On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 3. Mass Inflation and Extendibility of the Solutions

Abstract

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein–Maxwell-scalar field system with a cosmological constant $\Lambda$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner–Nordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a “suitably regular” Lorentzian manifold. In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon. In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when $\Lambda >0$.

Appendices

Appendix A: On the Choice of the Parameters and Its Consequences

The objective of this appendix is to study the behavior of $\rho$ (defined in (42), the quotient of the surface gravities at $r_-$ and at $r_+$ of the reference subextremal Reissner–Nordström black hole) as a function of the parameters $\Lambda$, $\varpi _0$ and e. It turns out that it is easiest to express $\rho$ in terms of the new parameters $\sigma$ and $\Upsilon$, defined in (108). The formula for $\rho$ in terms of $\sigma$ and $\Upsilon$ is given in (110). At the end of this appendix, the reader can find a figure showing the behavior of $\rho$ in the $(\sigma ,\Upsilon )$ plane.

We consider the fourth order polynomial

$$\begin{aligned} p(r):=r^2(1-\mu )(r,\varpi _0)=-\,\frac{\Lambda }{3}r^4+r^2-2\varpi _0r+e^2. \end{aligned}$$

Since we assume p has zeros at $r_-$ and $r_+$, it can be factored as

$$\begin{aligned} p(r)=[r^2-(r_++r_-)r+r_-r_+]\left[ {\textstyle -\,\frac{\Lambda }{3}r^2+cr+\frac{e^2}{r_-r_+}}\right] . \end{aligned}$$

The constant c can be computed by imposing that the coefficient of p in $r^3$ is equal to zero. We obtain $c=-\,\frac{\Lambda }{3}(r_-+r_+)$. Hence, p can be factored as

$$\begin{aligned} p(r)=[r^2-(r_++r_-)r+r_-r_+]\left[ {\textstyle -\,\frac{\Lambda }{3}r^2-\,\frac{\Lambda }{3}(r_-+r_+)r+\frac{e^2}{r_-r_+}}\right] . \end{aligned}$$

Since the coefficient of p in r is equal to $-2\varpi _0$, we must have

$$\begin{aligned} \varpi _0=\frac{e^2}{2r_-}+\frac{e^2}{2r_+}+\frac{\Lambda }{6}(r_-+r_+)r_-r_+. \end{aligned}$$

On the other hand, since the coefficient of p in $r^2$ is equal to 1, we must have

$$\begin{aligned} \frac{e^2}{r_-r_+}=1-\frac{\Lambda }{3}(r_-^2+r_-r_++r_+^2). \end{aligned}$$

We define

$$\begin{aligned} \sigma :=\frac{r_+}{r_-}\qquad \mathrm{and}\qquad \Upsilon :=\frac{\Lambda r_-^2}{3}. \end{aligned}$$

(108)

Then

$$\begin{aligned} \frac{e^2}{r_-r_+}=1-\Upsilon (\sigma ^2+\sigma +1). \end{aligned}$$

A simple computation shows that

$$\begin{aligned} \frac{\varpi _0}{r_-}=\frac{1}{2}(\sigma +1)[1-\Upsilon (\sigma ^2+1)]. \end{aligned}$$

Of course, we could think of $\Lambda$, $\varpi _0$ and e as the independent parameters, and use the equation $p(r)=0$ to determine $r_-$ and $r_+$. Instead, we think of $r_-$, $r_+$ and $\Lambda$ as the independent parameters, and $\varpi _0$ and e as the dependent ones. More precisely, we regard $r_-$, $\sigma$ and $\Upsilon$ as the independent parameters and $\frac{e^2}{r_-r_+}$ and $\frac{\varpi _0}{r_-}$ as the dependent ones. Clearly, $\sigma >1$.

When $\Lambda >0$, the polynomial p has a third positive root $r_c$, the radius of the Reissner–Nordström de Sitter cosmological event horizon. This is the positive solution of

$$\begin{aligned} r^2+(r_-+r_+)r-\frac{3e^2}{\Lambda r_-r_+}=0. \end{aligned}$$

The value of $r_c$ is given by

$$\begin{aligned} r_c=\frac{-(r_-+r_+)+\sqrt{(r_-+r_+)^2+\frac{12e^2}{\Lambda r_-r_+}}}{2}\,. \end{aligned}$$

The fact that $r_+<r_c$ imposes a restriction on our independent parameters, namely

$$\begin{aligned} \frac{3e^2}{\Lambda r_-r_+}>2r_+^2+r_-r_+. \end{aligned}$$

In terms of $\sigma$ and $\Upsilon$, this can be written as

$$\begin{aligned} \frac{1-\Upsilon (\sigma ^2+\sigma +1)}{\Upsilon }>2\sigma ^2+\sigma , \end{aligned}$$

or

$$\begin{aligned} \Upsilon <\frac{1}{3\sigma ^2+2\sigma +1}. \end{aligned}$$

(109)

If $\Lambda \le 0$, condition (109) is also trivially satisfied. We say that a choice of parameters $(\sigma ,\Upsilon )$ is admissible if $\sigma >1$ and (109) holds.

