Boundedness of Massless Scalar Waves on Reissner ... - CERN

Digital Object Identifier (DOI) 10.1007/s00220-015-2440-7Commun. Math. Phys. 343, 601–650 (2016) Communications in

MathematicalPhysics

Boundedness of Massless Scalar Waves onReissner-Nordström Interior Backgrounds

Anne T. Franzen1,2

1 Department ofMathematics, Fine Hall, Princeton University,Washington Road, Princeton, NJ 08544, USA.E-mail: [email protected]; [email protected]

2 Universiteit Utrecht, Leuvenlaan 4, 3584 CE, Utrecht, The Netherlands

Received: 26 March 2015 / Accepted: 11 May 2015Published online: 28 December 2015 – © The Author(s) 2015. This article is published with open access atSpringerlink.com

Abstract: We consider solutions of the scalar wave equation�gφ = 0, without symme-try, on fixed subextremal Reissner-Nordström backgrounds (M, g) with nonvanishingcharge. Previously, it has been shown that for φ arising from sufficiently regular dataon a two ended Cauchy hypersurface, the solution and its derivatives decay suitably faston the event horizonH+. Using this, we show here that φ is in fact uniformly bounded,|φ| ≤ C , in the black hole interior up to and including the bifurcate Cauchy horizon CH+,to which φ in fact extends continuously. The proof depends on novel weighted energyestimates in the black hole interior which, in combination with commutation by angularmomentum operators and application of Sobolev embedding, yield uniform pointwiseestimates. In a forthcoming companion paper we will extend the result to subextremalKerr backgrounds with nonvanishing rotation.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6021.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6021.2 Motivation and strong cosmic censorship . . . . . . . . . . . . . . . . 6041.3 A first look at the analysis . . . . . . . . . . . . . . . . . . . . . . . . 6041.4 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6062.1 Energy currents and vector fields . . . . . . . . . . . . . . . . . . . . 6062.2 The Reissner-Nordström solution . . . . . . . . . . . . . . . . . . . . 607

2.2.1 The metric and ambient differential structure. . . . . . . . . . . . 6072.2.2 Eddington-Finkelstein coordinates. . . . . . . . . . . . . . . . . . 608

2.3 Further properties of Reissner-Nordström geometry . . . . . . . . . . 6102.3.1 (t, r�) and (t, r) coordinates. . . . . . . . . . . . . . . . . . . . . 6102.3.2 Angular momentum operators. . . . . . . . . . . . . . . . . . . . 6112.3.3 The redshift, noshift and blueshift region. . . . . . . . . . . . . . 611

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602 A. T. Franzen

2.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6123. The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

3.1 Horizon estimates and Cauchy stability . . . . . . . . . . . . . . . . . 6123.2 Statement of the theorem and outline of the proof in the neighbourhood

of i+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6144. Energy and Pointwise Estimates in the Interior . . . . . . . . . . . . . . . 616

4.1 Energy estimates in the neigbourhood of i+ . . . . . . . . . . . . . . 6164.1.1 Propagating through R fromH+ to r = rred. . . . . . . . . . . . 6164.1.2 Propagating through N from r = rred to r = rblue. . . . . . . . . 6214.1.3 Propagating through B from r = rblue to the hypersurface γ . . . . 6244.1.4 Propagating through B from the hypersurface γ to CH+ in the neigh-

bourhood of i+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6274.2 Pointwise estimates from higher order energies . . . . . . . . . . . . 635

4.2.1 The � notation and Sobolev inequality on spheres. . . . . . . . . 6354.2.2 Higher order energy estimates in the neighbourhood of i+. . . . . 6354.2.3 Pointwise boundedness in the neighbourhood of i+. . . . . . . . . 636

4.3 Energy along the future boundaries ofRV . . . . . . . . . . . . . . . 6374.4 Propagating the energy estimate up to the bifurcation sphere . . . . . 6424.5 Global higher order energy estimates, pointwise boundedness and conti-

nuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6445. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

5.1 Scalar waves on subextremal Kerr interior backgound . . . . . . . . . 6455.2 Mass inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6465.3 Extremal black holes . . . . . . . . . . . . . . . . . . . . . . . . . . 6465.4 Einstein vacuum equations . . . . . . . . . . . . . . . . . . . . . . . 647

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

1. Introduction

The Reissner-Nordström spacetime (M, g) is a fundamental 2-parameter family ofsolutions to the Einstein field equations coupled to electromagnetism, cf. Fig. 1 for theconformal representation of the subextremal case, M > |e| �= 0, with e the charge andM the mass of the black hole. The problem of analysing the scalar wave equation

�gφ = 0 (1)

on a Reissner-Nordström background is intimately related to the stability properties ofthe spacetime itself and to the celebrated Strong Cosmic Censorship Conjecture. Theanalysis 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. Thepurpose of the present work is to extend the investigation to the interior of the blackhole, up to and including the Cauchy horizon CH+.

