1.2.1 Review of spherically symmetric linear waves on extremal Reissner–Nordström and the Aretakis instability

Here we briefly review the theory of spherically symmetric solutions to the linear wave equation (1.5) on extremal Reissner–Nordström. Aretakis initiated the study of this problem in [Reference Aretakis10, Reference Aretakis11] but we will make use of technical advances made by the first-named author, Aretakis, and Gajic in [Reference Angelopoulos, Aretakis and Gajic5, Reference Angelopoulos, Aretakis and Gajic4, Reference Angelopoulos, Aretakis and Gajic6]. For a brief review of the geometry of extremal Reissner–Nordström, we refer the reader to Section 2.2 of the present paper, [Reference Aretakis10, Section 2], and the appendix of [Reference Aretakis9].

The general strategy to prove energy decay statements for waves on extremal Reissner–Nordström consists of, as in the subextremal case, deriving a hierarchy of weighted energy boundedness inequalities and time-integrated energy decay estimates. This hierarchy takes the form

(1.9) $$ \begin{align} &\int_{C(\tau_2)}r^p(\partial_v\psi)^2\, dv+ \int_{\underline C(\tau_2)} (r-M)^{2-p} \frac{(\partial_u\psi)^2}{-\partial_ur}\,du \nonumber\\& \quad\lesssim \int_{C(\tau_1)}r^p(\partial_v\psi)^2\, dv+ \int_{\underline C(\tau_1)} (r-M)^{2-p} \frac{(\partial_u\psi)^2}{-\partial_ur}\,du+\mathrm{l.o.t.}, \end{align} $$

(1.10) $$ \begin{align} \int_{\tau_1}^{\tau_2}\int_{C(\tau)} r^{p-1}(\partial_v\psi)^2\, dvd\tau&\lesssim \int_{C(\tau_1)}r^p(\partial_v\psi)^2\, dv + \text{l.o.t.}, \end{align} $$

(1.11) $$ \begin{align} \int_{\tau_1}^{\tau_2}\int_{\underline C(\tau)} (r-M)^{3-p} \frac{(\partial_u\psi)^2}{-\partial_ur}\,dud\tau&\lesssim\int_{\underline C(\tau_1)} (r-M)^{2-p} \frac{(\partial_u\psi)^2}{-\partial_ur}\,du + \text{l.o.t.}, \end{align} $$

where $(u,v)$ denote Eddington–Finkelstein double null coordinates on the domain of outer communication, $\tau $ is proper time along a timelike curve $\Gamma $ with constant area-radius, $\tau _1\le \tau _2$ , $p\in [0,3)$ in (1.9), $p\in [1,3)$ in (1.10) and (1.11), “l.o.t.” denotes terms lower in the p-hierarchy, the foliations $C(\tau )$ and $\underline C(\tau )$ are defined pictorially in Fig. 2, and $\psi $ denotes the radiation field

$$ \begin{align*} \psi\doteq r\phi. \end{align*} $$

Figure 2 A Penrose diagram of extremal Reissner–Nordström depicting the foliations $C(\tau )$ and $\underline{C}(\tau )$ used in the estimates (1.9)–(1.11). The region of integration in (1.10) and (1.11) is shaded darker.

The inequality (1.10) is the celebrated $r^p$ -weighted estimate of Dafermos and Rodnianski [Reference Dafermos and Rodnianski31] and relies only on the asymptotic flatness of the extremal Reissner–Nordström metric. The estimate (1.11) is specific to extremal Reissner–Nordström and can be thought of as a “horizon analogue” of the $r^p$ estimates at infinity: The event horizon $\mathcal H^+$ of extremal Reissner–Nordström is located at $r=M$ and hence $r-M$ is a degenerate weight on $\mathcal H^+$ . The estimate (1.11) states that the time integral of the $(p-1)$ -weighted horizon flux is bounded by the initial value of the p-weighted horizon flux. This horizon hierarchy – some special cases of which were introduced in [Reference Aretakis10] and in full generality was presented in [Reference Angelopoulos, Aretakis and Gajic6] – replaces the fundamental redshift estimate from [Reference Dafermos and Rodnianski30, Reference Dafermos and Rodnianski32] which holds for any stationary event horizon with positive surface gravity. Note that the $p=0$ horizon flux is equivalent to the T-energy, where T is the time-translation Killing field. The duality between r and $(r-M)^{-1}$ is related to the so-called Couch–Torrence conformal isometry [Reference Couch and Torrence20] that exchanges $\mathcal H^+$ and $\mathcal I^+$ in extremal Reissner–Nordström.

Using the pigeonhole principle as in [Reference Dafermos and Rodnianski31], (1.9)–(1.11) can be used to prove the energy decay estimate

(1.12) $$ \begin{align} \int_{C(\tau)} r^p(\partial_v\psi)^2\, dv+\int_{\underline C(\tau)} (r-M)^{2-p} \frac{(\partial_u\psi)^2}{-\partial_ur}\,du \le C_\star\tau^{-3+\delta+p}, \end{align} $$

for every $p\in [0,3-\delta ]$ and $\tau \ge \tau _0$ , where $C_\star $ is a constant depending on $\delta $ and the data at $\tau =\tau _0$ . This estimate can then be used to prove pointwise decay of $\psi $ itself. By commuting the wave equation, higher order versions of (1.9)–(1.12) can be proved, but we shall not need to do so in the present paper.