Now we compute $\rho$ as defined in (42), obtaining

$$\begin{aligned} \rho= & {} \left( \frac{r_+}{r_-}\right) ^2\frac{\frac{e^2}{r_-}+\frac{\Lambda }{3}r_-^3-\varpi _0}{-\,\frac{e^2}{r_+}-\,\frac{\Lambda }{3}r_+^3+\varpi _0}\nonumber \\= & {} \left( \frac{r_+}{r_-}\right) ^2\frac{\frac{e^2}{r_-r_+}\frac{r_+}{r_-}+\frac{\Lambda r_-^2}{3}-\frac{\varpi _0}{r_-}}{-\,\frac{e^2}{r_-r_+}-\,\frac{\Lambda r_-^2}{3}\frac{r_+^3}{r_-^3}+\frac{\varpi _0}{r_-}}\nonumber \\= & {} \sigma ^2\frac{(1-\Upsilon (\sigma ^2+\sigma +1))\sigma +\Upsilon -\frac{1}{2}(\sigma +1)[1-\Upsilon (\sigma ^2+1)]}{-(1-\Upsilon (\sigma ^2+\sigma +1))-\Upsilon \sigma ^3+\frac{1}{2}(\sigma +1)[1-\Upsilon (\sigma ^2+1)]}\nonumber \\= & {} \sigma ^2\frac{1-\Upsilon (\sigma ^2+2\sigma +3)}{1-\Upsilon (3\sigma ^2+2\sigma +1)}. \end{aligned}$$

(110)

Taking into account (109), in the region of interest, the condition $\rho >1$ is equivalent to

$$\begin{aligned} \Upsilon<\frac{1}{3\sigma ^2+2\sigma +1}\quad \mathrm{and}\quad \Upsilon <\frac{1}{(\sigma +1)^2}. \end{aligned}$$

As the first upper bound is smaller than the second, we conclude that for all admissible choices of parameters we have $\rho >1$, that is

$$\begin{aligned} -\partial _r(1-\mu )(r_-,\varpi _0)>\partial _r(1-\mu )(r_+,\varpi _0). \end{aligned}$$

We prove mass inflation in the region $\rho >2$. Using (109) and (110), the condition $\rho >2$ is equivalent to

$$\begin{aligned} \frac{\sigma ^2-2}{\sigma ^4+2\sigma ^3-3\sigma ^2-4\sigma -2}<\Upsilon <\frac{1}{3\sigma ^2+2\sigma +1} \end{aligned}$$

if

$$\begin{aligned} \sigma <\sigma _0:=\frac{1}{2}\left( -1+{\textstyle \sqrt{9+4\sqrt{6}}}\right) \approx 1.66783. \end{aligned}$$

The value $\sigma _0$ is the only positive solution of $\sigma ^4+2\sigma ^3-3\sigma ^2-4\sigma -2=0$. For $\sigma \ge \sigma _0$, the condition $\rho >2$, with the restriction (109), is always satisfied. Indeed, for $\sigma >\sigma _0$, we have

$$\begin{aligned} \frac{\sigma ^2-2}{\sigma ^4+2\sigma ^3-3\sigma ^2-4\sigma -2}>\frac{1}{3\sigma ^2+2\sigma +1} \end{aligned}$$

because the difference

$$\begin{aligned} \frac{\sigma ^2-2}{\sigma ^4+2\sigma ^3-3\sigma ^2-4\sigma -2}-\frac{1}{(\sigma +1)^2} \end{aligned}$$

is equal to

$$\begin{aligned} \frac{2\sigma ^2}{(\sigma ^4+2\sigma ^3-3\sigma ^2-4\sigma -2)(\sigma +1)^2}, \end{aligned}$$

and this is positive for $\sigma >\sigma _0$.

In the next figure we sketch part of the $(\sigma ,\Upsilon )$-plane. As we just saw, the restriction $r_+<r_c$ translates into (109) and this region (shaded in the figure) is the only relevant one for our purposes. We remark that the limit value of $\rho$ on the line $\sigma =1$ is one.

Appendix B: Proof of Theorem 3.1

We start by establishing the following useful result.

Lemma 6.10

Assume that $\zeta _0(u)>0$ for $u>0$. Then $\theta >0$ and $\zeta >0$ in $\mathcal{P}\setminus \{0\}\times [0,\infty [$.

Proof

The proof proceeds in three steps.

Step 1 If $\theta _0>0$ and $\zeta _0>0$, then $\theta >0$ and $\zeta >0$ in $\mathcal{P}$. Otherwise, there would exist a point $(u,v)\in \mathcal{P}$ such that $\theta (u,v)=0$ or $\zeta (u,v)=0$ but $\theta >0$ and $\zeta >0$ in $J^-(u,v)$. Integrating (25) and (26), we obtain a contradiction.