1.1. Main result. The main result of this paper can be stated as follows.

Theorem 1.1. On subextremal Reissner-Nordström spacetime (M, g), with mass Mand charge e and M > |e| �= 0, let φ be a solution of the wave equation �gφ = 0

Boundedness of Massless Scalar Waves on Reissner-Nordström Interior Backgrounds 603

Fig. 1. Maximal development of Cauchy hypersurface � in Reissner-Nordström spacetime (M, g)

arising from smooth compactly supported Cauchy data on a two-ended asymptoticallyflat Cauchy surface �. Then

|φ| ≤ C (2)

globally in the black hole interior, in particular up to and including the Cauchy horizonCH+, to which φ extends in fact continuously.

The constant C is explicitly computable in terms of parameters e and M and a suitablenorm on initial data.1 The above theorem will follow, after commuting (1) with angularmomentum operators and applying Sobolev embedding, from the following theorem,expressing weighted energy boundedness.

Theorem 1.2. On subextremal Reissner-Nordström spacetime (M, g), with mass Mand charge e and M > |e| �= 0, let φ be a solution of the wave equation �gφ = 0arising from smooth compactly supported Cauchy data on a two-ended asymptoticallyflat Cauchy surface �. Then

∫

S2

∞∫

vfix

[v p(∂vφ)2(u, v, θ, ϕ) + 2|∇/φ|2(u, v, θ, ϕ)

]r2dvdσS2 ≤ E, for vfix ≥ 1, u > −∞

(3)

∫

S2

∞∫

ufix

[u p(∂uφ)2(u, v, θ, ϕ) + 2|∇/φ|2(u, v, θ, ϕ)

]r2dudσS2 ≤ E, for ufix ≥ 1, v > −∞,

(4)

where p > 1 is an appropriately chosen constant, and (u, v) denote Eddington-Finkelstein coordinates in the black hole interior, where by dσS2 we denote the volume

element of the unit two-sphere and |∇/φ|2 = 1r2

[(∂θφ)2 + 1

sin2 θ(∂ϕφ)2

].

1 The Theorem holds more generally for data in the weighted Sobolev space defined in this norm. We shallnot explicitly give the optimal norm here, but it can be deduced from the proof of Theorem 3.1.

604 A. T. Franzen

1.2. Motivation and strong cosmic censorship. Ourmotivation for proving Theorem 1.1is the Strong Cosmic Censorship Conjecture. The mathematical formulation of thisconjecture, here applied to electrovacuum, is given in [8] by Christodoulou as

“Generic asymptotically flat initial data for Einstein-Maxwellspacetimes have a maximal future development which is inex-tendible as a suitably regular Lorentzian manifold.”

(5)

Reissner-Nordström spacetime serves as a counterexample to the inextendibility state-ment, since it is (in fact smoothly) extendable beyond the Cauchy horizon CH+.2 Thus,for the above conjecture to be true, this property ofReissner-Nordströmmust in particularbe unstable.

Originally it was suggested by Penrose and Simpson that small perturbations ofReissner-Nordström would lead to a spacetime whose boundary would be a spacelikesingularity as in Schwarzschild and such that the spacetime would be inextendable as aC0 metric, cf. [43]. On the other hand, a heuristic study of a spherically symmetric butfully 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 aCauchy horizon, which however would be singular in a weaker sense, see also [37] byOri. Considering the spherically symmetric Einstein-Maxwell-scalar field equations asa toy model, Dafermos proved that the solution indeed exists up to a Cauchy horizonand moreover is extendible as a C0 metric but generically fails to be extendible as a C1

metric beyond CH+, cf. [12,13]. For more recent extensions see [9–11,27].In this work, as a first attempt towards investigation of the stability of theCauchy hori-

zon under perturbationswithout symmetry, we employ (1) on a fixedReissner-Nordströmbackground (M, g) as a toymodel for the full nonlinear Einstein field equations, cf. (10).The result of uniform pointwise boundedness of φ and continuous extension to CH+ isconcordant with the work of Dafermos [12]. This suggests that the non-spherically sym-metric perturbations of the astrophysically more realistic Kerr spacetime may indeedexist up to CH+. See Sect. 5.4.