While certain energies for $\phi $ do indeed decay by (1.12), the Aretakis instability states that nondegenerate ingoing null derivatives of $\phi $ on $\mathcal H^+$ do not decay, or can even grow polynomially. We now briefly explain this mechanism. Let $Y\doteq \partial _r$ in ingoing Eddington–Finkelstein coordinates $(v,r)$ .Footnote ¹ An elementary calculation using the wave equation (1.5) shows that

(1.13) $$ \begin{align} \partial_v\big(Y\psi|_{\mathcal H^+}\big)=0, \end{align} $$

i.e., $Y\psi $ is constant along $\mathcal H^+$ . Therefore, in sharp contrast to the subextremal case, $Y\psi $ does not decay along $\mathcal H^+$ . This constant is written as $H_0[\phi ]$ and is called the (zeroth) Aretakis charge of $\phi $ .

By commuting the wave equation with Y, we can likewise derive an evolution equation for $Y^2\psi $ along $\mathcal H^+$ . If $H_0[\phi ]\ne 0$ , we have that

(1.14) $$ \begin{align} \partial_v\big(Y^2\psi|_{\mathcal H^+}\big)= -2M^{-2}H_0[\phi]+\text{decaying terms}, \end{align} $$

so upon integrating in v we conclude that $|Y^2\psi |\gtrsim |H_0[\phi ]|v$ on $\mathcal H^+$ for v large. In fact, we have $|Y^k\psi |\gtrsim |H_0[\phi ]|v^{k-1}$ on $\mathcal H^+$ for any $k\ge 1$ and v large.

1.2.2 Prescription of seed data and the modulation parameter $\alpha $

Keeping in mind the ideas of Section 1.2.1, we now turn to the outline of the proofs of Theorems I and II.

We recall the notion of renormalized Hawking mass, which is the appropriate analogue of the usual Hawking mass $m\doteq \frac {r}{2}(1-g(\nabla r,\nabla r))$ for solutions of the spherically symmetric Einstein–Maxwell-neutral scalar field model:

$$ \begin{align*} \varpi\doteq m+\frac{e^2}{2r}, \end{align*} $$

where e is the constant charge of the solution. In Reissner–Nordström with parameters M and e, $\varpi =M$ .

We refer back to the Penrose diagram of the setup of our main results, Fig. 1. Bifurcate characteristic seed data for the spherically symmetric Einstein–Maxwell-neutral scalar field model on $C_{\mathrm{out}}\cup \underline C{}_{\mathrm{in}}$ consists of the restriction of $\phi $ to $C_{\mathrm{out}}\cup \underline C{}_{\mathrm{in}}$ , denoted by $\mathring \phi $ , the area-radius $\Lambda $ of $C_{\mathrm{out}}\cap \underline C{}_{\mathrm{in}}$ , the renormalized Hawking mass $\varpi _0$ of $C_{\mathrm{out}}\cap \underline C{}_{\mathrm{in}}$ , and the electric charge e of the Maxwell field F, which is defined in (2.4) below. Since the scalar field is neutral, e is in fact globally conserved and is thus fixed once and for all by the initial data. The collection of such quadruples $(\mathring \phi ,\Lambda ,\varpi _0,e)$ is what we call the moduli space of initial data ${\mathfrak{M}}$ .

Fix extremal parameters $M_0=|e_0|$ and an area-radius $100M_0$ (which lies far outside the horizon located at $r=M_0$ ). We consider perturbations of the bifurcate cone in the extremal Reissner–Nordström solution with parameters $(M_0,e_0)$ with bifurcation sphere at $r=100M_0$ . We consider the seed data norm

(1.15) $$ \begin{align} \mathfrak D\approx |\Lambda-100M_0|+|\varpi_0-M_0|+|e-e_0|+ \sup_{\underline C_{\mathrm{in}}}\big(|\phi|+|\partial_{\hat u}\phi|\big)+\sup_{ C_{\mathrm{out}}}\left(|\psi|+|r^2\partial_{\hat v}\psi|\right) \end{align} $$

and define a master smallness parameter $\varepsilon \ge \mathfrak D$ . Here $(\hat u,\hat v)$ are “initial data normalized” coordinates such that $\partial _{\hat u}r=-1$ on $\underline C{}_{\mathrm{in}}$ and $\partial _{\hat v}r=1$ on $C_{\mathrm{out}}$ . In particular, $\hat u$ will be regular across the event horizon.

As in [Reference Dafermos, Holzegel, Rodnianski and Taylor27], we in fact consider a family of initial data which are indexed by a real parameter $\alpha $ such that $\varpi _0=M_0+\alpha $ . Therefore, for $(\mathring \phi ,\Lambda ,e)$ fixed, the “codimension-one” aspect of Theorem I means that we find (at least one) $\alpha =\alpha _\star $ such that the solution generated by the seed data $(\mathring \phi ,\Lambda ,M_0+\alpha _\star ,e)$ converges to extremal Reissner–Nordström with parameters $M=|e|$ and e. The critical parameter $\alpha _\star $ is determined in evolution and cannot be read off from the initial data. See already Section 1.2.6.