Step 2 Since in Part 1 we proved continuous dependence of the solution on $\theta _0$ and $\zeta _0$, if $\theta _0\ge 0$ and $\zeta _0\ge 0$, then $\theta \ge 0$ and $\zeta \ge 0$.

Step 3 Suppose that $(u,v)\in \mathcal{P}\setminus \{0\}\times [0,\infty [$. Since $\zeta _0(u)>0$ for $u>0$, (26) implies that $\zeta (u,v)>0$, because, from the previous step, $\theta \ge 0$. So $\zeta >0$ in $\mathcal{P}\setminus \{0\}\times [0,\infty [$. Now (25) implies that $\theta$ is positive on $\mathcal{P}\setminus \{0\}\times [0,\infty [$ because $\lambda$ is negative on this set. $\square$

Corollary 6.11

Under the hypotheses of Lemma 6.10, for $u>\bar{u}$ and $v>\bar{v}$, we have

$$\begin{aligned} \varpi (u,v)-\varpi (u,\bar{v})\ge \varpi (\bar{u},v)-\varpi (\bar{u},\bar{v}). \end{aligned}$$

Proof

This is an easy consequence of the fact that

$$\begin{aligned} \partial _u\partial _v\varpi =-\frac{\theta \zeta \lambda }{\kappa r}-\frac{\theta ^2\zeta ^2}{2\kappa r \nu }\ge 0. \end{aligned}$$

$\square$

Proof of Theorem 3.1

We follow the argument on pages 493–497 of [10]. We consider the same three cases as in the proof of Lemma 2.5, presented in Part 2.

Case 1. If (44) holds, there is nothing to prove.

Case 2. If

$$\begin{aligned} \lim _{u\searrow 0}\varpi (u,\infty )>\varpi _0, \end{aligned}$$

(111)

then (44) holds. This was proven on page 494 of [10] and is repeated here for the convenience of the reader. Suppose that (111) holds. Then there exists $\varepsilon >0$ such that

$$\begin{aligned} \lim _{u\searrow 0}\varpi (u,\infty )>\varpi _0+3\varepsilon . \end{aligned}$$

Since $\lim _{v\rightarrow \infty }\varpi (u_\gamma (v),v)=\varpi _0$ (see (37)), we have

$$\begin{aligned} \varpi (u_\gamma (v),v)<\varpi _0+\varepsilon \end{aligned}$$

for, say, $v\ge V$. Hence, if u is sufficiently small, there exists $v=v_{\gamma ,\varepsilon }(u)$ so that $\varpi (u,v_{\gamma ,\varepsilon }(u))-\varpi (u,v_\gamma (u))=\varepsilon$. We construct a sequence $(u_n,v_n)$ in the following way. Starting with $(u_0,v_0)=(u_\gamma (V),V)$, we let $(u_{n+1},v_{n+1})$ be such that $v_{n+1}=v_{\gamma ,\varepsilon }(u_n)$ and $u_{n+1}=u_\gamma (v_{n+1})$. According to Lemma 6.11,

$$\begin{aligned} \varpi (u_n,v_{n+1})-\varpi (u_n,v_n)\ge \varpi (u_{n+1},v_{n+1})-\varpi (u_{n+1},v_n)=\varepsilon \end{aligned}$$

for all n,

$$\begin{aligned} \varpi (u_n,v_{n+2})-\varpi (u_n,v_{n+1})\ge \varpi (u_{n+1},v_{n+2})-\varpi (u_{n+1},v_{n+1})\ge \varepsilon \end{aligned}$$

for all n, and

$$\begin{aligned} \varpi (u_n,v_{n+k})-\varpi (u_n,v_{n+k-1})\ge \varpi (u_{n+1},v_{n+k})-\varpi (u_{n+1},v_{n+k-1})\ge \varepsilon \end{aligned}$$

for all n and all $k\ge 1$. Hence,

$$\begin{aligned} \varpi (u_n,v_{n+k})-\varpi (u_n,v_n)\ge k\varepsilon \end{aligned}$$

for all n and all $k\ge 1$. This implies (44) because $\partial _v\varpi \ge 0$.

Case 3. Suppose now that $\lim _{u\searrow 0}\varpi (u,\infty )=\varpi _0$. As in the proof of Lemma 2.5, we have $\left( \frac{e^2}{r}+\frac{\Lambda }{3}r^3-\varpi \right) (u,v)\ge 0$ for $(u,v) \in J^+(\Gamma _{{\check{r}}_-})$ and u sufficiently small. Then, from (21) it follows that $\partial _u(-\lambda )\le 0$ in $J^+(\Gamma _{{\check{r}}_-})$, whereas from Lemma 6.10 it follows $\partial _u\theta \ge 0$. As a consequence, the integral

$$\begin{aligned} I(u):=\int _{v_{{\check{r}}_-}(u)}^\infty \left[ \frac{\theta ^2}{-\lambda }\right] (u,\tilde{v})\,d{\tilde{v}} \end{aligned}$$

is a nondecreasing function of u. Therefore we have two alternatives to consider.