1.3. A first look at the analysis. The proof of Theorems 1.1 and 1.2 involves first con-sidering a characteristic rectangle � within the black hole interior, whose future rightboundary 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 entireproof. Once that is done, analogous results hold for a characteristic rectangle � tothe left depicted in Fig. 2a). Hereafter, boundedness of the energy is easily propa-gated to regions RV , RV and RV I as depicted, giving Theorem 1.2. Commutationby angular momentum operators and application of Sobolev embedding then yieldsTheorem 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+, whichcan 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 weshall need will be stated in Sect. 3.1.

2 Outside the future maximal domain of dependenceD+(M) in the future of the Cauchy horizon J+(CH+)

the spacetime shows the peculiar feature that uniqueness of the solutions of the initial value problem is lostwithout loss of regularity. It is precisely the undesirability of this feature that motivates the conjecture.


Fig. 2. a Penrose diagram depicting regions considered in the proof. b � with redshift R, noshift N andblueshift regions B

In � the proof involves distinguishing redshiftR, noshiftN and blueshift B regions,as shown in Fig. 2b).

Some of these regions have appeared in previous analysis of the wave equation, espe-cially R = {rred ≤ r ≤ r+}. Region N = {rblue ≤ r ≤ rred} and region B = {r− ≤ r≤ rblue} were studied in [13] in the spherically symmetric self-gravitating case, butusing techniques which are very special to 1+1 dimensional hyperbolic equations.3 Theanalogous regions R and N were also considered by Luk [30] on an interior Schwarz-schild background. We will discuss this separation into R, N and B regions furtherin Sect. 3.2. One of the main analytic novelties of this paper is the introduction of anew vector field energy identity constructed for analyses in region B. In particular, theweighted vector field is given in Eddington-Finkelstein coordinates (u, v) by

S = |u|p∂u + v p∂v,

for p > 1 as appearing in Theorem 1.2. This vector field associated to region B willallow 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 ofthe 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 previouswork, [6,15] and [41], for φ arising from sufficiently regular initial data on a Cauchyhypersurface. This is stated as Theorem 3.1. Further, we state our main result specializedto the rectangle � in the neighbourhood of i+ (see Theorem 3.3) and give an outline ofits proof.

In Sect. 4 we prove boundedness of φ up to and including H+. In particular, inSect. 4.1 we first propagate the decay bound for the energy flux of φ from H+ upto CH+ in the neighbourhood of i+. The investigation is divided into considerationswithin the redshift R, noshift N , and blueshift B regions. Section 4.2 reveals howcommutation with angular momentum operators and applying Sobolev embedding willreturn us pointwise boundedness for |φ|.Wenowmust extend our result to the full interior

3 Let us note that the result of Theorem 1.1 for spherically symmetric solutions φ can be obtained byspecializing [13–15] to the uncoupled case. Restricted results for fixed spherical harmonics can be in principlealso inferred from [33].

606 A. T. Franzen

region. In Sect. 4.3 we propagate the energy estimates further along CH+ in the depictedregion RV . Eventually, in subsection 4.4 we propagate the estimate to the region RV Iup to the bifurcation two-sphere, and thus obtain a bound for the energy flux globallyin the black hole interior completing the proof of Theorem 1.2. In subsection 4.5 weprove boundedness of the weighted higher order energies. Using the conclusion of thistheorem, we apply again Sobolev embedding as before and thus obtain the boundednessstatement of Theorem 1.1.

An Outlook of open problems will be given in Sect. 5. We first state an analogousresult to our Theorem1.1 for general subextremalKerr black holes (to appear as Theorem1.1 of the follow up paper). The conjectured blow up of the transverse derivatives4

along the Cauchy horizon for generic solutions of (1) will also be discussed, as well asthe peculiar extremal case. Finally, we will discuss what our results suggest about thenonlinear dynamics of the Einstein equations themselves.