The setup for the initial data is given in Section 3.1.

1.2.3 Setup of the bootstrap argument and the teleological gauge

Our general setup for the bootstrap argument (depicted in Fig. 3 below) is inspired by [Reference Dafermos, Holzegel, Rodnianski and Taylor27] and the work of Luk–Oh [Reference Luk and Oh56] on stability of subextremal Reissner–Nordström in spherical symmetry.

Figure 3 A Penrose diagram of one of the bootstrap domains $\mathcal{D}{}_{\tau{}_{f}}{\doteq} J^-(\Gamma(\tau_f))$ used in the proof of Theorem I. Here $\Gamma {\doteq}\{r=\Lambda \}$ is the timelike curve which anchors the bootstrap domains and $(u,v)$ are double null coordinates teleologically normalized as depicted.

In Fig. 3, the timelike curve $\Gamma $ has constant area-radius $r=\Lambda $ (from the seed data) and is parametrized by proper time $\tau \ge 1$ . The parameter $\tau _f$ determines the bootstrap domain $\mathcal D_{\tau _f}$ and is sent to infinity in the course of the proof. The double null coordinates $(u,v)$ are teleologically normalized according to the conditions $\partial _ur=-(1-\frac {2m}{r})$ on the ingoing future boundary of $\mathcal D_{\tau _f}$ , where m is the Hawking mass of the spacetime, and $\partial _vr=1-\frac {2m}{r}$ on $\Gamma $ . This latter gauge condition for the v coordinate is nonstandard and marks a crucial difference with the subextremal case in [Reference Luk and Oh56, Reference Dafermos, Holzegel, Rodnianski and Taylor27]. We extend $\tau $ to a function on $\mathcal D_{\tau _f}$ by setting it equal to proper time along $\Gamma $ and then declaring it to be constant along ingoing cones to the left of $\Gamma $ and constant along outgoing cones to the right of $\Gamma $ .

Since the charge e is conserved, we know a priori that we should aim to converge to extremal Reissner–Nordström with parameters $M\doteq |e|$ and e. We anchor a comparison extremal Reissner–Nordström solution with area-radius function $\bar r$ and parameters $(M,e)$ in Eddington–Finkelstein double null gauge to the bootstrap domain $\mathcal D_{\tau _f}$ by setting $\bar r(\Gamma (\tau _f))=\Lambda $ . Relative to this background solution, we define energy norms motivated by the uncoupled case (recall Section 1.2.1),

$$ \begin{align*} \mathcal E_p(\tau) &\doteq \int_{C_{\Gamma^u(\tau)}\cap\mathcal D_{\tau_f}}r^p(\partial_v\psi)^2\, dv+\cdots ,\\ \underline{\mathcal E}{}_p(\tau) & \doteq\int_{\underline C{}_{\Gamma^u(\tau)}\cap\mathcal D_{\tau_f}} (\bar r-M)^{2-p}\frac{(\partial_u\psi)^2}{-\partial_u\bar r} \,du+\cdots, \end{align*} $$

where we have only written the most important terms for now.

For the modulation parameter $\alpha $ lying in an appropriate range (to be explained in Section 1.2.6 below), we make the bootstrap assumptions

(1.16) $$ \begin{align} \left|\frac{\partial_ur}{\partial_u\bar r}-1\right|\lesssim \varepsilon^{3/2}\tau^{-1+\delta},\quad |r-\bar r|\lesssim \varepsilon^{3/2}\tau^{-2+\delta},\quad |\varpi-M|\lesssim \varepsilon^{3/2}\tau^{-3+\delta} \end{align} $$

on $\mathcal D_{\tau _f}$ for the geometry and

(1.17) $$ \begin{align} \mathcal E_p(\tau)+ \underline{\mathcal E}{}_p(\tau)\lesssim \varepsilon^2\tau^{-3+\delta+p}\end{align} $$

for $\tau \in [1,\tau _f]$ for the scalar field; compare with (1.12).Footnote ² The main analytic content of Theorem I consists of recovering the bootstrap assumptions (1.16) and (1.17).

The setup for the bootstrap argument is given in Section 3.2, the anchoring procedure is given in Section 3.3, and the bootstrap assumptions are precisely stated in Section 4.1.

1.2.4 Estimates for the geometry: the role of the degenerate redshift

We utilize the well-known quantities

$$ \begin{align*} \kappa\doteq \frac{\partial_vr}{1-\frac{2m}{r}},\quad \gamma\doteq \frac{\partial_ur}{1-\frac{2m}{r}}, \end{align*} $$

which satisfy good evolution equations in u and v, respectively. The background extremal Reissner–Nordström solution $\bar r $ has $\bar \kappa =-\bar \gamma =1$ globally. Therefore, the anchoring and gauge conditions ensure that $r(\Gamma (\tau _f))=\bar r(\Gamma (\tau _f))$ , $\kappa =\bar \kappa =1$ along $\Gamma \cap \mathcal D_{\tau _f}$ , and $\gamma =\bar \gamma =-1$ along the final ingoing cone in $\mathcal D_{\tau _f}$ .

