Boundedness of Massless Scalar Waves on Reissner-Nordström Interior Backgrounds

We consider solutions of the scalar wave equation □gϕ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Box_g\phi=0}$$\end{document}, without symmetry, on fixed subextremal Reissner-Nordström backgrounds (M,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathcal{M}, g)}$$\end{document} with nonvanishing charge. Previously, it has been shown that for ϕ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}^+}$$\end{document}. Using this, we show here that ϕ is in fact uniformly bounded, |ϕ|≤C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|\phi| \leq C}$$\end{document}, in the black hole interior up to and including the bifurcate Cauchy horizon CH+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}\mathcal{H}^+}$$\end{document}, to which ϕ in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.


Introduction
The Reissner-Nordström spacetime (M, g) is a fundamental 2-parameter family of solutions to the Einstein field equations coupled to electromagnetism, cf. Fig. 1 for the conformal representation of the subextremal case, M > |e| = 0, with e the charge and M the mass of the black hole. The problem of analysing the scalar wave equation on a Reissner-Nordström background is intimately related to the stability properties of the spacetime itself and to the celebrated Strong Cosmic Censorship Conjecture. The analysis of (1) in the exterior region J − (I + ) has been accomplished already, cf. [6] and [21] for an overview and references therein for more details, as well as Sect. 3.1. The purpose of the present work is to extend the investigation to the interior of the black hole, up to and including the Cauchy horizon CH + .

Main result.
The main result of this paper can be stated as follows. arising from smooth compactly supported Cauchy data on a two-ended asymptotically flat Cauchy surface . Then |φ| ≤ C (2) globally in the black hole interior, in particular up to and including the Cauchy horizon CH + , to which φ extends in fact continuously.
The constant C is explicitly computable in terms of parameters e and M and a suitable norm on initial data. 1 The above theorem will follow, after commuting (1) with angular momentum operators and applying Sobolev embedding, from the following theorem, expressing weighted energy boundedness.

Motivation and strong cosmic censorship.
Our motivation for proving Theorem 1.1 is the Strong Cosmic Censorship Conjecture. The mathematical formulation of this conjecture, here applied to electrovacuum, is given in [8] by Christodoulou as "Generic asymptotically flat initial data for Einstein-Maxwell spacetimes have a maximal future development which is inextendible as a suitably regular Lorentzian manifold." (5) Reissner-Nordström spacetime serves as a counterexample to the inextendibility statement, since it is (in fact smoothly) extendable beyond the Cauchy horizon CH + . 2 Thus, for the above conjecture to be true, this property of Reissner-Nordström must in particular be unstable. Originally it was suggested by Penrose and Simpson that small perturbations of Reissner-Nordström would lead to a spacetime whose boundary would be a spacelike singularity as in Schwarzschild and such that the spacetime would be inextendable as a C 0 metric, cf. [43]. On the other hand, a heuristic study of a spherically symmetric but fully nonlinear toy model by Israel and Poisson cf. [39], led to an alternative scenario, which suggested that spacetimes resulting from small perturbations would exist up to a Cauchy horizon, which however would be singular in a weaker sense, see also [37] by Ori. Considering the spherically symmetric Einstein-Maxwell-scalar field equations as a toy model, Dafermos proved that the solution indeed exists up to a Cauchy horizon and moreover is extendible as a C 0 metric but generically fails to be extendible as a C 1 metric beyond CH + , cf. [12,13]. For more recent extensions see [9][10][11]27].
In this work, as a first attempt towards investigation of the stability of the Cauchy horizon under perturbations without symmetry, we employ (1) on a fixed Reissner-Nordström background (M, g) as a toy model for the full nonlinear Einstein field equations, cf. (10). The result of uniform pointwise boundedness of φ and continuous extension to CH + is concordant with the work of Dafermos [12]. This suggests that the non-spherically symmetric perturbations of the astrophysically more realistic Kerr spacetime may indeed exist up to CH + . See Sect. 5.4.

A first look at the analysis.
The proof of Theorems 1.1 and 1.2 involves first considering a characteristic rectangle within the black hole interior, whose future right boundary coincides with the Cauchy horizon CH + in the vicinity of i + , cf. Fig. 2a).
Establishing boundedness of weighted energy norms in is the crux of the entire proof. Once that is done, analogous results hold for a characteristic rectangle˜ to the left depicted in Fig. 2a). Hereafter, boundedness of the energy is easily propagated to regions R V ,R V and R V I as depicted, giving Theorem 1.2. Commutation by angular momentum operators and application of Sobolev embedding then yields Theorem 1.1.
Let us return to the discussion of since that is the most involved part of the proof. In order to prove Theorem 1.2 (and hence Theorem 1.1) restricted to we will begin with an upper decay bound for |φ| and its derivatives on the event horizon H + , which can be deduced by putting together preceding work of Blue-Soffer cf. [6], Dafermos-Rodnianski cf. [15] and Schlue cf. [41]. The precise result from previous work that we shall need will be stated in Sect. 3.1. In the proof involves distinguishing redshift R, noshift N and blueshift B regions, as shown in Fig. 2b).
Some of these regions have appeared in previous analysis of the wave equation, especially R = {r red ≤ r ≤ r + }. Region N = {r blue ≤ r ≤ r red } and region B = {r − ≤ r ≤ r blue } were studied in [13] in the spherically symmetric self-gravitating case, but using techniques which are very special to 1 + 1 dimensional hyperbolic equations. 3 The analogous regions R and N were also considered by Luk [30] on an interior Schwarzschild background. We will discuss this separation into R, N and B regions further in Sect. 3.2. One of the main analytic novelties of this paper is the introduction of a new vector field energy identity constructed for analyses in region B. In particular, the weighted vector field is given in Eddington-Finkelstein coordinates (u, v) by for p > 1 as appearing in Theorem 1.2. This vector field associated to region B will allow us to prove uniform boundedness despite the blueshift instability.

