,v_1)$, then the future of this curve would either be part of the apparent horizon or be within the trapped region.

Last but not least, we review some preliminaries for the (future) singular boundary of the spacetime.

In view of Theorem <ref>, there exists a point $P = (0,v_0)$ on $C_0^+$, such that the incoming null curve $C_0^-$

\begin{equation}\label{Setup45}
\mbox{with } C_0^- := \{ (u,v) \in \mathcal{Q}: v = v_0 \text{  and  } 0 \leq u \leq v_0\}
\end{equation}

cannot terminate at a regular vertex on $\Gamma$. Denote this vertex to be $\mathcal{O}' = (v_0,v_0)$. In addition, we call $P$ as the maximal sphere of development, and we say that the past of $C_0^-$ is a terminal indecomposable past set in the terminology of [29]. The vertex $\mathcal{O}'$, also known as the unique future limit point of $\Gamma$ in $\overline{\mathcal{Q}} \setminus \mathcal{Q}$, marks the start of the singular boundary $\mathcal{B}$, which consists of boundary points of $\mathcal{Q}$ lying in the ambient manifold $\mathbb{R}^{1+1}$. Note that $\mathcal{B}$ contains two components, the central component $\mathcal{B}_0$ (composed by possibly empty null segment and its past endpoint $\mathcal{O}'$) and the noncentral component $\mathcal{B} \setminus \mathcal{B}_0$ (being achronal, with $r$ vanishing on it). According to [35], we can decompose the null segment $\mathcal{B}_0$ as $\mathcal{B}_0 = \mathcal{S}^1_{\Gamma} \cup \mathcal{CH}_\Gamma \cup \mathcal{S}^2_{\Gamma}$. Moreover, the radius function $r$ extends continuously to zero on $\mathcal{S}^1_{\Gamma}$ and $\mathcal{S}^2_{\Gamma}$, but extends continuously to possibly non-zero values on $\mathcal{CH}_\Gamma$. Furthermore, $\mathcal{A}$ necessarily issues from $\mathcal{CH}_\Gamma$, and in particular, from $\mathcal{B}_0$. The relative positions of the aforementioned points are illustrated in Figure <ref>.

[-] (0,0) node[anchor=north]$\mathcal{O}$ –  (0,8) node[anchor = east]$\mathcal{O}'$;
[-] (0,0) – (8,8);
[rotate=45] at (5.2,4.8) $C_0^+$;
[] at (-0.3,5.0) $\Gamma$;

[gray!50] (1,9) to (2,10) to[out = 45,in=135] (6,10) to[out = -135, in = 45] (1,9);
[] (0,8) – (4,4);
[densely dashed] (0,4) – (6,10);
[rotate=45] at (4.2,7.8) $\mathcal{EH}$;
[] (6,10) – (8,8);
[rotate = -45] at (7.2,9.2)$\mathcal{I}^+$;
[thick, densely dotted] (0,8) to (2,10);
[thick, loosely dotted] (2,10) to [out = 45,in=135] (6,10);
[fill=white] (2,10) circle (2pt);
[fill=white] (0.7,8.7) circle (2pt);
[rotate=45] at (0.1,8.5) $\mathcal{S}^1_\Gamma$;
[fill=white] (1.4,9.4) circle (2pt);
[rotate=45] at (0.8,9.2) $\mathcal{CH}_\Gamma$;
[rotate=45] at (1.5,9.9) $\mathcal{S}^2_\Gamma$;
[thick, dashdotted] (1,9) to[out = 45, in = - 135] (6,10);
[] at (4.1,9.7) $\mathcal{A}$;
[] at (4.2,10.3) $\mathcal{BH}$;

[anchor = west] at (9.8,10.2) Legend:;
[anchor = west] at (11,9.5) $\mathcal{B}_0$;
[thick, densely dotted] (10,9.3) to (11,9.7);
[anchor = west] at (11,9) $\mathcal{B} \setminus \mathcal{B}_0$;
[thick, loosely dotted] (10,8.8) to (11,9.2);
[anchor = west] at (11,8.5) $\mathcal{A}$;
[thick, dashdotted] (10,8.3) to (11,8.7);
[fill=white] (0,8) circle (2pt);
[fill=white] (6,10) circle (2pt);

Let $\mathcal{U}$ be a subset of the regular region $\mathcal{R}$. With regards to the regularity of solutions to the spherically symmetric Einstein-Maxwell-charged scalar field system, we first define the notion of a $C^1$ solution as follows:

A solution to equations (<ref>)-(<ref>) is called a $C^1$ solution in $\mathcal{U}$ if
$$\inf_{\mathcal{U}} \du r > - \infty$$

and $\dv r, \du r, \phi, r \dv \phi, r \du \phi, Q/r$ are $C^1$ functions on $\overline{\mathcal{U}}$.