2. Preliminaries

2.1. Energy currents and vector fields. The essential tool used throughout this workis the so called vector field method. Let (M, g) be a Lorentzian manifold. Let φ bea solution to the wave equation �gφ = 0. A symmetric stress-energy tensor can beidentified from variation of the massless scalar field action by

Tμν(φ) = ∂μφ∂νφ − 1

2gμνgαβ∂αφ∂βφ,

and this satisfies the energy-momentum conservation law

∇μTμν = 0. (6)

By contracting the energy-momentum tensor with a vector field V , we define the current

J Vμ (φ)

.= Tμν(φ)V ν . (7)

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 weintegrate J V

μ contracted with the normal vector field over an associated hypersurface wewill often refer to the integral as energy flux. Note that J V

μ (φ)nμ� ≥ 0 if V is future

directed timelike and � spacelike, where nμ� 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

K V (φ).= T (φ)(∇V ) = (πV )μνTμν(φ), (8)

by (6) it follows that

∇μ J Vμ (φ) = K V (φ). (9)

4 Note in contrast that the tangential derivatives of φ can be shown to be uniformly bounded up to CH+

(away from the bifurcation sphere) from the energy estimates proven in this paper together with commutingwith angular momentum operators, cf. Theorem 1.2.


Further, (πV )μν .= 12 (LV g)μν is the so called deformation tensor of V . Therefore,

∇μ J Vμ (φ) = 0 if V is 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 alreadythat in Reissner-Nordström spacetime we have �gT φ = 0 and �g�iφ = 0, where Tand �i with i = 1, 2, 3 are Killing vector fields that will be defined in Sects. 2.3.1and 2.3.2, respectively.

For a more detailed discussion see [21] by Dafermos and Rodnianski [26] by Klain-erman and [7] by Christodoulou.

2.2. The Reissner-Nordström solution. In the following we will briefly recall theReissner-Nordström solution5 which is a family of solutions to the Einstein-Maxwellfield equations

Rμν − 1

2gμν R = 2T EM

μν , (10)

with Rμν the Ricci tensor, R the Ricci scalar and the units chosen such that 8πGc4

= 2.The Maxwell equations are given by

∇α Fαβ = 0, ∇[λFαβ] = 0, (11)

and the energy-momentum tensor by

T EMμν = Fα

μ Fαν − 1

4gμν Fαβ Fαβ. (12)

The system (10)–(12) describes the interaction of a gravitational field with a sourcefree electromagnetic field. The Reissner-Nordström solution represents a charged blackhole as an isolated system in an asymptotically Minkowski spacetime. The causal struc-ture is similar to the structure of the astrophysically more realistic axisymmetric Kerrblack 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 maximaldomain 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) withcoordinates U, V ∈ (−1, 1) and thus

M = (−1, 1) × (−1, 1) × S2. (13)

The metric in global double null coordinates then takes the form

g = −Ω2(U, V )dUdV + r2(U, V )[dθ2 + sin2 θdϕ2

], (14)

where Ω2 and r will be described below.

5 The reader unfamiliar with this solution may for example consult [25] for a more detailed review.

608 A. T. Franzen

As a gauge condition we choose the hypersurface U = 0 and V = 0 to coincide withwhat will be the event horizons and we set

Ω2(0, V ) = 1

1 − V 2 , Ω2(U, 0) = 1

1 − U 2 , (15)

consistent with the fact that these hypersurfaces are to have infinite affine length. Fixparameters M > |e| �= 0. The Reissner-Nordström metric (14) in our gauge is uniquelydetermined from (10) to (12) by setting

r(0, V ) = r |HA+ = M +

√M2 − e2 = r+, r(U, 0) = r |HB

+ = M+√

M2 − e2 = r+.

(16)

Rearranging the Einstein-Maxwell equations (10) using (14) we obtain the followingHessian equation

∂U ∂V r = e2Ω2

4r3− Ω2

4r− ∂U r∂V r

r, (17)

from the U, V component. From the θ, θ or equivalently φ, φ component we obtain

∂U ∂V logΩ2 = −2∂U ∂V r

r(17)= −e2Ω2

2r4+

Ω2

2r2+2∂U r∂V r

r2, (18)

In fact, all relevant information about Reissner-Nordström geometry can be under-stood directly from (15) to (18) without explicit expressions forΩ2(U, V ) and r(U, V ).In particular, one can derive the Raychaudhuri equations