1.4.
Outline of the paper. The paper is organized as follows.
In Sect. 2 we introduce the basic tools needed to derive energy estimates from the energy momentum tensor associated to (1) and an appropriate vector field. A review of the Reissner-Nordström solution and the coordinates used in this paper will be given. Moreover, we will discuss further features of Reissner-Nordström geometry.
In Sect. 3 we give a brief review of estimates obtained along H + from previous work, [6,15] and [41], for φ arising from sufficiently regular initial data on a Cauchy hypersurface. This is stated as Theorem 3.1. Further, we state our main result specialized to the rectangle in the neighbourhood of i + (see Theorem 3.3) and give an outline of its proof.
In Sect. 4 we prove boundedness of φ up to and including H + . In particular, in Sect. 4.1 we first propagate the decay bound for the energy flux of φ from H + up to CH + in the neighbourhood of i + . The investigation is divided into considerations within the redshift R, noshift N , and blueshift B regions. Section 4.2 reveals how commutation with angular momentum operators and applying Sobolev embedding will return us pointwise boundedness for |φ|. We now must extend our result to the full interior region. In Sect. 4.3 we propagate the energy estimates further along CH + in the depicted region R V . Eventually, in subsection 4.4 we propagate the estimate to the region R V I up to the bifurcation two-sphere, and thus obtain a bound for the energy flux globally in the black hole interior completing the proof of Theorem 1.2. In subsection 4.5 we prove boundedness of the weighted higher order energies. Using the conclusion of this theorem, we apply again Sobolev embedding as before and thus obtain the boundedness statement of Theorem 1.1.
An Outlook of open problems will be given in Sect. 5. We first state an analogous result to our Theorem 1.1 for general subextremal Kerr black holes (to appear as Theorem 1.1 of the follow up paper). The conjectured blow up of the transverse derivatives 4 along the Cauchy horizon for generic solutions of (1) will also be discussed, as well as the peculiar extremal case. Finally, we will discuss what our results suggest about the nonlinear dynamics of the Einstein equations themselves.

Energy currents and vector fields.
The essential tool used throughout this work is the so called vector field method. Let (M, g) be a Lorentzian manifold. Let φ be a solution to the wave equation g φ = 0. A symmetric stress-energy tensor can be identified from variation of the massless scalar field action by and this satisfies the energy-momentum conservation law By contracting the energy-momentum tensor with a vector field V , we define the current In this context we call V a multiplier vector field. If the vector field V is timelike, then the one-form J V μ can be interpreted as the energy-momentum density. When we integrate J V μ contracted with the normal vector field over an associated hypersurface we will often refer to the integral as energy flux. Note that J V μ (φ)n μ ≥ 0 if V is future directed timelike and spacelike, where n μ is the future directed normal vector on the hypersurface . Since we will frequently use versions of the divergence theorem, we are interested in the divergence of the current (7). Defining by (6) it follows that Further, (π V ) μν . = 1 2 (L V g) μν is the so called deformation tensor of V . Therefore, Killing. For a Killing vector field W we have in addition the commutation relation [ g , W ] = 0. In that context W is called a commutation vector field. In particular, we note already that in Reissner-Nordström spacetime we have g T φ = 0 and g i φ = 0, where T and i with i = 1, 2, 3 are Killing vector fields that will be defined in Sects. 2.3.1 and 2.3.2, respectively.

The Reissner-Nordström solution.
In the following we will briefly recall the Reissner-Nordström solution 5 which is a family of solutions to the Einstein-Maxwell field equations with R μν the Ricci tensor, R the Ricci scalar and the units chosen such that 8π G c 4 = 2. The Maxwell equations are given by and the energy-momentum tensor by The system (10)-(12) describes the interaction of a gravitational field with a source free electromagnetic field. The Reissner-Nordström solution represents a charged black hole as an isolated system in an asymptotically Minkowski spacetime. The causal structure is similar to the structure of the astrophysically more realistic axisymmetric Kerr black holes. Since spherical symmetry can often simplify first investigations, Reissner-Nordström spacetime is a popular proxy for Kerr.
2.2.1. The metric and ambient differential structure. To set the semantic convention, whenever we refer to the Reissner-Nordström solution (M, g) we mean the maximal domain of dependence D( ) = M of complete two-ended asymptotically flat data . The manifold M can be expressed by M = Q × S 2 , and Q = (−1, 1) × (−1, 1) with coordinates U, V ∈ (−1, 1) and thus The metric in global double null coordinates then takes the form where Ω 2 and r will be described below.
As a gauge condition we choose the hypersurface U = 0 and V = 0 to coincide with what will be the event horizons and we set consistent with the fact that these hypersurfaces are to have infinite affine length. Fix parameters M > |e| = 0. The Reissner-Nordström metric (14) in our gauge is uniquely determined from (10) to (12) by setting Rearranging the Einstein-Maxwell equations (10) using (14) we obtain the following Hessian equation from the U, V component. From the θ, θ or equivalently φ, φ component we obtain In fact, all relevant information about Reissner-Nordström geometry can be understood directly from (15) to (18) without explicit expressions for Ω 2 (U, V ) and r (U, V ). In particular, one can derive the Raychaudhuri equations from the above. We can illustrate the 2-dimensional quotient spacetime Q as a subset of an ambient R 1+1 : Identifying U , V with ambient null coordinates of R 1+1 , the boundary of Q ⊂ R 1+1 is given by ±1 × [−1, 1] ∪ [−1, 1] × ±1. Let us further define the darker shaded region I I of Fig. 3 by which is the future boundary of the interior of region I I . We define M| I I = π −1 (Q| I I ), where π is the projection π : M → Q.