The v-equation for $\gamma $ sees the flux $\mathcal E_0$ and hence gives us that $\gamma =-1 + O(\varepsilon ^2\tau ^{-3+\delta })$ , which is the best that can be expected. On the other hand, the u-equation for $\kappa $ reads $\partial _u\kappa = r\kappa (\partial _ur)^{-1}(\partial _u\phi )^2$ , which sees $\underline {\mathcal E}{}_2$ , and hence decays much slower.Footnote ³ On the other hand, the quantity $(\bar r-M)^{2-p}(\kappa -\bar \kappa )$ , for $p\in [0,2]$ , obeys a u-evolution equation that sees the flux $\underline {\mathcal E}{}_p$ when integrating to the left of $\Gamma $ , and hence we show that

(1.18) $$ \begin{align} (\bar r-M)^{2-p}\kappa = (\bar r-M)^{2-p}+O(\varepsilon^2\tau^{-3+\delta+p}) \end{align} $$

to the left of $\Gamma $ . This hierarchy of decay rates for $\kappa $ is a fundamental aspect of our geometric estimates.

Because of this, it is important not to waste any powers of $\bar r-M$ in the system and our scheme is essentially sharp at the horizon in this regard. Of particular importance are the quantities $1-\frac {2m}{r}$ and $\partial _vr$ , which we show admit “Taylor expansions” of the form

(1.19) $$ \begin{align}\kern-6pt 1-\frac{2m}{r} &= \overline{1-\frac{2m}{r}} + \frac{2M}{\bar r^3}(\bar r-M)(r-\bar r)+O(\varepsilon^{3/2}\tau^{-3+\delta}), \end{align} $$

(1.20) $$ \begin{align} \partial_vr &=\partial_v\bar r + \frac{2M\kappa}{\bar r^3}(\bar r-M)(r-\bar r)+ O(\varepsilon^{3/2}\tau^{-3+\delta}). \end{align} $$

In the background solution, the quantities $ \overline {1-\frac {2m}{r}} $ and $\partial _v\bar r$ vanish quadratically on $\mathcal H^+$ , so these are to be viewed as expansions in powers of $\bar r-M$ with rapidly decaying error.

The coefficients of the terms which are linear in $\bar r-M$ and the strong decay of the error terms in (1.19) and (1.20) are important. For example, the function $1-\frac {2m}{r}$ appears naturally in the u-equation for $\varpi $ , which reads $\partial _u\varpi = \tfrac 12 (1-\frac {2m}{r})r^2(\partial _ur)^{-1}(\partial _u\phi )^2$ , and proving $\tau ^{-3+\delta }$ decay for this requires taking advantage of the linear term in (1.19). In the subextremal case, the redshift effect implies one can prove $\tau ^{-3+\delta }$ decay for $\varpi $ while completely ignoring the $1-\frac {2m}{r}$ weight, so no expansion of the form (1.19) is required (or helpful). Moreover, we use the linear term in (1.19) to control the error term in (1.20).

The sign of the linear term in (1.20) is crucial: it is positive in the domain of outer communication, which reflects the global redshift effect on extremal Reissner–Nordström. However, unlike the subextremal case, where the redshift leads to an exponentially decaying integrating factor for the analogue of (1.20) (compare [Reference Luk and Oh56, Lemma 8.19]), on an asymptotically extremal black hole the effect degenerates on the horizon and the rapidly decaying error term becomes important, since integrating (1.20) in v now loses a power of $\tau $ . See also Remark 5.12 below.

The geometric estimates are proved in Section 5.

1.2.5 Integrated local energy decay and the $r^p$ and $(\bar r-M)^{2-p}$ energy hierarchies

To recover the bootstrap assumption (1.17) for the scalar field, we follow the general strategy outlined in Section 1.2.1. We prove the estimates

(1.21) $$ \begin{align} \mathcal E_p(\tau_2)+ \underline{\mathcal E}{}_p(\tau_2) \lesssim \mathcal E_p(\tau_1)+ \underline{\mathcal E}{}_p(\tau_1)+ \text{decaying nonlinear error} \end{align} $$

for $1\le \tau _1\le \tau _2$ and $p\in [0,3-\delta ]$ , and

(1.22) $$ \begin{align} \int_{\tau_1}^{\tau_2}\big( \mathcal E_{p-1}(\tau)+ \underline{\mathcal E}{}_{p-1}(\tau)\big)\,d\tau \lesssim \mathcal E_p(\tau_1)+ \underline{\mathcal E}{}_p(\tau_1)+ \text{decaying nonlinear error} \end{align} $$

for $p\in [1,3-\delta ]$ . The choice of multiplier vector fields and lower order currents is inspired by work in the uncoupled case and [Reference Luk and Oh56]. We utilize a mix of “dynamical” (i.e., defined relative to the dynamical metric) and “background” (i.e., defined relative to the background extremal Reissner–Nordström metric $\bar r$ ) multipliers to minimize the number of error terms. The expansion (1.19) is used crucially to estimate nonlinear errors in the near-horizon region that are specific to the extremal case. Once (1.21) and (1.22) have been proved, decay is inferred by a standard application of the pigeonhole principle as in [Reference Dafermos and Rodnianski31].