∂U

(∂U r

Ω2

)= 0, ∂V

(∂V r

Ω2

)= 0, (19)

from the above.We can illustrate the 2-dimensional quotient spacetime Q as a subset of an ambient

R1+1:Identifying U , V with ambient null coordinates of R1+1, the boundary of Q ⊂

R1+1 is given by ±1 × [−1, 1] ∪ [−1, 1] × ±1. Let us further define the darker

shaded region I I of Fig. 3 by Q|I I = [0, 1) × [0, 1). Particularly important isCH+ = CH+

A ∪ CH+B = 1 × (0, 1] ∪ (0, 1] × 1,which is the future boundary of the inte-

rior of region I I .We defineM|I I = π−1(Q|I I ), whereπ is the projectionπ : M → Q.

2.2.2. Eddington-Finkelstein coordinates. It will be convenient to rescale the globaldouble null coordinates and define

u = f (U ) = 2r+r+2 − e2

ln

∣∣∣∣ln∣∣∣∣ 1 + U

1 − U

∣∣∣∣∣∣∣∣ , v = h(V ) = 2r+

r+2 − e2ln

∣∣∣∣ln∣∣∣∣ 1 + V

1 − V

∣∣∣∣∣∣∣∣ .(20)

Note that u is the retarded and v is the advanced Eddington-Finkelstein coordinate. Thesecoordinates are both regular in the interior ofQ|I I , cf. Fig. 3. Nonetheless, we can viewthe whole of Q|I I as

Q|I I = [−∞,∞) × [−∞,∞), (21)


Fig. 3. Conformal diagram of the maximal domain of dependence of Reissner-Nordström spacetime

Fig. 4. Conformal diagram of darker shaded region I I , compare Fig. 3, with the ranges of (u, v) depicted

where we have formally parametrized by

HA+ = {−∞} × [−∞,∞), HB

+ = [−∞,∞) × {−∞} ,

as depicted in Fig. 4, see also (20).In u, v coordinates the metric is given by

g = −2(u, v)dudv + r2(u, v)[dθ2 + sin2 θdϕ2

], (22)

with

2(u, v) = Ω2(U, V )

∂U f ∂V h= −

(1 − 2M

r+

e2

r2

), (23)

where the unfamiliar minus sign on the right hand side arises since all definitions havebeen made suitable for the interior. We will often make use of the fact that by the choice

610 A. T. Franzen

of Eddington-Finkelstein coordinates (20) for the interior we have scaled our coordinatessuch that

∂ur

2 = −1

2,

∂vr

2 = −1

2. (24)

[The fact that the above expressions are constants follows from the Raychaudhuri equa-tions (19) and (19)]. Taking the derivatives of (23) with respect to u and v and using(24) it follows that

∂u

(u, v) = 1

2r2

(M − e2

r

),

∂v

(u, v) = 1

2r2

(M − e2

r

). (25)

2.3. Further properties of Reissner-Nordström geometry.

2.3.1. (t, r�) and (t, r) coordinates. It is useful to define the function t :M|I I → R by

t (u, v) = v − u

2, (26)

where M|I I = M|I I \∂M|I I is the interior ofM|I I . Moreover, we define the functionr� : M|I I → R by

r�(u, v) = v + u

2, (27)

where r� is usually referred to as theRegge-Wheeler coordinate.Note that for coordinates(t, r�, ϕ, θ) defined in M|I I we have that ∂

∂t is a spacelike Killing vector field whichextends to the globally defined Killing vector field T on M. By ϕτ we denote a 1-parameter group of diffeomorphisms generated by the Killing field T . We can moreoverrelate the functions r and r� by

dr� = dr

1 − 2Mr + e2

r2

, ⇒ r� = r +1

κ+ln

∣∣∣∣r − r+r+

∣∣∣∣ + 1

κ−ln

∣∣∣∣r − r−r−

∣∣∣∣ + C, (28)

where C is constant which is implicitly fixed by previous definitions,

r− = M −√

M2 − e2, (29)

and the surface gravities are given by

κ± = r± − r∓2r2±

. (30)

Note that κ+ is the surface gravity at H+ and κ− is the surface gravity at CH+. Thefunction r(u, v) extends continuously and is monotonically decreasing in both u and v

towards CH+ such that we have

r(u,∞) = r |CHA+ = r−, r(∞, v) = r |CHB

+ = r−. (31)


