Mass Inflation and the $C^2$-inextendibility of Spherically Symmetric Charged Scalar Field Dynamical Black Holes

Abstract

It has long been suggested that the Cauchy horizon of dynamical black holes is subject to a weak null singularity, under the mass inflation scenario. We study in spherical symmetry the Einstein–Maxwell–Klein–Gordon equations and while we do not directly show mass inflation, we obtain a “mass inflation/ridigity” dichotomy. More precisely, we prove assuming (sufficiently slow) decay of the charged scalar field on the event horizon, that the Cauchy horizon emanating from time-like infinity $\mathcal {CH}_{i^+}$ can be partitioned as $\mathcal {CH}_{i^+}= {\mathcal {D}} \cup {\mathcal {S}}$ for two (possibly empty) disjoint connected sets ${\mathcal {D}}$ and ${\mathcal {S}}$ such that

• ${\mathcal {D}}$ (the dynamical set) is a future set on which the Hawking mass blows up (mass inflation scenario).

• ${\mathcal {S}}$ (the static set) is a past set isometric to a Reissner–Nordström Cauchy horizon i.e. the radiation is zero on ${\mathcal {S}}$.

As a consequence of this result, we prove that the entire Cauchy horizon $\mathcal {CH}_{i^+}$ is globally $\underline{C^2-{ inextendible}}$, extending a previous local result established by the author. To this end, we establish a novel classification of Cauchy horizons into three types: dynamical (${\mathcal {S}}=\emptyset$), static (${\mathcal {D}}=\emptyset$) or mixed. As a side benefit, we prove that there exists a trapped neighborhood of the Cauchy horizon, thus the apparent horizon cannot cross the Cauchy horizon, which is a result of independent interest. Our main motivation is to prove the $C^2$ Strong Cosmic Censorship Conjecture for a realistic model of spherical collapse in which charged matter emulates the repulsive role of angular momentum. In our case, this model is the Einstein–Maxwell–Klein–Gordon system on space-times with one asymptotically flat end. As a consequence of the $C^2$-inextendibility of the Cauchy horizon, we prove the following statements, in spherical symmetry:

• 1. Two-ended asymptotically flat space-times are $C^2$-future-inextendible i.e. $C^2$ Strong Cosmic Censorship is true for Einstein–Maxwell–Klein–Gordon, assuming the decay of the scalar field on the event horizon at the expected rate.

• 2. In the one-ended case, under the same assumptions, the Cauchy horizon emanating from time-like infinity is $C^2$-inextendible. This result suppresses the main obstruction to $C^2$ Strong Cosmic Censorship in spherical collapse.

The remaining obstruction in the one-ended case is associated to “locally naked” singularities emanating from the center of symmetry, a phenomenon which is also related to the Weak Cosmic Censorship Conjecture.

Appendix A Construction of a Cauchy Horizon of Mixed Type for the Einstein–Null–Dust Model

In this appendix, we construct an example of a Cauchy horizon of mixed type, following Definition 3.3. In the second part of our development, we prove the blow up of the Hawking mass for mixed and dynamical type Cauchy horizons in the Einstein–Maxwell-null-dust model, following Poisson and Israel [43] and²⁶ Ori [37], the first instances of the mass inflation scenario in the literature. Notice that their model is very elementary, as the dust clouds are simply transported linearly in the null directions, and only interact indirectly, via the metric: thus, the “scattering theory” is trivial. Additionally, such a model does not allow for one-ended regular solutions, unlike the charged/massive scalar field model. In this appendix we sketch a mathematical proof of mass inflation, using the methods of [47]; this is also an opportunity to illustrate the terminology of the present paper in the simpler setting of dust, where little analysis is required and monotonicity suffices. The system (1.7), (1.8), (1.9), (1.10), (1.11), (1.12), (1.13) can be expressed in spherical symmetry as

$$\begin{aligned} \partial _{u}\partial _{v} r= & {} \partial _u \lambda =\partial _v \nu = -\frac{\Omega ^2}{2r} \cdot (\frac{\varpi }{r}-\frac{e^2}{r^2}) =\frac{\lambda \nu }{r}-\frac{\Omega ^2}{4r} \cdot (1- \frac{e^2}{r^ 2}) \end{aligned}$$