The energy hierarchies are defined in Section 6 and energy decay is proved in Section 7.

1.2.6 The codimension-one “submanifold” ${\mathfrak{M}}_{\mathrm{stab}}$ and modulation on dyadic timescales

While our bootstrap argument is performed continuously in time, the choice of allowed modulation parameters $\alpha $ is only decided when $\tau _f$ is dyadic, i.e., a power of 2. This approach is motivated by the purely dyadic approach of Dafermos–Holzegel–Rodnianski–Taylor in [Reference Dafermos, Holzegel, Rodnianski and Taylor28] and turns out to be quite fortuitous compared to the continuous-in-time modulation theory employed in [Reference Dafermos, Holzegel, Rodnianski and Taylor27].

In practice, we construct a sequence of nested compact intervals $\mathfrak A_0\supset \mathfrak A_1\supset \cdots $ such that $\mathfrak A_i$ consists of those $\alpha $ ’s which we consider for bootstrap time $\tau _f\in [2^i,2^{i+1})$ . These sets are defined implicitly by the requirement that the renormalized Hawking mass $\varpi $ at the point $\Gamma (2^i)$ must lie within $\varepsilon ^{3/2}2^{(-3+\delta )i}$ of M for $\alpha \in \mathfrak A_i$ . By a shooting argument at each dyadic time $2^i$ , we show that the implicitly defined set $\mathfrak A_i$ is nonempty. Therefore, as $\tau _f\to \infty $ (and, consequently, $i\to \infty $ ), we extract (at least one) critical parameter $\alpha _\star \in \bigcap _{i\ge 0}\mathfrak A_i$ for which the solution tends to extremality. The stable “submanifold” ${\mathfrak{M}}_{\mathrm{stab}}\subset {\mathfrak{M}}$ is the collection of all seed data sets of the form $(\mathring \phi ,\Lambda ,M_0+\alpha _\star ,e)$ and is codimension-one in the sense that every “line” of constant $\mathring \phi $ , $\Lambda $ , and e intersects it at least once.Footnote ⁴ Refer to Fig. 4 below.

Figure 4 A schematic depiction of our modulation scheme. Let $p\mapsto \mathring \phi (p)$ be a one-parameter family of characteristic initial data for the scalar field with $\mathring \phi (0)=0$ . We can then consider the plane in ${\mathfrak{M}}$ parametrized by $(p,\alpha )$ . Each p generates a line segment $\mathcal L$ in ${\mathfrak{M}}$ which intersects the “submanifold” ${\mathfrak{M}}_{\mathrm{stab}}$ at least once. The horizontal line ${\mathfrak{M}}_0$ denotes the hyperplane in ${\mathfrak{M}}$ consisting of data sets with $\varpi _0=M_0$ . On the three $\mathcal L$ ’s depicted here, we have also drawn three of the nested modulation sets $\mathfrak A_i$ which converge to $\mathcal L\cap {\mathfrak{M}}_{\mathrm{stab}}$ . Note that we have drawn ${\mathfrak{M}}_{\mathrm{stab}}$ as a smooth, connected curve here, which is in line with our conjectures in Section 1.3, but we do not prove any such fine structure of it in this paper.

Since we construct $\mathfrak A_i$ only after each dyadic time, we may always assume a $\tau ^{-3+\delta }$ decay rate for the difference $\varpi -M$ , which is sharp. This should be contrasted with [Reference Dafermos, Holzegel, Rodnianski and Taylor27], where the assumption on angular momentum in the modulation set has a decay rate of $\tau _f^{-1}$ , but the improvement (i.e., the change in angular momentum along $\mathcal I^+$ ) is required to decay like $\tau _f^{-2}$ , which is a significant technical issue. In our work, both the assumption and the improvement have the sharp decay rate $\tau _f^{-3+\delta }$ and the gain is purely in the power of $\varepsilon $ ( $\varepsilon ^{3/2}$ vs. $\varepsilon ^2$ ); see already (8.1). This strong decay rate for $\varpi -M$ is important, in particular because of the absence of redshift in the geometric estimates.

We have depicted our modulation scheme in Fig. 4 above. The modulation sets $\mathfrak A_i$ are defined in Section 4.1 and the modulation argument is performed in Section 8.1.2. Modulation on dyadic timescales, motivated by [Reference Dafermos, Holzegel, Rodnianski and Taylor28], has also been performed by Kádár in [Reference Kádár47, Reference Kádár46].

1.2.7 Construction of the eschatological gauge and background solution

Recall from Section 1.2.3 that the teleologically normalized coordinate u and the background extremal Reissner–Nordström solution $\bar r$ depend on the bootstrap time $\tau _f$ . Therefore, an important final step when taking $\tau _f\to \infty $ is to prove that we can define a unique “final background solution” which we converge to and an “eschatological gauge” (i.e., final teleological gauge) in which we converge to it. The regularity of the limiting gauge is actually related to the decay assumptions on initial data,Footnote ⁵ and the best that can be hoped for given only the finiteness of (1.15) is that the final u coordinate is a $C^2$ function of the initial data coordinate $\hat u$ . These limiting arguments are carried out in Sections 8.2 and 8.3.