Eddington-Finkelstein coordinates.
It will be convenient to rescale the global double null coordinates and define Note that u is the retarded and v is the advanced Eddington-Finkelstein coordinate. These coordinates are both regular in the interior of Q| I I , cf. Fig. 3. Nonetheless, we can view the whole of Q| I I as  where we have formally parametrized by as depicted in Fig. 4, see also (20). In u, v coordinates the metric is given by where the unfamiliar minus sign on the right hand side arises since all definitions have been made suitable for the interior. We will often make use of the fact that by the choice of Eddington-Finkelstein coordinates (20) for the interior we have scaled our coordinates such that [The fact that the above expressions are constants follows from the Raychaudhuri equations (19) and (19)]. Taking the derivatives of (23) with respect to u and v and using (24) it follows that whereM| I I = M| I I \ ∂M| I I is the interior of M| I I . Moreover, we define the function r : where r is usually referred to as the Regge-Wheeler coordinate. Note that for coordinates (t, r , ϕ, θ) defined inM| I I we have that ∂ ∂t is a spacelike Killing vector field which extends to the globally defined Killing vector field T on M. By ϕ τ we denote a 1parameter group of diffeomorphisms generated by the Killing field T . We can moreover relate the functions r and r by dr = dr where C is constant which is implicitly fixed by previous definitions, and the surface gravities are given by Note that κ + is the surface gravity at H + and κ − is the surface gravity at CH + . The function r (u, v) extends continuously and is monotonically decreasing in both u and v towards CH + such that we have

The redshift, noshift and blueshift region.
As we have already mentioned in the introduction, in the interior we can distinguish subregions, as shown in Fig. 5, for values r red , r blue to be defined immediately below.
In the redshift region R we make use of the fact that the surface gravity κ + of the event horizon is positive. The region is then characterized by the fact that there exists a vector field N such that its associated current J N μ n μ v=const on a v = const hypersurface can be controlled by the related bulk term K N , cf. Proposition 4.1. This positivity of the bulk term K N is only possible sufficiently close to H + . In particular we shall define with > 0 and small enough such that Proposition 4.1 is applicable. (Furthermore, note that the quantity M − e 2 r is always positive in R.) As defined in (35) the r coordinate in the noshift region N ranges between r red defined by (36) and r blue , defined below, strictly bigger than r − . In N we exploit the fact that J −∂ r and K −∂ r are invariant under translations along ∂ t . For that reason we can uniformly control the bulk by the current along a constant r hypersurface. This will be explained further in Sect. 4.1.2. The blueshift region B is characterized by the fact that the bulk term K S 0 associated to the vector field S 0 to be defined in (44) is positive. We define with˜ > 0 for an˜ such that M − e 2 r blue carries a negative sign and such that (for convenience) r (r blue ) > 0. (38) In particular, in view of (25) and (25) for˜ sufficiently small the following lower bound holds in B with β a positive constant.

Notation.
We will describe certain regions derived from the hypersurfaces r = r red , r = r blue and in addition the hypersurface γ which will be defined in Sect. 4.1.3.1. For example given the hypersurface r = r red and the hypersurface u =ũ we define the v value at which these two hypersurfaces intersect by a function v red (ũ) evaluated forũ. Let us therefore introduce the following notation: For a better understanding the reader may also refer to Fig. 6. Note that the above functions are well defined since r = r red , r = r blue and γ are spacelike hypersurfaces terminating at i + .

Horizon estimates and Cauchy stability.
Our starting point will be previously proven decay bounds for φ and its derivatives in the black hole exterior up to and including the event horizon; in particular we can state:   Fig. 1. Then, there exists δ > 0 such that on H A + , for all v and some positive constants C k depending on the initial data. 6 Proof. The Theorem follows by putting together work of Blue and Soffer [6] on integrated local energy decay, Dafermos and Rodnianski [15] on the redshift and Schlue [41] on improved decay using the method of [17] in the exterior region. Specifically, [6] proves integrated local energy decay with degeneration on the horizon. Adding the redshift vector field energy identity of [15] yields integrated local energy decay without degeneration on the horizon. One can then apply the black box result of [17] to obtain polynomial decay on H + A of the form (41) with δ = 0. One finally applies [41] to obtain (41) for δ > 0.
The assumption of smoothness and compact support in Theorem 3.1 can be weakened. In fact, we shall only need (41) for k = 0, 1, 2. Moreover, as shown in [41], we can in fact take δ arbitrarily close to 1 2 , but δ > 0 is sufficient for our purposes and allows in principle for a larger class of data on .
On the other hand, trivially from Cauchy stability, boundedness of the energy along the second component of the past boundary of the characteristic rectangle , cf. Sect. 1.3, which we have picked to be v = 1, can be derived. More generally we can state the following proposition.