(A.1)

$$\begin{aligned} \partial _{u}\partial _{v} \log (\Omega ^2)= & {} \frac{\Omega ^2}{2r^2} \cdot ( \frac{2\varpi }{r}-\frac{3e^2}{r^2})= \frac{2 \nu \lambda }{r^{2}}+ \frac{ \Omega ^{2}}{2r^{2}} \cdot (1- \frac{ 2 e^{2}}{r^{2}}),\end{aligned}$$

(A.2)

$$\begin{aligned} \partial _{u} \varpi= & {} \frac{r^2}{2} \kappa ^{-1}| f_R|^{2}, \end{aligned}$$

(A.3)

$$\begin{aligned} \partial _{v} \varpi= & {} \frac{r^2}{2} \iota ^{-1}| f_L|^{2}, \end{aligned}$$

(A.4)

$$\begin{aligned} \partial _v (r f_R)= & {} 0, \end{aligned}$$

(A.5)

$$\begin{aligned} \partial _u (r f_L)= & {} 0, \end{aligned}$$

(A.6)

$$\begin{aligned} \partial _{u}(\frac{-4\nu }{\Omega ^{2}})= & {} \partial _{u}(\kappa ^{-1})=\frac{4r}{\Omega ^{2}}| f_R|^{2}, \end{aligned}$$

(A.7)

$$\begin{aligned} \partial _{v}(\frac{-4\lambda }{\Omega ^{2}})= & {} \partial _v(\iota ^{-1})=\frac{4r}{\Omega ^{2}}|f_L|^{2}, \end{aligned}$$

(A.8)

$$\begin{aligned} F_{\mu \nu }= & {} \frac{e^2 \Omega ^2}{ 2r^2} du \wedge dv. \end{aligned}$$

(A.9)

Proposition A.1

We consider the maximal development of smooth, spherically symmetric and admissible (in the sense of Definition 2.6) two-ended initial data $(M={\mathcal {Q}}^+ \times _r {\mathcal {S}}^2,g_{\mu \nu }, \phi ,F_{\mu \nu })$ satisfying the Einstein–Maxwell-null-dust system. Suppose that Assumptions 1, 2 and 3 of Theorem 3.1 are satisfied, and we also work under the same (U, v) gauge choice. We also choose $U_{max} \in {\mathbb {R}}$ such that $[0,U_{max}]\times \{v=+\infty \} \subset \mathcal {CH}_{i^+}$ in the Penrose diagram ${\mathcal {Q}}^+$. We consider data $f_L^0$ on the event horizon ${\mathcal {H}}^+= \{0\} \times [v_0,+\infty )$ and $f_R^0$ on an ingoing cone $\underline{C}_{v_0}= [0,U_{max}]\times \{v_0\}$; we assume that both $f_R^0$ and $f_L^0$ are smooth in the (U, v) coordinate system. We also make assumptions analogous to Assumptions 4 and 5 of Theorem 3.1 i.e. for some $s>\frac{1}{2}$ and some $C>0$:

$$\begin{aligned} \sup _{v \ge v_0 }|f_L^0|_{|{\mathcal {H}}^+}(v) \cdot v^{s} \le C \cdot , \\ \sup _{0 \le U \le U_{max}}|f_R^0| (U, v_0) \le C. \end{aligned}$$

Then by continuity of $f^0_R$ on $[0,U_{max})$, there are three possibilities:

• (a) For all $0<U_s<U_{max}$, there exists $0 \le U \le U_s$ such that $f^0_R(U) \ne 0$. We call this the dynamical case.

• (b) For all $U\in [0,U_{max})$, $f^0_R(U) = 0$. We call this the static case.

• (c) There exists $U_T \in (0,U_{max})$ such that for all $U\in [0,U_{T}]$, $f^0_R(U) = 0$, and such that for every $\epsilon >0$, there exists $U\in (U_{T},U_{T}+\epsilon ]$ such that $f^0_R(U) \ne 0$. We call this the mixed case.