1.2.8 Absence of trapped surfaces and rigidity of the apparent horizon

After we have constructed the dynamical extremal spacetimes, it remains to prove part (iv) of Theorem I. This argument is originally due to Kommemi (in unpublished work) and a very similar argument appears in [Reference Luk and Oh55, Appendix A]. The first aspect of the argument is a proof that any point on the outermost apparent horizon $\mathcal A'$ must lie on $\mathcal H^+$ because of simple monotonicities to the right of $\mathcal A'$ and the fact that r and $\varpi $ both asymptote to M along $\mathcal H^+$ . The second part involves Taylor expanding $\partial _vr$ in u along $\mathcal H^+$ to show that there are no trapped or marginally trapped spheres behind $\mathcal H^+$ . This second part of the argument is reminiscent of the proof of “Israel’s observation” (Proposition 1.1) in [Reference Kehle and Unger48]. We carry out these arguments in Section 8.3.5.

1.2.9 The Aretakis instability on the dynamical geometry

Using the geometric estimates proved in Section 5, the exact conservation law (1.13) is replaced by the “almost conservation law”

$$ \begin{align*} \partial_v\big(Y\psi|_{\mathcal H^+}\big) = O\big(\varepsilon^{3}\tau^{-2+\delta}\big). \end{align*} $$

Note that the error is integrable in $\tau $ (again, thanks to going up to $p=3-\delta $ in the hierarchy!) and much better in $\varepsilon $ than $Y\psi $ . Therefore, integrating this immediately gives (i) of Theorem II. Part (ii) is completely soft and is essentially an immediate consequence of (i). To prove part (iii), we commute the wave equation on the dynamical background with Y and show that the nonlinear error terms are decaying, which results in an estimate entirely analogous to (1.14). The instability of the geometry is obtained by directly inserting this behavior for $\phi $ into the Einstein equations. We do not prove growth of derivatives higher than second order because this would require proving higher order estimates for the geometry, which is not necessary for the proof of Theorem I. These arguments are carried out in Section 9.1.

1.3.1 Asymptotically extremal black holes as a locally separating hypersurface between collapse and dispersion

In the following conjectures, we will write the symbol ${\mathfrak{M}}$ to mean a moduli space of initial data posed “in the same way” as in our Theorem I, but possibly topologized by a different norm than in Section 3.1 below. In particular, it could be that the following conjectures are only true under stronger asymptotic decay assumptions. However, we will always suppose that ${\mathfrak{M}}$ carries a Banach space structure, so that we have access to the notion of $C^1$ submanifolds of ${\mathfrak{M}}$ .

For ${\mathfrak{r}}\in [0,1]$ , we can consider the subset ${\mathfrak{M}}^{\mathfrak{r}}_{\mathrm{stab}}\subset {\mathfrak{M}}$ consisting of initial data which form a black hole with asymptotic parameter ratio $|e|/M_\infty ={\mathfrak{r}}$ , where e is the conserved charge of the solution and $M_\infty \doteq \lim _{v\to \infty }\varpi |_{\mathcal H^+}$ is the teleologically determined final mass of the black hole. In this notation, the stable set from Theorem I is ${\mathfrak{M}}_{\mathrm{stab}}^1$ and $\bigcup _{{\mathfrak{r}}<1}{\mathfrak{M}}_{\mathrm{stab}}^{\mathfrak{r}}$ was studied by Dafermos and Rodnianski in [Reference Dafermos and Rodnianski29].

In particular, we conjecture that the stable “submanifold” from Theorem I is in fact a connected, regular hypersurface in ${\mathfrak{M}}$ . Our next conjecture addresses what happens on “either side” of ${\mathfrak{M}}_{\mathrm{stab}}^1$ ; see Fig. 5 above.

Figure 5 A cartoon depiction of the conjectured structure of a neighborhood of extremal Reissner–Nordström in the moduli space ${\mathfrak{M}}$ of initial data posed as in Fig. 1. We have suppressed infinitely many dimensions and emphasize the codimension-one property of the submanifolds ${\mathfrak{M}}_{\mathrm{stab}}^{\mathfrak{r}}$ . We have drawn a distinguished point, which is extremal Reissner–Nordström. We have also drawn one of the lines $\mathcal L$ from Fig. 4 with the natural orientation given by increasing modulation parameter $\alpha $ . The solid point on $\mathcal L$ corresponds to $\alpha =\alpha _\star $ (recall Section 1.2.6). See also Fig. 6 below.