Case 3.1. $I(u)=+\infty$ for all small u, say $0<u\le U$. Consider such a u. We observe that the following limit exists and is finite:

$$\begin{aligned} \lim _{v\nearrow \infty }(1-\mu )(u,v)=:(1-\mu )(u,\infty )=1-\frac{2\varpi (u,\infty )}{r(u,\infty )}+\frac{e^2}{r^2(u,\infty )}-\frac{\Lambda }{3}r^2(u,\infty ). \end{aligned}$$

Equation (22) and $\left( \frac{e^2}{r}+\frac{\Lambda }{3}r^3-\varpi \right) (u,v)\ge 0$ imply that $v\mapsto \nu (u,v)$ is a nondecreasing function in $J^+(\gamma )$. So we may define

$$\begin{aligned} \nu (u,\infty )= & {} \lim _{v\nearrow +\infty }\nu (u,v). \end{aligned}$$

Integrating (122) we get $\lim _{v\nearrow \infty }\frac{\nu }{1-\mu }(u,v)=0$. Therefore, $\nu (u,\infty )=0$. Let $0<\delta <u\le U$. Clearly,

$$\begin{aligned} r(u,v)=r(\delta ,v)+\int _\delta ^u\nu (s,v)\,ds. \end{aligned}$$

Thus, by Lebesgue’s Monotone Convergence Theorem,

$$\begin{aligned} r(u,\infty )=r(\delta ,\infty )+\int _\delta ^u\nu (s,\infty )\,ds=r(\delta ,\infty ). \end{aligned}$$

Letting $\delta$ decrease to zero, due to (31), we obtain $r(u,\infty )\equiv r_-$. This contradicts Theorem 2.4.

Case 3.2. $I(u)<+\infty$ for all small u, say $0<u\le U$. Arguing as in pages 495–496 of [10], we know $\lim _{u\searrow 0}I(u)=0$. We will use this information to improve our upper bound on $-\lambda$ in the region $J^+(\gamma )$. Then we will obtain a lower bound for $\theta$ in this region. Finally, we use these bounds to arrive at the contradiction that $I(u)=+\infty$.

Let $\varepsilon >0$. As $\lim _{u\searrow 0}I(u)=0$, we may choose $U>0$ sufficiently small, so that for all $(u,v)\in J^+(\gamma )$ with $0<u\le U$,

$$\begin{aligned} e^{\frac{1}{r(U,\infty )}\int _{v_{{\check{r}}_-}({\bar{u}})}^v\bigl [\big |\frac{\theta }{\lambda }\bigr ||\theta |\bigr ]({\bar{u}},{\tilde{v}})\,d{\tilde{v}}}\le 1+\varepsilon , \end{aligned}$$

(112)

for ${\bar{u}}\in [u_{\gamma }(v),u]$.

Next we use (122), (134) and (112). We may bound the integral of $\nu$ along $\Gamma _{{\check{r}}_-}$ in terms of the integral of $\frac{\nu }{1-\mu }$ on the segment $\bigl [u_{\gamma }(v),u\bigr ]\times \{v\}$ in the following way:

$$\begin{aligned}&-\int _{u_{\gamma }(v)}^u\nu ({\tilde{u}},v_{{\check{r}}_-}({\tilde{u}}))\,d{\tilde{u}} \le \nonumber \\&\qquad \qquad \qquad -\min _{\Gamma _{{\check{r}}_-}}(1-\mu ) \int _{u_{\gamma }(v)}^u\frac{\nu }{1-\mu }({\tilde{u}},v_{{\check{r}}_-}({\tilde{u}}))\,d{\tilde{u}}\le \nonumber \\&\qquad \qquad \qquad -(1+\varepsilon )\min _{\Gamma _{{\check{r}}_-}}(1-\mu ) \int _{u_{\gamma }(v)}^u\frac{\nu }{1-\mu }({\tilde{u}},v)\,d{\tilde{u}}. \end{aligned}$$

(113)

Applying successively (113), (125), (131), and (138),

$$\begin{aligned}&\int _{u_{\gamma }(v)}^u\frac{\nu }{1-\mu }({\tilde{u}},v)\,d{\tilde{u}}\nonumber \\&\qquad \qquad \ge {\tiny \frac{1}{-(1+\varepsilon )\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\int _{u_{\gamma }(v)}^u-\nu ({\tilde{u}},v_{{\check{r}}_-}({\tilde{u}}))\,d{\tilde{u}}\nonumber \\&\qquad \qquad = {\tiny \frac{1}{-(1+\varepsilon )\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\int _{v_{{\check{r}}_-}(u)}^{v_{{\check{r}}_-}(u_{\gamma }(v))}-\lambda (u_{{\check{r}}_-}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}\nonumber \\&\qquad \qquad \ge {\tiny \frac{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}{(1+\varepsilon )\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\int _{\frac{v_{\gamma }(u)}{1+\beta }}^{\frac{v}{1+\beta }}\kappa (u_{{\check{r}}_-}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}\nonumber \\&\qquad \qquad \ge {\tiny \frac{(1-\varepsilon )\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}{(1+\varepsilon )\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\bigl ({\textstyle \frac{v-v_{\gamma }(u)}{1+\beta }}\bigr ). \end{aligned}$$