Proposition 3.2. Let u , v ∈ (−∞, ∞). Under the assumption of Theorem 3.1, the energy at advanced Eddington-Finkelstein coordinate
and further with D k (u , v ) positive constants depending on the initial data on .
Proof. This follows immediately from local energy estimates in a compact spacetime region. Note the −2 and 2 weights which arise since u is not regular at H + A .
Remark. We will see that C depends only on the initial data. We will consider a characteristic rectangle extending from H A + as shown in Fig. 7. We pick the characteristic rectangle to be defined by where u ✂ is sufficiently close to −∞ for reasons that will become clear later on, cf. Proposition 4.11. As described in Sect. 3.1, from bounds of data on bounds on the solution on the lower segments follow according to Theorem 3.1 and Proposition 3.2.
In order to prove Theorem 3.3 we distinguish the redshift R, the noshift N and the blueshift B region, with the properties as explained in Sect. 2.3.3, cf. Fig. 7b). This distinction is made since different vector fields have to be employed in the different regions. 7 In the redshift region R we will make use of the redshift vector field N of [21] on which we will elaborate more in Sect. 4.1.1. Proposition 4.1 gives the positivity of the bulk K N which thus bounds the current J N μ n μ v=const from above. Applying the divergence theorem, decay up to r = r red will be proven.
In the noshift region N we can simply appeal to the fact that the future directed timelike vector field −∂ r is invariant under the flow of the spacelike Killing vector field ∂ t . It is for that reason that the bulk term K −∂ r can be uniformly controlled by the energy flux J −∂ r μ n μ r =r through the r =r hypersurface. Decay up to r = r blue will be proven by making use of this together with the uniform boundedness of the v length of N .
To understand the blueshift region B, we will partition it by the hypersurface γ admitting logarithmic distance in v from r = r blue , cf. Sect. 4.1.3.1. We will then separately consider the region to the past of γ , J − (γ ) ∩ B and the region to the future of γ , J + (γ ) ∩ B. The region to the future of γ is characterized by good decay bounds on 2 (implying for instance that the spacetime volume is finite, Vol(J + (γ )) < C). 7 The reader may wonder why the noshift region N is introduced instead of just separating the red-and the blueshift regions along the r hypersurface whose value renders the quantity M − e 2 r equal zero. This was to ensure strict positivity/negativity of the quantity in the redshift/blueshift region.
In J − (γ ) ∩ B we use a vector field where q is sufficiently large, cf. Sect. 4.1.3. We will see that for the right choice of q we can render the associated bulk term K S 0 positive which is the "good" sign when using the divergence theorem.
In order to complete the proof, we consider finally the region J + (γ ) ∩ B and propagate the decay further from the hypersurface γ up to the Cauchy horizon in a neighbourhood of i + . For this, we introduce a new timelike vector field S defined by for an arbitrary p such that where δ is as in Theorem 3.1. We use pointwise estimates on 2 in J + (γ ) as a crucial step, cf. Sect. 4.1.4.1.
Putting everything together, in view of the geometry and the weights of S, we finally obtain for all v * ≥ 1 for the weighted flux. Using the above, the uniform boundedness for φ stated in Theorem 3.3 then follows from an argument that can be sketched as follows.
Let us first see how we get an integrated bound on the spheres of symmetry. By the fundamental theorem of calculus and the Cauchy-Schwarz inequality one obtains where the first factor of the first term is controlled by (47). Therefore, we further get where we have used ∞ 1 v − p dv < ∞ which followed from the first inequality of (46).
Obtaining a pointwise statement from the above will be achieved by commuting (1) with symmetries as well as applying Sobolev embedding. As outlined in Sect. 2.1 in Reissner-Nordström geometry we have g i φ = 0, where i with i = 1, 2, 3 are the 3 spacelike Killing vector fields resulting from the spherical symmetry. Thus one obtains the analogue of (48) but with i φ and i j φ in place of φ. Using Sobolev embedding on S 2 thus leads immediately to the desired bounds. See Sect. 4.2.3. This will close the proof of Theorem 3.3.

Energy and Pointwise Estimates in the Interior
In this section we will derive the proof of Theorem 1.2. For this we will first state the decay bound for the energy flux of φ given on the event horizon H + , cf. Theorem 3.1. Using this we propagate the decay rate through the entire interior up to the Cauchy horizon. As outlined in Sect. 1.3, we first consider a characteristic rectangle in the neighbourhood of i + . Within we need to separate the interior into different subregions. We then apply suitable vector fields according to the specific properties of the underlying subregion, which will be described further in the following subsections.

Energy estimates in the neigbourhood of i + .
4.1.1. Propagating through R from H + to r = r red . The estimates in this and the following section are motivated by work of Luk [30]. He proves that any polynomial decay estimate that holds along the event horizon of Schwarzschild black holes can be propagated to any constant r hypersurface in the black hole interior. This followed a previous spherically symmetric argument of [13]. See also Dyatlov [23].
As outlined in Sect. 3.1, we will first propagate energy decay from H + up to the r = r red hypersurface.
The rough idea can be understood with the help of Fig. 8. By Theorem 3.1 we are given energy decay on the event horizon H + , see dash-dotted line. By using the energy identity for the vector field N in region , we obtain decay of the flux through constant v hypersurfaces throughout the entire region. Using this result and considering the energy identity once again in regioñ we eventually obtain decay on the r = r red hypersurface, note the dashed line.
The redshift vector field was already introduced by Dafermos and Rodnianski in [16] and elaborated on again in [21]. The existence of such a vector field in the neighbourhood of a Killing horizon H + depends only on the positivity of the surface gravity, in this case κ + . Thus by (30) the following proposition follows by Theorem 7.1 of [21].
for all solutions φ of g φ = 0.
The decay bound along r = r red can now be stated in the following proposition.
Remark 1. The decay in Proposition 4.2 matches the decay on H + of Theorem 3.1.