Contrasting with the corresponding definitions in [19], here we also require $Q/r$ being $C^1$ on $\overline{\mathcal{U}}$. This follows from considering (<ref>), (<ref>) and the fact that $\dv r, \du r, \phi, r \dv \phi, r \du \phi$ being $C^1$ on $\overline{\mathcal{U}}$. Furthermore, note that $Q/r$ is a dimensionless quantity, which is consistent with the other terms mentioned above.

In this paper, when we refer to $C^1$ initial data, we mean
$\alpha:=\frac{\partial_v(r\phi)}{\partial_v r}$ is $C^1$ along $C_0^+$.

We proceed to define the notion of a solution of bounded variation in $\mathcal{U}$ as in [19] below:

A solution to equations  (<ref>)-(<ref>) is called a solution of bounded variation in $\mathcal{U}$ if

\begin{equation}\label{Setup77}
\inf_{\mathcal{U}} \du r > - \infty,
\end{equation}

* $\dv r$ is of bounded variation along each outgoing null curve and $\du r$ is of bounded variation along each incoming null curve;

* For each $p$ with $(p,p) \in \Gamma$, we have

\begin{equation}\label{Setup78}
\lim_{\varepsilon \rightarrow 0^+}(\dv r + \du r)(a,a+\varepsilon) = 0;
\end{equation}

* $\phi$ is an absolutely continuous function on each outgoing null curve and on each incoming null curve;

* For each $p$ with $(p,p) \in \Gamma$, the following conditions hold:

\begin{equation}\label{Setup79}
\begin{aligned}
\lim_{\varepsilon \rightarrow 0} \sup_{0 < \delta \leq \varepsilon} TV_{\{p - \delta\} \times (p-\delta,p)}[\phi] = 0, \qquad \lim_{\varepsilon \rightarrow 0} \sup_{0 \leq \delta < \varepsilon} TV_{ (p-\varepsilon,p-\delta) \times \{p-\delta\} }[\phi] = 0, \\
\lim_{\varepsilon \rightarrow 0} \sup_{0 < \delta \leq \varepsilon} TV_{ (p,p+\delta) \times \{p+\delta\} }[\phi] = 0, \qquad \lim_{\varepsilon \rightarrow 0} \sup_{0 \leq \delta < \varepsilon} TV_{\{p+\delta\} \times (p+\delta,p+\varepsilon) }[\phi] = 0;
\end{aligned}
\end{equation}

* $r\dv \phi$ is of bounded variation on each outgoing null curve, and $r \du \phi$ is of bounded variation on each incoming null curve;

* For each $p$ with $(p,p) \in \Gamma$, we have

\begin{equation}\label{Setup80}
\lim_{\varepsilon \rightarrow 0^+} \ml \dv(r\phi) + \du(r\phi) \mr (p,p+\varepsilon) = 0.
\end{equation}

Here, we retain the definition of a solution of  bounded variation solution as in [19]. In Section <ref>, we will deduce the additional total-variation estimates for the charged terms and observe that the above is a proper definition.

§ NOVEL ESTIMATES FOR Q AND  PHI

We now prove several crucial estimates for $Q$ and $\phi$ which are employed throughout the paper. We start off with a useful lemma for $\dv r/(1-\mu)$ and $\du r/(1-\mu)$ below.

In the regular region $\mathcal{R}$, the function $\frac{\dv r}{1 - \mu}(u,v)$ is decreasing along any future-directed incoming null curve and the function $\frac{-\du r}{1-\mu}$ is increasing along any future-directed outgoing null curve. Furthermore, if $(u,v) \in \mathcal{R}$ and there exists $0 \leq \overline{u} < u$ such that $(u',v) \in \mathcal{R}$ for all $u' \in [\overline{u},u]$, then we have

\begin{equation}\label{Estimates1}
\frac{\dv r}{1-\mu} (u,v) \leq \frac{\bar{B}}{(1-\overline{\mu_*})}(\overline{u},v)
\end{equation}

\begin{equation}\label{Bandmu*}
\bar{B}(\overline{u},v) := \sup_{v'\in(\overline{u},v]} \dv r(\overline{u},v') \quad \text{ and } \quad \overline{\mu_*}(\overline{u},v) := \sup_{v'\in (\overline{u},v]}\mu(\overline{u},v') \in [0,1).
\end{equation}

[->] (0,0) node[anchor=north]$\mathcal{O}$ –  (0,6) node[anchor = east]$\Gamma$;
[->] (0,0) – (5,5) node[anchor = west]$C_0^+$;
[thick, dashdotted]  (4.9,5.1) to[in=-120,out= -140] (2.4,2.9) to[out=60,in=0] (2,4) to[out=180,in=-45] (0,4.6);
[thick] (1.2,3.8) node[anchor = east]$p$ – (2,3)node[anchor = south]$\overline{p}$;
[thick] (0,1) – (2,3);
[dotted] (2,3) – (2.5,2.5);
[] at (0.5,2.5) $\mathcal{R}$;
[] at (3.5,4.5) $\mathcal{T}$;
[] at (2,4.3) $\mathcal{A}$;
at (1.2,3.8)[circle,fill,inner sep=1.5pt];
at (2,3)[circle,fill,inner sep=1.5pt];