(114)

Thus,

$$\begin{aligned}&e^{\int _{u_{\gamma }(v)}^{u}\bigl [\frac{\nu }{1-\mu }\partial _r(1-\mu )\bigr ]({\tilde{u}},v)\,d{\tilde{u}}}\nonumber \\&\qquad \qquad \le e^{\bigl [\max _{J^+(\gamma )}\partial _r(1-\mu )\bigr ] \displaystyle \int _{u_{\gamma }(v)}^{u}\frac{\nu }{1-\mu }({\tilde{u}},v)\,d{\tilde{u}}}\nonumber \\&\qquad \qquad \le e^{\bigl [\max _{J^+(\gamma )}\partial _r(1-\mu )\bigr ] {\tiny \frac{(1-\varepsilon )}{(1+\varepsilon )}\frac{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}{\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\, \bigl ({\textstyle \frac{v-v_{\gamma }(u)}{1+\beta }}\bigr )}\nonumber \\&\qquad \qquad \le e^{\bigl [\partial _r(1-\mu )({\check{r}}_-,\varpi _0)+\max _{J^+(\gamma )}{\tiny \frac{2(\varpi -\varpi _0)}{r^2}}\bigr ] {\tiny \frac{(1-\varepsilon )}{(1+\varepsilon )}\frac{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}{\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\,\bigl ({\textstyle \frac{v-v_{\gamma }(u)}{1+\beta }}\bigr )}\nonumber \\&\qquad \qquad \le e^{\bigl [\partial _r(1-\mu )({\check{r}}_-,\varpi _0)+{\tiny \frac{\varepsilon }{(r_--\varepsilon _0)^2}}\bigr ] {\tiny \frac{(1-\varepsilon )}{(1+\varepsilon )}\frac{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}{\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\, \bigl ({\textstyle \frac{v-v_{\gamma }(u)}{1+\beta }}\bigr )}. \end{aligned}$$

(115)

We integrate (21) and we use (143) and (115) to obtain

$$\begin{aligned} -\lambda (u,v)= & {} -\lambda (u_{\gamma }(v),v)e^{\int _{u_{\gamma }(v)}^{u}\bigl [\frac{\nu }{1-\mu }\partial _r(1-\mu )\bigr ]({\tilde{u}},v)\,d{\tilde{u}}}\end{aligned}$$

(116)

$$\begin{aligned}\le & {} C e^{(1-{\tilde{\delta }})\partial _r(1-\mu )({\check{r}}_-,\varpi _0)\bigl (\frac{\beta v}{1+\beta }+\frac{v}{1+\beta }-\frac{v_{\gamma }(u)}{1+\beta }\bigr )}\nonumber \\= & {} C(u) e^{(1-{\tilde{\delta }})\partial _r(1-\mu )({\check{r}}_-,\varpi _0)v}. \end{aligned}$$

(117)

The value of ${\tilde{\delta }}$ can be made small by choosing U sufficiently small. Here $C(u)=Ce^{-(1-{\tilde{\delta }})\partial _r(1-\mu )({\check{r}}_-,\varpi _0)\frac{v_{\gamma }(u)}{1+\beta }}$. This is the desired upper estimate for $-\lambda$.

Now we turn to obtaining the lower estimate for $\theta$. Combining (127) with (128), for $(u,v)\in \Gamma _{r_+-\delta }$ we have

$$\begin{aligned} -\lambda (u,v)\ge & {} \Bigl (\frac{r_+-\delta }{r_+}\Bigr )\frac{\partial _r(1-\mu )(r_+,\varpi _0)}{1+\varepsilon }u\,e^{[\partial _r(1-\mu )(r_+,\varpi _0)-\varepsilon ]v}\nonumber \\\ge & {} Cu\,e^{[\partial _r(1-\mu )(r_+,\varpi _0)-\varepsilon ]v}. \end{aligned}$$

(118)

Note that C can be chosen independently of $\delta$. Using (26), Lemma 6.10 and (43),

$$\begin{aligned} \zeta (u,v)\ge cu^s\ \mathrm{for\ all}\ (u,v). \end{aligned}$$

(119)

We take into account that (118) and (119) are valid for arbitrary $\delta$, small, and that $J^-(\Gamma _{{\check{r}}_+})$ is foliated by curves $\Gamma _{r_+-\delta }$ for $0< \delta < r_+ - {\check{r}}_+$. Therefore, integrating (25), for $(u,v)\in \Gamma _{{\check{r}}_+}$ we have

$$\begin{aligned} \theta (u,v)\ge & {} Cu^{s+2}e^{[\partial _r(1-\mu )(r_+,\varpi _0)-\varepsilon ]v}\nonumber \\\ge & {} Ce^{[-(s+1)\partial _r(1-\mu )(r_+,\varpi _0)-{\tilde{\varepsilon }}]v}. \end{aligned}$$

(120)

For the last inequality, we used (128). The constant C depends on ${\check{r}}_+$. The value of ${\tilde{\varepsilon }}$ can be made small by choosing ${\check{r}}_+$ sufficiently close to $r_+$. We know that $\partial _u\theta \ge 0$. Thus, (120) also holds in $J^+(\Gamma _{{\check{r}}_+})$. This is the desired lower estimate for $\theta$.