To motivate the following conjecture, it is helpful to consider the Reissner–Nordström family itself for a moment. If we consider characteristic data for the spherically symmetric Einstein–Maxwell system posed on a bifurcate null hypersurface as in Fig. 1 and then vary the renormalized Hawking mass $\varpi $ of the bifurcation sphere, the resulting developments sweep out a portion of the family of Reissner–Nordström solutions. In particular, we can observe a transition from dispersion within the domain of dependence of the data (when the parameters are superextremal) to existence of a black hole (when the parameters are extremal or subextremal). Extremal Reissner–Nordström is the critical solution in this phase transition. Note that none of these developments contain naked singularities. We conjecture that this critical behavior is preserved when we pass to the spherically symmetric Einstein–Maxwell-neutral scalar field model:

This conjecture relates to Fig. 4 and our modulation scheme in the present paper as follows: We expect there to exist exactly one $\alpha _\star $ for which the modulation argument outlined in Section 1.2.6 applies (as opposed to at least one, which is what is shown in the present paper). For $\alpha>\alpha _\star $ , we expect the seed data set $(\mathring \phi ,\Lambda ,M_0+\alpha ,e)$ to lie in ${\mathfrak{M}}_{\mathrm{sub}}$ and if $\alpha <\alpha _\star $ , we expect it to lie in ${\mathfrak{M}}_{\mathrm{disp}}$ . We note that these conjectures are consistent with the numerical findings in [Reference Murata, Reall and Tanahashi58].

Figure 6 Penrose diagrams depicting evolutions of seed data in the sets $\mathfrak{M}_{\mathrm{disp}}$ , $\mathfrak{M}_{\mathrm{stab}}^{1}$ , and $\mathfrak{M}_{\mathrm{sub}}$ . One can think of these as arising from a one-parameter family of seed data crossing the extremal threshold in Fig. 5. Spacetimes arising from $\mathfrak{M}_{\mathrm{disp}}$ have incomplete null infinity $\mathcal {I}^+$ because the ingoing cone $\underline{C}_{\mathrm{in}}$ is incomplete and no black hole has formed. As proved in Theorem I, solutions on the critical threshold $\mathfrak{M}_{\mathrm{stab}}^1$ contain no trapped surfaces, but it follows from the work of Dafermos [Reference Dafermos22] that solutions arising from $\mathfrak{M}_{\mathrm{sub}}$ have a nonempty trapped region as depicted. The explicit Reissner–Nordström family itself already displays this transition behavior, in which case the trapped region in the third Penrose diagram would intersect the initial data, and the solid point would lie in the region $\{ r< M - \sqrt {M^2 - e^2}\}$ (in the left half of region III in [Reference Hawking and Ellis44, Fig. 25]) in the maximal analytic extension of subextremal Reissner–Nordström.

Since charge is conserved in the neutral scalar field model, it is not possible to have solutions with a nontrivial charge and a regular center. Therefore, Conjectures 1 and 2 are necessarily “local near the event horizon” because the ingoing cone $\underline C{}_{\mathrm{in}}$ must terminate at a symmetry sphere with positive area-radius. Therefore, even in the “dispersive” case, the solutions are necessarily geodesically incomplete. The second- and third-named authors of the present paper have shown that extremal Reissner–Nordström black holes can form dynamically from initial data posed on $\mathbb R^3$ in the Einstein–Maxwell-charged scalar field and Einstein–Maxwell–charged Vlasov models [Reference Kehle and Unger48, Reference Kehle and Unger49]. Therefore, Conjectures 1 and 2 can be formulated for these models, where data in ${\mathfrak{M}}_{\mathrm{disp}}$ now really “exist globally” and “disperse” in the literal sense. We refer to the introduction of [Reference Kehle and Unger49] for more details.

1.3.2 The black hole interior at extremality

As explained in Section 1.2.8, the black hole regions of the spacetimes evolving from data in ${\mathfrak{M}}_{\mathrm{stab}}$ are free of trapped surfaces, see Fig. 1. In particular, this shows that the interior is bounded to the future by a smooth outgoing null hypersurface (a trivial, smooth Cauchy horizon) emanating from the final sphere of $\underline C{}_{\mathrm{in}}$ (the solid point in Fig. 1) and a potentially singular ingoing Cauchy horizon $\mathcal {CH}^+$ emanating from $i^+$ . Analogously to the surface gravity of $\mathcal H^+$ defined in (1.4), one can define the surface gravity of $\mathcal {CH}^+$ in Reissner–Nordström for $0 < |e| \leq M$ . In the extremal case $|e| = M$ , this surface gravity also vanishes, resulting in the absence of Penrose’s blueshift instability [Reference Penrose61] and, somewhat ironically, renders the extremal Reissner–Nordström Cauchy horizon $\mathcal {CH}^+$ more stable than its subextremal variant [Reference Gajic33, Reference Gajic34]. Indeed, our decay estimates of Theorem I on the geometry and the scalar field along $\mathcal H^+$ satisfy the assumptions in [Reference Gajic and Luk36, Theorem 5.1] and therefore the spacetimes evolving from ${\mathfrak{M}}_{\mathrm{stab}}$ can be extended in $C^{0,1/2}\cap H_{\mathrm{loc}}^1$ across $\mathcal {CH}^+$ and the Hawking mass remains uniformly bounded. This is in sharp contrast to the subextremal case, where generically the Hawking mass blows up identically along $\mathcal {CH}^+$ [Reference Dafermos21, Reference Dafermos22, Reference Luk, Oh and Shlapentokh-Rothman57, Reference Gautam37] and extensions are believed to be less regular than $C^{0,1/2}\cap H^1_{\mathrm{loc}}$ [Reference Luk and Oh55, Reference Luk and Oh56]. In the extremal case, the existence of Cauchy data leading to any type of singularity at $\mathcal {CH}^+$ remains open – even for the linear wave equation!