Then we have, in the three different cases:

• (a) In the dynamical case, r extends to $\mathcal {CH}_{i^+}$ as function $r_{CH}$ which is strictly decreasing on $[0,U_{max})$.

• (b) In the static case, r extends to $\mathcal {CH}_{i^+}$ as a constant $r_{-}>0$.

• (c) In the mixed case, r extends to $\mathcal {CH}_{i^+}$ as a function $r_{CH}$, which is constant on $[0,U_T]$, and strictly decreasing on $(U_T,U_{max} )$.

If we additionally assume that $f_L^0(v)=C \cdot v^{-s}+o(v^{-s})$, for some $s>\frac{1}{2}$, where v is defined by gauge (3.1), then we have

• (a) In the dynamical case, the Hawking mass $\rho$ blows up on $\mathcal {CH}_{i^+}\cap [0,U_{max}]$.

• (b) In the static case, the Hawking mass $\rho$ is constant on $\mathcal {CH}_{i^+}\cap [0,U_{max}]$.

• (c) In the mixed case, the Hawking mass $\rho$ is constant on $\mathcal {CH}_{i^+}\cap [0,U_{T}]$ and blows up on $\mathcal {CH}_{i^+}\cap (U_T,U_{max}]$.

Remark A.1

Note that $\mathcal {CH}_{i^+}\ne \emptyset$, applying (an easier version of) the argument of [47], which proves the stability of the Reissner–Nordström Cauchy horizon for the Einstein–Maxwell–Klein–Gordon model in spherical symmetry.

Proof

We first prove that the possibilities reduce to the three cases a, b, c. Define the set $E:=\{U\in [0,U_{max}), f_R^0(U)\ne 0\}$. If $E=\emptyset$ then we are in case b. If $E \ne \emptyset$, define $U_T=\inf E$. There are two possibilities: either $U_T=0$ or $U_T>0$. If $U_T=0$ then we are in case  a by definition of E. If $U_T>0$ then for all $0 \le U < U_T$, $f_R^0(U)= 0$. By continuity of $f_R^0$ we have also $f_R^0(U_T)= 0$. Again by definition of E, we are therefore in case c.

We work in the gauge $\kappa _{|{\mathcal {H}}^+} \equiv 1$ and we denote $\varphi _R(u)= r(u,v) f_R(u,v)$, $\varphi _L(v)= r(u,v) f_L(u,v)$ by (A.5), (A.6).

By the stability estimates²⁷ of [47] (see Sect. 4), one can show that for some sub-extremal parameters $0<|e|<M$,

$$\begin{aligned}&|\varpi (U_{\gamma }(v),v)-M|+ |r(U_{\gamma }(v),v)-r_-(M,e)|\\&\quad + |\partial _v \log (\Omega ^2)(U_{\gamma }(v),v)-2K_-(M,e)| + v \cdot |\partial _v r| (U_{\gamma }(v),v)\lesssim v^{1-2s} \end{aligned}$$

holds on some space-like curve $\gamma$ terminating at $i^+$, for v large enough. We will then establish estimates for all $U \in [0,U_{max}]$: combining (A.1) and (A.2), we get $\partial _U \partial _v \log (r \Omega ^2) = \frac{ \Omega ^{2}}{2r^{2}} \cdot (1- \frac{ 3 e^{2}}{r^{2}})$. In view of $|e| > r_-(M,e)$ and the monotonicity of r (which implies that $r(U,v)\le r(U_{\gamma }(v),v)$ in our region of interest), we have $\partial _U \partial _v \log (r \Omega ^2)(U,v) <0$ for all $U \in [0,U_{max}]$. Integrating in U from $\gamma$, assuming v large enough (in particular $v\ge v_{\gamma }(U_s)$), we have, for all $U \in [0,U_{max}]$:

$$\begin{aligned} \partial _v\log (r\Omega ^2)(U,v) \le \partial _v\log (r\Omega ^2)(U_{\gamma }(v),v) \le 2K_-(M,e) - C \cdot v^{1-2s} \le K_-(M,e)<0, \end{aligned}$$

where the last inequality follows from v being large enough. This proves that for all $U \in [0,U_{max}]$

$$\begin{aligned} \Omega ^2(U,v) \lesssim e^{K_-(M,e) \cdot v }. \end{aligned}$$

(A.10)