We can now obtain a lower bound for I(u) using (117) and (120):

$$\begin{aligned} I(u)\ge & {} \int _{v_{\gamma }(u)}^\infty \left[ \frac{\theta ^2}{-\lambda }\right] (u,{\tilde{v}})\,d{\tilde{v}}\\\ge & {} C(u)\int _{v_{\gamma }(u)}^\infty \frac{e^{[-2(s+1)\partial _r(1-\mu )(r_+,\varpi _0)-2{\tilde{\varepsilon }}]{\tilde{v}}}}{e^{(1-{\tilde{\delta }})\partial _r(1-\mu )({\check{r}}_-,\varpi _0){\tilde{v}}}}\,d{\tilde{v}}. \end{aligned}$$

This integral is infinite if

$$\begin{aligned} -2(s+1)\partial _r(1-\mu )(r_+,\varpi _0)-\partial _r(1-\mu )({\check{r}}_-,\varpi _0)>0, \end{aligned}$$

or, equivalently,

$$\begin{aligned} s<\frac{1}{2}\frac{-\partial _r(1-\mu )({\check{r}}_-,\varpi _0)}{\partial _r(1-\mu )(r_+,\varpi _0)}-1, \end{aligned}$$

(121)

provided that ${\tilde{\varepsilon }}$ and ${\tilde{\delta }}$ are chosen sufficiently small (which we can achieve by decreasing U and $\delta$, if necessary). To complete the proof of Theorem 3.1 we just have to note that given $s<\frac{\rho }{2}-1$ we can always choose ${\check{r}}_-$ so that (121) holds, contradicting $I(u)<\infty$. $\square$

Appendix C: Some Useful Formulas

Here we collect some formulas that were obtained in Part 2 and that are needed to study the behavior of the solution at the Cauchy horizon.

The Raychaudhuri Equations Written in Terms of $\kappa$ and $\frac{\nu }{1-\mu }$

Using equations (20), (22), (24) and (28), we get

$$\begin{aligned} \partial _v\left( \frac{\nu }{1-\mu }\right) =\frac{\nu }{1-\mu }\left( \frac{\theta }{\lambda }\right) ^2\frac{\lambda }{r}. \end{aligned}$$

(122)

The equations (27) and (122) are the Raychaudhuri equations.

Evolution Equations for $\frac{\theta }{\lambda }$ and $\frac{\zeta }{\nu }$

Using equations (21), (25) and (22), (26) we obtain, respectively,

$$\begin{aligned} \partial _u\frac{\theta }{\lambda }= & {} -\,\frac{\zeta }{r}-\frac{\theta }{\lambda }\frac{\nu }{1-\mu }\partial _r(1-\mu ), \end{aligned}$$

(123)

$$\begin{aligned} \partial _v\frac{\zeta }{\nu }= & {} -\,\frac{\theta }{r}-\frac{\zeta }{\nu }\frac{\lambda }{1-\mu }\partial _r(1-\mu ). \end{aligned}$$

(124)

The Integrals of $\nu$ and $\lambda$ Along a Curve $\Gamma _{{\check{r}}}$

Equation (118) in Part 2 is

$$\begin{aligned} \int _{u_{{\check{r}}}(v)}^u\nu ({\tilde{u}},v_{{\check{r}}}({\tilde{u}}))\,d\tilde{u}=\int _{v_{{\check{r}}}(u)}^v\lambda (u_{{\check{r}}}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}. \end{aligned}$$

(125)

Estimates in $J^-(\Gamma _{{\check{r}}_+})$

Estimates (42) and (50) in Part 2 are

$$\begin{aligned}&\left| \frac{\zeta }{\nu }\right| (u,v)\le C\max _{{\bar{u}}\in [0,u]}|\zeta _0|({\bar{u}}).\nonumber \\&\qquad -\,\frac{\lambda }{1-\mu }\partial _r(1-\mu )\le -\Bigl (\frac{{\check{r}}_+}{r_+}\Bigr )^{\hat{\delta }^2}\min _{r\in [{\check{r}}_+,r_+]}\partial _r(1-\mu )(r,\varpi _0)\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \quad =-\alpha <0, \end{aligned}$$

(126)

where $\hat{\delta }$ is a bound for $\bigl |\frac{\zeta }{\nu }\bigr |$ in $J^-({\check{r}}_+)$.