2.3.2. Angular momentum operators. We have already mentioned the generators ofspherical symmetry �i , i = 1, 2, 3, in Sect. 2.1. They are explicitly given by

�1 = sin ϕ∂θ + cot θ cosϕ∂ϕ, �2 = − cosϕ∂θ + cot θ sin ϕ∂ϕ, �3 = −∂ϕ,

(32)

which satisfy

3∑i=1

(�iφ)2 = r2|∇/φ|2,3∑

i=1

3∑j=1

(�i� jφ)2 = r4|∇/ 2φ|2, (33)

where we define

|∇/φ|2 = 1

r2

[(∂θφ)2 +

1

sin2 θ(∂ϕφ)2

]. (34)

2.3.3. The redshift, noshift and blueshift region. As we have already mentioned in theintroduction, in the interior we can distinguish

redshift R = {rred ≤ r ≤ r+}, noshift N = {rblue ≤ r ≤ rred},blueshift B = {r− ≤ r ≤ rblue} (35)

subregions, as shown in Fig. 5, for values rred, rblue 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 avector field N such that its associated current J N

μ nμv=const on a v = const hypersurface

can be controlled by the related bulk term K N , cf. Proposition 4.1. This positivity of thebulk term K N is only possible sufficiently close to H+. In particular we shall define

rred = r+ − ε, (36)

with ε > 0 and small enough such that Proposition 4.1 is applicable. (Furthermore, notethat the quantity M − e2

r is always positive in R.)As defined in (35) the r coordinate in the noshift region N ranges between rred

defined by (36) and rblue, defined below, strictly bigger than r−. In N we exploit thefact that J−∂r and K −∂r are invariant under translations along ∂t . For that reason we canuniformly control the bulk by the current along a constant r hypersurface. This will beexplained further in Sect. 4.1.2.

Fig. 5. Region I I with distinction into redshift R, noshiftN and blueshift B regions

612 A. T. Franzen

The blueshift region B is characterized by the fact that the bulk term K S0 associatedto the vector field S0 to be defined in (44) is positive. We define

rblue = r− + ε, (37)

with ε > 0 for an ε such that M − e2rblue

carries a negative sign and such that (forconvenience)

r�(rblue) > 0. (38)

In particular, in view of (25) and (25) for ε sufficiently small the following lower boundholds in B

0 < β ≤ −∂u

, 0 < β ≤ −∂v

, (39)

with β a positive constant.

2.4. Notation. Wewill describe certain regions derived from the hypersurfaces r = rred,r = rblue and in addition the hypersurface γ which will be defined in Sect. 4.1.3.1. Forexample given the hypersurface r = rred and the hypersurface u = u we define the v

value at which these two hypersurfaces intersect by a function vred(u) evaluated for u.Let us therefore introduce the following notation:

vred(u) is determined by r(vred(u), u) = rred, ured(v) is determined by r(ured(v), v) = rred,vγ (u) is determined by (vγ (u), u) ∈ γ , uγ (v) is determined by (uγ (v), v) ∈ γ ,vblue(u) is determined by r(vblue(u), u) = rblue, ublue(v) is determined by r(ublue(v), v) = rblue.

(40)

For a better understanding the reader may also refer to Fig. 6.Note that the above functions are well defined since r = rred, r = rblue and γ are

spacelike hypersurfaces terminating at i+.

3. The Setup

3.1. Horizon estimates and Cauchy stability. Our starting point will be previouslyproven decay bounds for φ and its derivatives in the black hole exterior up to andincluding the event horizon; in particular we can state:

Fig. 6. Sketch of blueshift region B, with quantities depicted a dependent on u and b dependent on v


Theorem 3.1. Let φ be a solution of the wave equation (1) on a subextremal Reissner-Nordström background (M, g), with mass M and charge e and M > |e| �= 0, arisingfrom smooth compactly supported initial data on an arbitrary Cauchy hypersurface �,cf. Fig. 1. Then, there exists δ > 0 such that

∫

S2

v+1∫

v

[(∂v�kφ)2(−∞, v) + 2|∇/�kφ|2(−∞, v)

]r2dvdσS2 ≤ Ckv−2−2δ, for all k ∈ N

0,

(41)

on HA+, for all v and some positive constants Ck depending on the initial data.6

Proof. The Theorem follows by putting together work of Blue and Soffer [6] on inte-grated 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 theredshift vector field energy identity of [15] yields integrated local energy decay withoutdegeneration on the horizon. One can then apply the black box result of [17] to obtainpolynomial decay onH+

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 infact take δ arbitrarily close to 1

2 , but δ > 0 is sufficient for our purposes and allows inprinciple for a larger class of data on �.