Note that according to Conjecture 2, asymptotically extremal Reissner–Nordström black holes only form from the (manifestly nongeneric) “codimension 1 submanifold” ${\mathfrak{M}}_{\mathrm{stab}}$ , so their more regular Cauchy horizons do not endanger Penrose’s strong cosmic censorship conjecture.

1.4.1 Observational signatures and late-time tails

In Section 9.2, we prove asymptotics for the scalar field in the near-horizon region, using the energy decay estimates from the proof of Theorem I (in particular the $(\bar {r}-M)^{2-p}$ hierarchy up to $p=3-\delta $ ). In Section 9.3, we then use this sharp decay estimate to show that the range of the $(\bar {r}-M)^{2-p}$ hierarchy is indeed sharp, as long as the asymptotic Aretakis charge is not too small compared to the master smallness parameter $\varepsilon $ of the problem. Namely, we show that the nondegenerate integrated energy is unbounded in any infinite area close to the horizon. This is in sharp contrast to the subextremal case, where the celebrated redshift estimate [Reference Dafermos and Rodnianski30] implies that this integrated energy is finite near the horizon.

We note that compared to the uncoupled case, the leading order term in the decay for $\psi $ does not only include the asymptotic Aretakis charge $H_0[\phi ]$ but also an error of order $\varepsilon ^3$ that comes from the geometric estimates. This marks a difference with the uncoupled case and is a manifestation of the failure of the Couch–Torrence conformal isometry in the dynamical case. This creates a marked difference in the proof of the failure of the redshift estimate compared to the uncoupled case; see already Remark 9.9.

In the present paper, we only obtain sharp leading asymptotics for the scalar field near the horizon. It would be an interesting problem to obtain asymptotics for the scalar field everywhere for our asymptotically extremal black holes, depending on the initial assumptions for the behavior of the scalar field at infinity. In the uncoupled case, assuming either compactly supported initial data or sufficiently fast decay so that the so-called Newman–Penrose constant vanishes, one can extend the $r^p$ hierarchy to the range of $p\in (0,5)$ . After some technical work (commutation and a “time inversion” process), one can show that the radiation field decays like $u^{-2}$ in a region close to $\mathcal I^+$ , where the constant of the leading order decaying term depends not only on the Newman–Penrose constant but also on the Aretakis charge. This can be considered an observational signature for extremal Reissner–Nordström black holes and it would be interesting to extract this signature in our dynamical setting. For more details on observational signatures for extremal Reissner–Nordström black holes, see [Reference Angelopoulos, Aretakis and Gajic4, Reference Angelopoulos, Aretakis and Gajic4] and the numerical results of [Reference Aretakis, Khanna and Sabharwal14].

1.4.2 Outside of symmetry

The present work in symmetry is the first step towards addressing the stability problem for (and, more generally, the structure of moduli space near) extremal black holes in Einstein(–Maxwell) theory outside of symmetry. We wish to consider the extremal Kerr–Newman family which consists of axisymmetric, rotating charged black holes with mass M, charge e, and specific angular momentum a satisfying the relation $M^2=e^2+a^2$ . When $a=0$ , this reduces to extremal Reissner–Nordström and when $e=0$ , to extremal Kerr.

In the very slowly rotating case ( $|a|\ll M$ ), we have the following:

Significant progress on linear stability in the subextremal case has been made by Giorgi and Wan in [Reference Giorgi40, Reference Giorgi42, Reference Giorgi41, Reference Giorgi and Wan43]. For extremal Reissner–Nordström, a complete understanding of the Teukolsky equation was recently obtained by Apetroaie [Reference Apetroaie8], who showed both decay statements and that a version of the Aretakis instability is present (see also the earlier [Reference Lucietti, Murata, Reall and Tanahashi52]).Footnote ⁶ Unfortunately, an understanding of even the linear wave equation on very slowly rotating extremal Kerr–Newman (with $a\ne 0$ ) remains open. In addition to requiring a detailed understanding of linear theory around very slowly rotating extremal Kerr–Newman, we expect that resolving Conjecture 3 will also rely on a nontrivial understanding of the null structure of the Einstein equations in the near-horizon region, analogous to the nonlinear results of [Reference Angelopoulos1, Reference Angelopoulos, Aretakis and Gajic3, Reference Angelopoulos, Aretakis and Gajic7] and the present work.

The case of extremal Kerr is considerably more speculative. While mode stability for the linear wave equation has been shown by Teixeira da Costa [Reference da Costa66], axisymmetric scalar perturbations have been shown to exhibit a similar nondecay and growth hierarchy as general scalar perturbations of extremal Reissner–Nordström [Reference Aretakis12]. Moreover, even pointwise boundedness for linear waves remains open. Recently, Gajic has shown the existence of stronger azimuthal instabilities (i.e., associated to $m\ne 0$ ) on extremal Kerr [Reference Gajic35] (see also the earlier heuristic analysis [Reference Casals, Gralla and Zimmerman16]). In particular, growth is already triggered at the first order of differentiability. It remains to be seen if these instabilities are compatible with global existence and stability for nonlinear wave equations! For more discussion, we refer to [Reference Dafermos25].