Remark 2. n
μ r =r red denotes the normal to the r = r red hypersurface oriented according to Lorentzian geometry convention. dVol denotes the volume element over the entire spacetime region and dVol r =r red denotes the volume element on the r = r red hypersurface. Similarly for all other subscripts. 9 Proof. Applying the divergence theorem, see e.g. [21] or [45], in region We immediately see that the second term on the left hand side is positive since r = r red is a spacelike hypersurface and N is a timelike vector field. Therefore, we write By Theorem 3.1 we have 9 Refer to "Appendix A" for further discussion of the volume elements.
Using that the energy current associated to the timelike vector field N is controlled by the deformation K N as shown in (49) and substituting into (50) as well as using the coarea formula for the bulk term, 10 we obtain for all v 0 ≥ 1 and v * > v 0 , the relation Note by Proposition 3.2 with k = 0, applied to u defined through the relation r red = r (u , 1), we have since the vector field N is regular at H + and thus E(φ; 1) is comparable to the left hand side of (42). In order to obtain estimates from (53) we appeal to the following lemma.
for allt ≥ 1, where C 0 ,p are positive constants. Then for any t ≥ 1 we have whereC depends only on f (1), b and C 0 .
Proof. For t > t 0 , we will show (56) by a continuity (bootstrap) argument. It suffices to show that leads to for some large enough constantC. 10 where f ∼ g means that there exist constants 0 We note first that given any t 0 , from (55) we obtain, Given t ≥ t 0 , chooset = t − L for an L to be determined later. Moreover, t 0 will have to be chosen large enough so that ∀ t ≥ t 0 , Given a t satisfying (57) applying (55) yields Further, by the pigeonhole principle, there exists t in ∈ [t, t] such that Since f (t) is a positive function (61) also leads to Thus, (62) and (63) yield Now lett = t in and use (64) in (55), then then (58) follows. Thus picking first L such that 1 − 4 bL > 0, and then t 0 such that t 0 ≥ L + 1 and satisfying (60), and finally choosingC asC = max f (1) 2C 0 (L+1) bL + 2C 0 (L + 1) (58) and thus (56) follows by continuity.
By Lemma 4.3 we obtain from (53) together with (54) withC depending onb 1 and D 0 (u , 1). Finally, in order to close the proof of Proposition 4.2 we perform again the divergence theorem but for regionR I = {r red ≤ r ≤ r + } ∩ {v * ≤ v ≤ v * + 1}: In view of the signs we obtain Due to (67) and Theorem 3.1 we are left with the conclusion of Proposition 4.2.
Note that the above also implies the following statement.
with C depending on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1) and v red (ũ) as in (40).
Proof. The conclusion of the statement follows by applying again the divergence theorem and using the results of the proof of Proposition 4.2.

4.1.2.
Propagating through N from r = r red to r = r blue . Now that we have obtained a decay bound along the r = r red hypersurface in the previous section, we propagate the estimate further inside the black hole through the noshift region N up to the r = r blue hypersurface. In order to do that we will use the future directed timelike vector field Using (70) in (B2) of "Appendix B" we obtain for the bulk current. It has the property that it can be estimated by where B 1 is independent of v * . Validity of the estimate can in fact be seen without computation from the fact that timelike currents, such as J −∂ r μ (φ)n μ r =r contain all derivatives. The uniformity of B 1 is given by the fact that K −∂ r and J −∂ r are invariant under translations along ∂ t , cf. Sect. 2.3.1 for definition of the t coordinate. Therefore, we can just look at the maximal deformation on a compact {t = const} ∩ {r blue ≤ r ≤ r red } hypersurface and get an estimate for the deformation everywhere.

Proof.
Given v * , we define regions R I I andR I I as in Fig. 9, where we use (40) and Thus the depicted regions are given by In the following we will apply the divergence theorem in region R I I ∪R I I to obtain decay on an arbitrary r =r hypersurface, dash-dotted line, forr ∈ [r blue , r red ), from the derived decay on the r = r red hypersurface. The second integral of the left hand side represents the current through the u = u blue (v * ) hypersurface, defined by (40). As u = u blue (v * ) is a null hypersurface and −∂ r is timelike, the positivity of that second term is immediate. Similarly, the fourth term of the left hand side of our equation is positive and we obtain Further, we use that the deformation K −∂ r is controlled by the energy associated to the timelike vector field −∂ r as stated in (72). Thus we obtain By the coarea formula we obtain Now let withr ∈ [r blue , r red ). Replacing r blue withr in the above, considering the future domain of dependence of {v 1 ≤ v ≤ v * + 1} ∩ {r = r red } up to the r =r hypersurface we obtain similarly to (75) Using Grönwall's inequality in (77) yields Finally, note that where k = 2[r (r blue ) − r (r red )] + 1. This can be seen since (24) and (28) yields Further, by using the conclusion of Proposition 4.2 and (79) we have We thus infer The above now also implies the following statement. Then, for all v * > 1 and allũ such that r (ũ, v * ) ∈ [r blue , r red ) with C depending on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1) and v red (ũ), v blue (ũ) are as in (40).
Proof. The conclusion of the statement follows by considering the divergence theorem for a triangular region J − (x) ∩ N with x = (ũ, v blue (ũ)), x ∈ J − (r = r blue ) and using the results of Proposition 4.5. Note that v * ∼ v blue (ũ) ∼ v red (ũ).
By the previous proposition we have successfully propagated the energy estimate further inside the black hole, up to r = r blue . To go even further will be more difficult and we will address this in the next section.

4.1.3.
Propagating through B from r = r blue to the hypersurface γ . In the following we want to propagate the estimates from the r = r blue hypersurface further into the blueshift region to a hypersurface γ which is located a logarithmic distance in v from the r = r blue hypersurface, cf. Fig. 10. We will define the hypersurface γ and its most basic properties in Sect Let α be a fixed constant satisfying with β as in (39) and (39). [The significance of the bound (82) will become clear later]. Let us for convenience also assume that and α(2 − log 2α) > 2r blue + 1.  H (u, v) by were r (r blue ) = r blue is the r value evaluated at r blue according to (28), and r blue > 0 according to the choice (38). We then define the hypersurface γ as the levelset Since we see that γ is a spacelike hypersurface and terminates at i + , cf. "Appendix A". (In the notation (40), u γ (v) → −∞ as v → ∞.) Note that by our choices u < −1 and v > |u| in D + (γ ).
Recall that in Sect. 2.3.1 we have defined the Regge-Wheeler tortoise coordinate r depending on u, v by (27). Using this for r blue we have with v blue (u) as in (40). Plugging this into (86) recalling v γ (u) defined in (40), we obtain the relation As we shall see in Sect. 4.1.4.1 the above properties of γ will allow us to derive pointwise estimates of 2 in J + (γ ) ∩ B. We first turn however to the region J − (γ ) ∩ B.