Using (A.10) with $\partial _U \partial _v \log (r \Omega ^2) = \frac{ \Omega ^{2}}{2r^{2}} \cdot (1- \frac{ 3 e^{2}}{r^{2}})$ and the fact that $C(M,e)^{-1} \lesssim |1- \frac{ 3 e^{2}}{r^{2}}| \lesssim C(M,e)$ by monotonicity of r and the fact that r is lower bounded, we get an improved estimate, still for all $U \in [0,U_{max}]$:

$$\begin{aligned} |\partial _v \log (r \Omega ^2)(U,v) -2K_-(M,e) | \lesssim v^{1-2s}, e^{3K_-(M,e) \cdot v } \lesssim \Omega ^2(U,v) \lesssim e^{K_-(M,e) \cdot v }.\nonumber \\ \end{aligned}$$

(A.11)

Now, we can write (A.1), since $\nu \le 0$, as $\partial _v( r|\nu |) =\Omega ^2 \cdot (1- \frac{e^2}{r^2})$. Since r is decreasing in U, $(1- \frac{e^2}{r^2(U,v)})<0$ for all $U \in [0,U_{max}]$, as $r<r_-(M,e)<|e|$. Hence $v \rightarrow r\nu (U,v)$ is strictly decreasing for all fixed U, thus has a limit $r\nu _{\mathcal {CH}_{i^+}}(U) \in {\mathbb {R}}_-$. Moreover, we establish, using (A.1), the following estimate for all $U \in [0,U_{max}]$:

$$\begin{aligned} | r\nu (U,v) - r\nu _{\mathcal {CH}_{i^+}}(U) | \lesssim \Omega ^2(U,v) . \end{aligned}$$

(A.12)

Now, writing (A.1) as $\partial _U(- r\lambda ) =\Omega ^2 \cdot (1- \frac{e^2}{r^2})$, we get decay for $\lambda$ so we can extend r to $\mathcal {CH}_{i^+}$ into a function $r_{\mathcal {CH}_{i^+}} >0$. Therefore, for any fixed $U<U_{max}$, $v \rightarrow \nu (U,v)$ has a limit as $v\rightarrow +\infty$ that we consistently denote $\nu _{\mathcal {CH}_{i^+}}(U)$.

Since r is lower bounded in this region, we have $\partial _U (\kappa ^{-1})(U,v) \gtrsim C \cdot \varphi _R^2(U) \cdot e^{|K_-|(M,e) v}$ hence, integrating from $\gamma$:

$$\begin{aligned} \kappa ^{-1}(U,v)\gtrsim & {} e^{|K_-|(M,e) v} \cdot \int _{U_{\gamma (v)}}^U \varphi _R^2(U')dU'\nonumber \\= & {} e^{|K_-|(M,e) v} \cdot \int _{U_{\gamma (v)}}^U r^2(U',v_0) \cdot (f_R^0(U'))^2dU' \nonumber \\\gtrsim & {} e^{|K_-|(M,e) v} \cdot \int _{U_{\gamma (v)}}^U (f_R^0(U'))^2dU'. \end{aligned}$$

(A.13)

Then, there are our three possibilities, starting with the easiest:

• 1. Dynamical case: then for all $U\in [0,U_{max})$, $\int _{0}^U(f^0_R)^2(U')dU' > 0$. Indeed by (a), for all $U\in [0,U_{max})$, there exists $U'<U$ with $(f^0_R(U'))^2 > 0$, hence by continuity there exists $\epsilon >0$ such that $(f^0_R(U'))^2 > 0$ on $[U',U'+\epsilon ]$, which implies $\int _{0}^U(f^0_R)^2(U')dU' > 0$ . In view of the limit $U_{\gamma }(v) \rightarrow 0$ as $v\rightarrow +\infty$ (recall that $\gamma$ terminates at $i^+$) then $\lim _{v \rightarrow +\infty } \int _{U_{\gamma (v)}}^U (f_R^0(U'))^2dU' >0$ for all $U\in [0,U_{max})$, hence $\kappa ^{-1}(U,v) \rightarrow +\infty$. If $\nu _{CH}(U)=0$ for some $0<U < U_{max}$, then (A.12) is contradicted (after dividing both sides of (A.12) by $r \Omega ^2$). Hence $\nu _{\mathcal {CH}_{i^+}}(U)<0$ for all $U\in [0,U_{max}]$ thus $r_{\mathcal {CH}_{i^+}}$ is strictly decreasing on $[0,U_{max})$ as desired . Moreover $\kappa ^{-1}(U,v) \sim \frac{4|\nu _l|(U)}{\Omega ^2(U,v)}$ by (A.12). If additionally we have a lower bound $f_L^0(v)=C \cdot v^{-s}+o(v^{-s})$ then by the stability estimates of [47], $-\lambda \sim C \cdot v^{-2s}$, at least for U small enough and in the future of the curve $\gamma$. Since $2\rho -r= \kappa ^{-1} \cdot (-\lambda )$, $\rho (U,v) \rightarrow +\infty$ as $v \rightarrow +\infty$, for small U. A forciciori, $\varpi (U,v) \rightarrow +\infty$ for small U, and by the monotonicity of (A.3), $\varpi (U,v) \rightarrow +\infty$ for all $U\in [0,U_{max})$. Hence, for all $U\in [0,U_{max})$, $\rho (U,v) \rightarrow +\infty$ as $v \rightarrow +\infty$.

• 2. Static case: for all $U\in [0,U_{max})$, $f^0_R(U) = 0$. Then $\kappa (U,v) = 1$ as $\partial _U (\kappa ^{-1})=0$. By (A.12), it means that $\nu _{\mathcal {CH}_{i^+}}(U)=0$ for all $U\in [0,U_{max})$ hence $r_{\mathcal {CH}_{i^+}}$ is constant and by (A.3) $\varpi _{\mathcal {CH}_{i^+}}$ is constant. Their values are $r_{\mathcal {CH}_{i^+}} \equiv r_-(M,e)>0$ and $\varpi _{\mathcal {CH}_{i^+}} \equiv M >0$ by the stability estimates of [47]. This implies that $\rho$ is constant on $\mathcal {CH}_{i^+}$: $\rho (U,v) \rightarrow M-\frac{e^2}{r_-(M,e)}$ as $v \rightarrow +\infty$ for all $U\in [0,U_{max})$.

• 3. Mixed case: $\int _{0}^{U_T}(f^0_R)^2(U')dU' = 0$ but $\int _{U_T}^{U}(f^0_R)^2(U')dU' > 0$ for all $U_T<U<U_{max}$. Indeed, the above statement follows from the same logic as for the dynamical case, see earlier discussion. Then similarly, $r_{\mathcal {CH}_{i^+}}(U)= r_-(M,e)$, $\varpi _{\mathcal {CH}_{i^+}}(U)= M$, and $\rho _{\mathcal {CH}_{i^+}}(U)= M-\frac{e^2}{r_-(M,e)}$ for all $U \in [0,U_T]$. Yet, $\kappa ^{-1}(U,v) \rightarrow +\infty$ for all $U \in (U_T,U_{max})$, as $\int _{0}^U (f_R^0(U'))^2dU' >0$. Thus $\nu _{\mathcal {CH}_{i^+}}(U)<0$ for all $U \in (U_T,U_{max})$ and $r_{\mathcal {CH}_{i^+}}$ is strictly decreasing on $(U_T,U_{max})$. If additionally we have a lower bound $f_L^0(v)=C \cdot v^{-s}+o(v^{-s})$, then we have the blow up of $\rho$, as in the dynamical case.

$\square$

Remark A.2

Notice that the proof was considerably easier for the uncharged dust than for a charged scalar field, as we used in this section some special monotonicity properties that are not exploitable in the charged case. In contrast, in the earlier sections, we proved estimates that were harder to obtain but also more robust as they do not rely on monotonicity.