Estimates for $(u,v)\in \Gamma _{r_+-\delta }$

Estimates (82) and (84) in Part 2 are

$$\begin{aligned}&-\,\Bigl (\frac{r_+}{r_+-\delta }\Bigr ) \left( \frac{\partial _r(1-\mu )(r_+,\varpi _0)}{1-\varepsilon }+\frac{4\tilde{\delta }}{r_+^2}\right) \,\delta \le \lambda \nonumber \\&\qquad \quad \le -\Bigl (\frac{r_+-\delta }{r_+}\Bigr )\frac{\partial _r(1-\mu )(r_+,\varpi _0)}{1+\varepsilon }\,\delta , \end{aligned}$$

(127)

$$\begin{aligned}&\delta \, e^{-[\partial _r(1-\mu )(r_+,\varpi _0)+\varepsilon ]\,v} \le u \le \delta \, e^{-[\partial _r(1-\mu )(r_+,\varpi _0)-\varepsilon ]\,v}, \end{aligned}$$

(128)

for $\delta > 0$ sufficiently small, where $\varepsilon >0$ and $\tilde{\delta }>0$ can be chosen arbitrarily close to zero if $\delta$ is small enough.

Estimate in $J^-(\Gamma _{{\check{r}}_-})\cap J^+(\Gamma _{{\check{r}}_+})$

Estimate (79) in Part 2 is

$$\begin{aligned} \Bigl (\frac{{\check{r}}_-}{r_+}\Bigr )^{\hat{\delta }^2} \frac{1-\varepsilon }{\partial _r(1-\mu )(r_+,\varpi _0)\,u}\le \frac{\nu }{1-\mu }(u,v)\le \frac{1+\varepsilon }{\partial _r(1-\mu )(r_+,\varpi _0)\,u}, \end{aligned}$$

(129)

where $\hat{\delta }>0$ can be chosen arbitrarily close to zero if U is small enough.

Estimate in $J^-(\Gamma _{{\check{r}}_-})$

Equation (93) in Part 2 is

$$\begin{aligned} \lim _{\mathop {{{\tiny {(u,v)\in J^-({\check{r}}_-)}}}}\limits ^{(u,v)\rightarrow (0,\infty )}}\,\Bigl |\frac{\theta }{\lambda }\Bigr |(u,v)=0. \end{aligned}$$

(130)

Relation Between the Integrals of $\lambda$ and $\kappa$ Along the Curve $\Gamma _{{\check{r}}_-}$

Estimates (119) and (120) in Part 2 are

$$\begin{aligned}&-\max _{\Gamma _{{\check{r}}_-}}(1-\mu ) \int _{v_{{\check{r}}_-}(u)}^v\kappa (u_{{\check{r}}_-}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}\end{aligned}$$

(131)

$$\begin{aligned}&\qquad \qquad \le -\int _{v_{{\check{r}}_-}(u)}^v\lambda (u_{{\check{r}}_-}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}\le \nonumber \\&\qquad \qquad \qquad \qquad -\min _{\Gamma _{{\check{r}}_-}}(1-\mu ) \int _{v_{{\check{r}}_-}(u)}^v\kappa (u_{{\check{r}}_-}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}. \end{aligned}$$

(132)

Relation Between the Integrals of $\nu$ and $\frac{\nu }{1-\mu }$ Along the Curve $\Gamma _{{\check{r}}_-}$

Estimates (121) and (122) in Part 2 are

$$\begin{aligned}&-\max _{\Gamma _{{\check{r}}_-}}(1-\mu ) \int _{u_{{\check{r}}_-}(v)}^u\frac{\nu }{1-\mu }({\tilde{u}},v_{{\check{r}}_-}({\tilde{u}}))\,d{\tilde{u}}\end{aligned}$$

(133)

$$\begin{aligned}&\qquad \ \ \le -\int _{u_{{\check{r}}_-}(v)}^u\nu ({\tilde{u}},v_{{\check{r}}_-}({\tilde{u}}))\,d{\tilde{u}}\le \nonumber \\&\qquad \qquad \qquad -\min _{\Gamma _{{\check{r}}_-}}(1-\mu ) \int _{u_{{\check{r}}_-}(v)}^u\frac{\nu }{1-\mu }({\tilde{u}},v_{{\check{r}}_-}({\tilde{u}}))\,d{\tilde{u}}. \end{aligned}$$

(134)

Relation Between the Integrals of $\frac{\nu }{1-\mu }$ and $\kappa$ Along the Curve $\Gamma _{{\check{r}}_-}$

Estimates (130) and (123) in Part 2 are

$$\begin{aligned}&{\tiny \frac{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}{(1+\varepsilon )\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\int _{v_{{\check{r}}_-}(u)}^v\kappa (u_{{\check{r}}_-}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}\end{aligned}$$

(135)

$$\begin{aligned}&\qquad \ \ \le \int _{u_{{\check{r}}_-}(v)}^u\frac{\nu }{1-\mu }({\tilde{u}},v)\,d{\tilde{u}}\le \nonumber \\&\qquad \qquad \qquad {\tiny \frac{\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\int _{v_{{\check{r}}_-}(u)}^v\kappa (u_{{\check{r}}_-}({\tilde{v}}),{\tilde{v}})\,d{\tilde{v}}. \end{aligned}$$

(136)

where $\varepsilon >0$ can be chosen arbitrarily close to zero for appropriate choices of the parameters $\beta _-$, $\beta _+$, ${\check{r}}_-$, ${\check{r}}_+$, $\varepsilon _0$ and U.