4.1.3.2.
Energy estimates from r = r blue to the hypersurface γ . Now we are ready to propagate the energy estimates further into the blueshift region B up to the hypersurface γ . We will in this part of the proof use the vector field which we have defined in (44). Let us now consider the bulk term and derive positivity properties which are needed later on. Plugging (44) in (B2) of "Appendix B" yields Our aim is to show that K S 0 is positive. All terms multiplied by the angular derivatives are manifestly positive in B, cf. (39), (39) together with (24) to (25). Therefore, it is only left to show that the first term on the right hand side dominates the last term. Since by the Cauchy-Schwarz inequality We show now that at the expense of one polynomial power, we can extend the local energy estimate on r = r blue to an energy estimate along γ which is valid for a dyadic length.

Proposition 4.7. Let φ be as in Theorem 3.1. Then, for all
on the hypersurface γ , with C depending on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1). Proof. In the following we will again make use of notation (40).

Remark. n
Let v * > 2α, such that γ is spacelike for v > v * , cf. Sect. 4.1.3.1. Define u 3 by (u 3 , v * ) ∈ γ , i.e. u γ (v * ) = u 3 and define v blue as the intersection of u 3 with r blue , i.e. v blue (u 3 ) = v blue . And similarly the hypersurfaces u = u 1 and u = u 2 as shown in Fig. 10 are given by u blue (2v * ) = u 1 and u γ (2v * ) = u 2 . Having defined the relations between all these quantities we can now carry out the divergence theorem for region R I I I : Positivity of the flux along the u = u 3 segment and the flux along the v = 2v * segment, as well as positivity of K S 0 for the choice q ≥ 2, which was derived in (90) and (91), leads to where the second step is implied by Proposition 4.5 and the last step follows from the inequality v * ≤ Cv blue which is implied by (89).
We have already mentioned in the introduction that we will use the vector field S, cf. (45) in the region J + (γ ) ∩ B. To control the initial flux term of S we require a weighted energy estimate along the hypersurface γ .
on the hypersurface γ , with C depending on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1) and p as in (46).
Proof. This follows by weighting (92) with v p * and summing dyadically.
Further, we can state the following.

Corollary 4.9.
Let φ be as in Theorem 3.1, r blue as in (37) and γ as in (86). Then, for all v * > 2α and for allũ with C depending on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1) and v γ (ũ), v blue (ũ) as in (40).
Proof. The proof is similar to the proof of Corollary 4.6 by considering the divergence theorem for a triangular region J − (x) ∩ B with x = (ũ, v γ (ũ)), x ∈ J − (γ ) and using the results of the proof of Proposition 4.7.

4.1.4.
Propagating through B from the hypersurface γ to CH + in the neighbourhood of i + . In order to prove our Theorem 3.3 and close our estimates up to the Cauchy horizon in the neighbourhood of i + we are interested in considering a region R I V within the trapped region whose boundaries are made up of the hypersurface γ , a constant u and a constant v segment, which can reach up to the Cauchy horizon, cf. Fig. 11.
Let v * > 2α and letv > v * . We may write Note that R I V lies entirely in the blueshift region, which was characterized by the fact that the quantity M − e 2 r takes the negative sign, cf. (39), (39) and (24) to (25).
In Sect. 4.1.4.1 we will derive pointwise estimates for 2 in the future of the hypersurface γ . With this estimate, the bulk term will be bounded in terms of the currents through the null hypersurfaces. Consequently, we will be able to absorb the bulk term and to show that the currents through the null hypersurfaces can be bounded by the current along the hypersurface γ , cf. Sect. 4.1.4.3. 2 in J + (γ ). In the following we will derive pointwise estimates on 2 in J + (γ ). We note that these will imply that the spacetime volume to the future of the hypersurface γ is finite, Vol(J + (γ )) < C. Fig. 11. Blueshift region of Reissner-Nordström spacetime from hypersurface γ onwards We first derive a future decay bound along a constant u hypersurface for the function 2 (u, v) for (u, v) ∈ B. Let x = (u fix , v fix ), x ∈ B, then, from (25) we can immediately see that

Pointwise estimates on
It then immediately follows that Analogously, we obtain and plugging (98) into (99) it yields From (98) and (89) we obtain a relation for 2 (u, v) on the hypersurface γ as follows For J + (γ ), using (98) we further get where we have used 2 (ū, v blue (ū)) ≤ C. Moreover, we may think of a parameterv which determines the associated u value via intersection with γ , we denote this value by the evaluation the function u γ (v) which was introduced in (40), cf. Fig. 6b). Moreover, by (25) we can also state Note that the choice (82) of α implies that βα > 1. From (103), the fact that |u γ (v)| ∼v, and the extra exponential factor, finiteness of the spacetime volume to the future of γ follows, See also [27].

Bounding the bulk term K S .
To derive energy estimates in R I V we use the timelike vector field multiplier which we have given before in (45). The weights of S are chosen such that they will allow us to derive pointwise estimates from energy estimates; see Sect. 4.2.3. In order to obtain our desired estimates first of all we need a bound on the scalar current K S , in terms of J S μ (φ)n μ v=v and J S μ (φ)n μ u=ū . In the following we will bound the occurring (u, v)-dependent weight functions by functions that depend on either u or v, respectively. Plugging the vector field S, cf. (45), into (B2) of "Appendix B" we obtain Recall (39) and (39). For large absolute values of v and u the first two terms multiplying the angular derivatives of φ dominate the last two terms, so in total the term multiplying the angular derivatives is always positive in D + (γ ). Consequently we will be able to use this property to derive an inequality by using the divergence theorem in the proof of Proposition 4.11. Let us therefore definẽ  Fig. 11, of the currentK S , defined by (106), can be estimated by where δ 1 and δ 2 are positive constants, with δ 1 → 0 and δ 2 → 0 as v * → ∞.
Remark. In the proof of Proposition 4.11 we will see that the above proposition determines u ✂ of Theorem 3.3, depicted in Fig. 7b). We have to choose u ✂ = u γ (v * ), with v * such that δ 1 is small.
Proof. Using the Cauchy-Schwarz inequality twice for the remaining part of the bulk term we obtain with the related volume element Note that the currents related to the vector field S with their related volume elements are given by cf. "Appendix A". Taking the integral over the spacetime region yields with u γ (v) and v γ (ū) in the integration limits as defined in (40). Note the following general relation for positive functions f (ū,v) and g(ū,v) Similarly, it immediately follows that Using (113) and (114) in (112) we obtain It remains to show finiteness and smallness of dv. Earlier we obtained the relation (103) for 2 in region R I V . Therefore, we can write where δ 1 → 0 for |u γ (v * )| → −∞ and thus for v * → ∞. Note that we have here used (82).
For finiteness of the second term in (115) we follow the same strategy and use (102) for the second term to obtain where δ 2 → 0 for v * → ∞. Therefore, we obtain the statement of Lemma 4.10.