On the other hand, trivially from Cauchy stability, boundedness of the energy alongthe 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 thefollowing proposition.

Proposition 3.2. Let u�, v� ∈ (−∞,∞). Under the assumption of Theorem 3.1, theenergy at advanced Eddington-Finkelstein coordinate {v = v�} ∩ {−∞ ≤ u ≤ u�} isbounded from the initial data

∫

S2

u�∫

−∞

[−2(∂u�kφ)2(u, v�) + 2|∇/�kφ|2(u, v�)

]r2dudσS2 ≤ Dk(u�, v�), (42)

and further

sup−∞≤u≤u�

∫

S2

(�kφ)2(u, v�)dσS2 ≤ Dk(u�, v�), for all k ∈ N0, (43)

with Dk(u�, v�) positive constants depending on the initial data on �.

Proof. This follows immediately from local energy estimates in a compact spacetimeregion. Note the −2 and 2 weights which arise since u is not regular at H+

A. �6 The notation�k denotes summation over angular momentum operators as defined in Sect. 4.2.1 by (128).

614 A. T. Franzen

Fig. 7. a Characteristic rectangle � in the interior of Reissner-Nordström spacetime, b region � zoomed in

3.2. Statement of the theorem and outline of the proof in the neighbourhood of i+. Themost difficult result of this paper can now be stated in the following theorem.

Theorem 3.3. On subextremal Reissner-Nordström spacetime with M > |e| �= 0, let φ

be as in Theorem 3.1, then

|φ| ≤ C

locally in the black hole interior up to CH+ in a “small neighbourhood” of timelikeinfinity i+, that is in (−∞, u✂] × [1,∞) for some u✂ > −∞.

Remark. We will see that C depends only on the initial data.Wewill consider a characteristic rectangle� extending fromHA

+ as shown in Fig. 7.We pick the characteristic rectangle to be defined by � = {(−∞ ≤ u ≤ u✂), (1 ≤ v

< ∞)}, where u✂ is sufficiently close to −∞ for reasons that will become clear lateron, cf. Proposition 4.11. As described in Sect. 3.1, from bounds of data on � bounds onthe 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 theblueshift B region, with the properties as explained in Sect. 2.3.3, cf. Fig. 7b). Thisdistinction is made since different vector fields have to be employed in the differentregions.7

In the redshift region R we will make use of the redshift vector field N of [21] onwhich we will elaborate more in Sect. 4.1.1. Proposition 4.1 gives the positivity of thebulk K N which thus bounds the current J N

μ nμv=const from above.Applying the divergence

theorem, decay up to r = rred 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 energyflux J−∂r

μ nμr=r through the r = r hypersurface. Decay up to r = rblue 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 = rblue, cf. Sect. 4.1.3.1. We will thenseparately consider the region to the past of γ , J−(γ ) ∩ B and the region to the futureof γ , J+(γ ) ∩ B. The region to the future of γ is characterized by good decay boundson 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 − e2r 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

S0 = rq∂r� = rq(∂u + ∂v), (44)

where q is sufficiently large, cf. Sect. 4.1.3. We will see that for the right choice of q wecan render the associated bulk term K S0 positive which is the “good” sign when usingthe divergence theorem.

In order to complete the proof, we consider finally the region J+(γ ) ∩ B and prop-agate the decay further from the hypersurface γ up to the Cauchy horizon in a neigh-bourhood of i+. For this, we introduce a new timelike vector field S defined by

S = |u|p∂u + v p∂v, (45)

for an arbitrary p such that

1 < p ≤ 1 + 2δ, (46)

where δ is as in Theorem 3.1. We use pointwise estimates on 2 in J+(γ ) as a crucialstep, cf. Sect. 4.1.4.1.