Estimates in $J^-(\gamma )\cap J^+(\Gamma _{{\check{r}}_-})$

Estimates (101), (105), (126) and (127) in Part 2 are

$$\begin{aligned}&\Bigl |\frac{\zeta }{\nu }\Bigr |(u,v) \le C\sup _{[0,u]}|\zeta _0|e^{-\bigl (\frac{\alpha }{1+\beta ^+}+\partial _r(1-\mu )(r_--\varepsilon _0,\varpi _0)\beta \bigr ) v}, \end{aligned}$$

(137)

$$\begin{aligned}&\kappa (u,v) \ge 1 - \varepsilon , \end{aligned}$$

(138)

$$\begin{aligned}&\Bigl |\frac{\theta }{\lambda }\Bigr |(u,v) \le C\sup _{[0,u]}|\zeta _0|e^{-\left( \frac{\alpha }{1+\beta ^+}+ \partial _r(1-\mu )(r_--\varepsilon _0,\varpi _0){\tiny \frac{\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\beta \right) v}, \end{aligned}$$

(139)

$$\begin{aligned}&e^{\frac{1}{r_--\varepsilon _0}\int _{v_{{\check{r}}_-}({\bar{u}})}^v\bigl [\big |\frac{\theta }{\lambda }\bigr ||\theta |\bigr ]({\bar{u}},{\tilde{v}})\,d{\tilde{v}}}\le 1+\varepsilon , \end{aligned}$$

(140)

where $\varepsilon >0$ can be chosen arbitrarily close to zero for appropriate choices of the parameters $\beta _-$, $\beta _+$, ${\check{r}}_-$, ${\check{r}}_+$, $\varepsilon _0$ and U.

Estimates for $(u,v)\in \gamma$

Estimates (131), (109), (110), (135) and (136) in Part 2 are

$$\begin{aligned}&e^{\int _{u_{{\check{r}}_-}(v)}^{u}\bigl [\frac{\nu }{1-\mu }\partial _r(1-\mu )\bigr ]({\tilde{u}},v)\,d{\tilde{u}}}\nonumber \\&\qquad \qquad \le e^{\bigl [\partial _r(1-\mu )({\check{r}}_-,\varpi _0)+{\tiny \frac{\varepsilon }{(r_--\varepsilon _0)^2}}\bigr ] {\tiny \frac{(1-\varepsilon )}{(1+\varepsilon )}\frac{\max _{\Gamma _{{\check{r}}_-}}(1-\mu )}{\min _{\Gamma _{{\check{r}}_-}}(1-\mu )}}\, \frac{\beta }{1+\beta } v}, \end{aligned}$$

(141)

$$\begin{aligned}&{\tilde{c}}e^{(1+\delta )\partial _r(1-\mu )(r_--\varepsilon _0,\varpi _0)\frac{\beta }{1+\beta }\,v}\end{aligned}$$

(142)

$$\begin{aligned}&\qquad \qquad \qquad \qquad \le -\lambda (u,v)\le \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad {\tilde{C}}e^{(1-\delta )\partial _r(1-\mu )({\check{r}}_-,\varpi _0)\frac{\beta }{1+\beta }\,v}, \end{aligned}$$

(143)

$$\begin{aligned}&ce^{-\partial _r(1-\mu )(r_+,\varpi _0)\frac{v}{1+\beta ^-}}\end{aligned}$$

(144)

$$\begin{aligned}&\qquad \qquad \qquad \qquad \le u\le \nonumber \\&\qquad \qquad \qquad \qquad \qquad Ce^{-\partial _r(1-\mu )(r_+,\varpi _0)\frac{v}{1+\beta ^+}}, \end{aligned}$$

(145)

for $\varepsilon <\varepsilon _0$. The bound (145) is actually valid in $J^-(\gamma )\cap J^+(\Gamma _{{\check{r}}_-})$.

Estimates in $J^+(\gamma )$

Lemmas 7.1 and 7.2 in Part 2 imply

$$\begin{aligned} -\lambda (u,v)\le & {} Ce^{(1-\delta )\partial _r(1-\mu )({\check{r}}_-,\varpi _0)\frac{\beta }{1+\beta }\,v},\end{aligned}$$

(146)

$$\begin{aligned} -\nu (u,v)\le & {} Cu^{{\tiny -\,\frac{1+\beta ^-}{1+\beta ^+}\frac{\partial _r(1-\mu )({\check{r}}_-,\varpi _0)}{\partial _r(1-\mu )(r_+,\varpi _0)}\beta }\,-1}, \end{aligned}$$

(147)

for appropriate choices of the parameters $\beta _-$, $\beta _+$, ${\check{r}}_-$, ${\check{r}}_+$, $\varepsilon _0$ and U.