4.1.4.3.
Energy estimates from γ up to CH + in the neighbourhood of i + . Now we come to the actual proof of weighted energy boundedness up to the Cauchy horizon. (46). Then, for u ✂ sufficiently

Proposition 4.11. Let φ be as in Theorem 3.1 and p as in
where C is a positive constant depending on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1).
Remark. Refer to (40) for the definition of u γ (v) and see Fig. 6b) for further clarification.  Let v * > 2α andv > v * . In order to obtain (118) we consider a region Fig. 11. Applying the divergence theorem we obtain In Sect. 4.1.4.2 we found that the angular part of K S (φ) is positive in R I V and we called the remaining partK S (φ) given in (106). Using (107) we can therefore write Using Lemma 4.10 we obtain Repeating estimate (123) withū,v in place of u γ (v * ),v and taking the supremum we have Recalling holds. The conclusion of Proposition 4.11 then follows by absorbing the first two terms of the right hand side of (124) by the two terms on the left and estimating the third from (120).

Energy estimates globally in the rectangle up to CH
where C is a positive constant depending on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1).
Proof. First of all we partition the integral of the statement into a sum of integrals of the different regions. That is to say For the integral in R and the integral in N we use Corollaries 4.4 and 4.6. (Note that the former has to be summed resulting in the loss of one polynomial power). Further, for the integral in region J − (γ ) ∩ B we apply Corollary 4.9 and for the integral in region J + (γ ) ∩ B we use Proposition 4.11. Putting all this together we arrive at the conclusion of Proposition 4.12.
In particular, we have Corollary 4.13. Let φ be as in Theorem 3.1 and p as in (46). Then, for u ✂ sufficiently close to −∞, for all v fix ≥ 1, andũ ∈ (−∞, u ✂ ), where C is a positive constant dependent on C 0 of Theorem 3.1 and D 0 (u , 1) of Proposition 3.2, where u is defined by r red = r (u , 1).
Proof. The conclusion of the proposition follows immediately examining the weights in Proposition 4.12. They were explicitly given in (32). Further, having the expressions of (33) in mind we introduce the following notation

Pointwise
with i j = 1, 2 or 3. By Sobolev embedding on the standard spheres we have in this notation which means that we can derive a pointwise estimate from an estimate of the integrals on the spheres, see e.g. [16].

4.2.2.
Higher order energy estimates in the neighbourhood of i + . We will need the following extension of Corollary 4.13 for higher order energies.
Theorem 4.14. Let φ be as in Theorem 3.1 and p as in (46). Then, for v fix ≥ 1 and Proof. Statement (130) with k = 0 was already derived in Corollary 4.13. Recall that i φ, i j φ etc. also satisfy the massless scalar wave equation, cf. Sect. 2.1. Summing over all angular momentum operators, keeping in mind notation (128), we therefore obtain (130) for all k ∈ N 0 .

4.2.3.
Pointwise boundedness in the neighbourhood of i + . We turn the discussion to the derivation of pointwise boundedness from energy estimates. In particular we prove Theorem 3.3 from Theorem 4.14 applied with k = 0, 1, 2.
By the fundamental theorem of calculus it follows for all v * > 1,v > v * and where we have used the Cauchy-Schwarz inequality in the last step. Squaring the entire expression, using Cauchy-Schwarz again and integrating over S 2 we obtain the expression that we had sketched in Sect. 3.2 already with p as in (46) and the first term on the right hand side controlled by the flux for which we derived boundedness in Sect. 4.1.4. Therefore, by using Theorem 4.14 and applying all our estimates to i φ, i j φ etc. and summing, we obtain in the notation of Sect. 4.2.1 the following: for all k ∈ N 0 . It is here that we have used the requirement p > 1 of (46).
with C depending on the initial data on . We therefore arrive at the statement given in Theorem 3.3. Fig. 12, and note that R V ⊂ B. We will apply the vector field

Energy along the future boundaries of
as a multiplier. The bulk can be calculated as Let us definẽ with K W ∇ / positive since the second term in the second equation dominates over the first for v * > 2α and ∂ u , ∂ v are negative in the blueshift region. We have therefore in R V . We aim for estimating it via the currents along v = constant and u = constant hypersurfaces.
Proof. Using the Cauchy-Schwarz inequality for the first equation of (138) we obtain with the related volume element Note that the currents related to the vector field W with their related volume elements are given by cf. "Appendix A". Taking the integral over the spacetime region therefore yields It remains to show finiteness and smallness of (1 +v p ) dv. Recall the properties of the hypersurface γ shown so that we obtain and moreover δ 1 → 0 for → 0. Further, in Sect. 4.1.4.1 we derived that similarly where where δ 2 → 0 for v * → ∞. Thus the conclusion of Lemma 4.15 is obtained.
From the above we obtain Proposition 4.16. Let φ be as in Theorem 3.1 and p as in (46).
Then for sufficiently small, the following is true. If where the last step follows by statement (42) of Proposition 3.2, with k = 0, and by (150). We have chosen sufficiently small and v * sufficiently close to ∞, such that δ 1 and δ 2 satisfy say The conclusion of Proposition 4.16 is obtained.
We are now ready to make a statement for the entire region R V . (46). Then, for all v * > v γ (u ✂ ) sufficiently large,v > v * , and u >û > u ✂ ,

Proposition 4.17. Let φ be as in Theorem 3.1 and p as in
where C depends on C 0 of Theorem 3.1 and D 0 (u , v * ) of Proposition 3.2.
Proof. Let be as in Proposition 4.16. We choose a sequence u i+1 − u i ≤ and i = {1, 2, .., n} such that u 1 = u ✂ and u n =û. Denote Fig. 13. Iterating the conclusion of Proposition 4.16 from u 1 up to u n then completes the proof. Note that n depends only on the smallness condition on from Proposition 4.16, since n u −u ✂ . In this section we will use both results from the right and left side on CH + . Fix u = v , such that moreover Proposition 4.17 holds with v = v * , and such that its left side analog holds with u = u * . We will consider a region Fig. 14 which we are going to use again to obtain an energy estimate up to the bifurcate twosphere. Recall K S given in (105), where the terms multiplying the angular derivatives are positive since R V I is located in the blueshift region. We further definedK S in (106) and stated (107) which will be useful to state the following proposition.

Proof
where p is as in (46) (46). and for all k ∈ N 0 .
Proof. This follows immediately from Corollary 4.20 by commutation.
Having proven Theorem 4.21, the pointwise boundedness of |φ| in all ofM| I I follows analogously to Sect. 4.2.3 but by integration in u-direction.
We estimate where u * ≥ u ✂ ,û ∈ (u * , ∞) and v ∈ (1, ∞) and k ∈ N 0 . By using the result (135) in (167) with C depending on the initial data. Inequalities (168) and (135) gives the desired (2) for all v ≥ 1. Interchanging the roles of u and v, likewise (2) follows for all u ≥ 1. The remaining subset of the interior has compact closure in spacetime for which (2) thus follows by Cauchy stability. We have thus shown (2) globally in the interior.
The continuity statement of Theorem 1.1 follows easily by estimating |φ(u, v, ϕ, θ)− φ(ũ, v, ϕ, θ)| via the fundamental theorem of calculus and Sobolev embedding, 12 and similarly for v, ϕ and θ in the role of of u.

Outlook
In the following section we give a brief overview of further related open problems.

Scalar waves on subextremal Kerr interior backgound.
All constructions of this paper have direct generalizations to the subextremal Kerr case. 13 The following theorem will appear in a subsequent companion paper. The above theorem of course depends on the fact that the analog of Theorem 3.1 has been proven in the full subextremal range |a| < M on Kerr backgrounds by Dafermos et al. cf. [19,22]; see also [1,18,20,21,31,35,44] for the |a| M case and [42,46] for mode stability.

Mass inflation.
In view of our stability result, what remains of the "blueshift instability"?
The result of Theorem 1.1 is still compatible with the expectation that the transverse derivatives (with respect to regular coordinates on CH + ) of φ will blow up along CH + , cf. the work of Simpson and Penrose [43]. In fact given a lower bound, say for some constant c > 0 and all sufficiently large v on H + , Dafermos has shown for the spherically symmetric Einstein-Maxwell-scalar field model that transverse derivatives blow up, cf. [13]. This blow up result of [14] would also apply to our setting here if the above lower bound (169) is assumed on the spherical mean of φ on H + . See also [34]. Such a lower bound however has not been proven yet for solutions arising from generic data on ; see also Sbierski [40]. One might therefore aim for proving the following conjecture.

Extremal black holes.
For a complete geometric understanding of black hole interiors one must also consider extremal black holes. Aretakis proved stability and instability properties for the evolution of a massless scalar field on a fixed extremal Reissner-Nordström exterior background. For data on a spacelike hypersurface intersecting the event horizon and extending to infinity he has proven decay of φ up to and including the horizon, cf. [2]. In subsequent work [3] Aretakis showed that first transverse derivatives of φ generically do not decay along the event horizon for late times. He further proved that higher derivatives blow up along the event horizon. The analysis of the evolution of the scalar wave in the region beyond the event horizon for the extreme case remains to be shown. Motivated by heuristics and numerics of Murata, Reall and Tanahashi [36], which suggest stronger stability results in the interior than in the subextremal case, we conjecture See work of Gajic [24]. Aretakis has further considered g φ on a fixed extremal Kerr background, c.f [4]. Analogously to the extremal Reissner-Nordström case, he shows decay up to and including the event horizon for axisymmetric solutions φ and in [5] shows instability properties when considering transverse derivatives. This has been further generalised in [28,29]. In analogy with Conjecture 5.3, we thus also make the following conjecture: The case of non axisymmetric φ seems to be rather complicated and we thus do not venture a conjecture here.

Einstein vacuum equations.
We now return to the problem that originally motivated our work, namely the dynamics of the Einstein vacuum equations. Our result and the forthcoming extension to subextremal Kerr backgrounds further support the expectation that the spherically symmetric toy models [12][13][14][37][38][39] are indeed indicative of what happens for the full nonlinear Einstein vacuum equations without symmetry. In particular our results support the following conjecture given in [14]. A proof of part (a) has recently been announced by Dafermos and Luk, given the conjectured stability of Kerr exterior (i.e. given the analog of Theorem 3.1 for the full nonlinear Einstein vacuum equations). Specific examples of vacuum spacetimes with null singularities as in (b) have been constructed by Luk [32]. For a discussion of what all this means to Strong Cosmic Censorship see [14].