Putting everything together, in view of the geometry and the weights of S, we finallyobtain for all v∗ ≥ 1

∫

S2

v∗∫

1

v p(∂vφ)2r2dvdσS2 ≤ Data, (47)

for the weighted flux. Using the above, the uniform boundedness for φ stated in Theo-rem 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 thefundamental theorem of calculus and the Cauchy-Schwarz inequality one obtains

∫

S2

φ2(u, v∗, θ, ϕ)dσS2 ≤ C∫

S2

⎛⎝

v∗∫

1

v p(∂vφ)2dv

⎞⎠⎛⎝

v∗∫

1

v−pdv

⎞⎠ r2dσS2 + data,

where the first factor of the first term is controlled by (47). Therefore, we further get

∫

S2

φ2dσS2(47)≤ Data

∫

S2

v∗∫

1

v−pdvdσS2 + data ≤ Data + data, (48)

where we have used∞∫1

v−pdv < ∞ 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 inReissner-Nordström geometry we have�g�iφ = 0, where�i with i = 1, 2, 3 are the 3spacelike Killing vector fields resulting from the spherical symmetry. Thus one obtainsthe analogue of (48) but with �iφ and �i� jφ in place of φ. Using Sobolev embeddingon S2 thus leads immediately to the desired bounds. See Sect. 4.2.3. This will close theproof of Theorem 3.3.

616 A. T. Franzen

Fig. 8. RegionsRI and RI

4. 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 thedecay 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 Cauchyhorizon. As outlined in Sect. 1.3, we first consider a characteristic rectangle � in theneighbourhood 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 underlyingsubregion, which will be described further in the following subsections.

4.1. Energy estimates in the neigbourhood of i+.

4.1.1. Propagating through R from H+ to r = rred. The estimates in this and thefollowing section are motivated by work of Luk [30]. He proves that any polynomialdecay estimate that holds along the event horizon of Schwarzschild black holes can bepropagated to any constant r hypersurface in the black hole interior. This followed aprevious 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 ther = rred hypersurface.

The rough idea can be understood with the help of Fig. 8. By Theorem 3.1 we aregiven energy decay on the event horizon H+, see dash-dotted line. By using the energyidentity for the vector field N in regionRI = {rred ≤ r ≤ r+} ∩ {1 < v ≤ v∗}, the coareaformula etc., we obtain decay of the flux through constant v hypersurfaces throughout theentire region. Using this result and considering the energy identity once again in regionRI = {rred ≤ r ≤ r+} ∩ {v∗ ≤ v ≤ v∗ + 1} we eventually obtain decay on the r = rredhypersurface, 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 neighbourhoodof a Killing horizonH+ depends only on the positivity of the surface gravity, in this caseκ+. Thus by (30) the following proposition follows by Theorem 7.1 of [21].


Proposition 4.1. (Dafermos and Rodnianski). For rred sufficiently close to r+ there existsaϕτ -invariant8 smooth future directed timelike vector field N on {rred ≤ r ≤ r+} ∩ {v ≥ 1}and a positive constant b1 such that

b1 J Nμ (φ)nμ

v ≤ K N (φ), (49)

for all solutions φ of �gφ = 0.

The decay bound along r = rred can now be stated in the following proposition.

Proposition 4.2. Let φ be as in Theorem 3.1. Then, for all r ∈ [rred, r+), with rred as inProposition 4.1 and for all v∗ > 1,∫

{v∗≤v≤v∗+1}J Nμ (φ)nμ

r=r dVolr=r ≤ Cv−2−2δ∗ ,

with C depending on C0 of Theorem 3.1 and D0(u�, 1) of Proposition 3.2, where u� isdefined by rred = r(u�, 1).

Remark 1. The decay in Proposition 4.2 matches the decay onH+ of Theorem 3.1.

Remark 2. nμr=rred denotes the normal to the r = rred hypersurface oriented according to

Lorentzian geometry convention. dVol denotes the volume element over the entire space-time region and dVolr=rred denotes the volume element on the r = rred hypersurface.Similarly for all other subscripts.9

Proof. Applying the divergence theorem, see e.g. [21] or [45], in regionRI = {rred ≤ r≤ r+} ∩ {v0 ≤ v ≤ v∗}, with v0 ≥ 1, we obtain∫

RI

K N (φ)dVol +∫

{v0≤v≤v∗}J Nμ (φ)nμ

r=rreddVolr=rred +∫

{rred≤r≤r+}J Nμ (φ)nμ

v=v∗dVolv=v∗

=∫


v=v0dVolv=v0 +

∫


H+dVolH+ .

We immediately see that the second term on the left hand side is positive since r = rredis a spacelike hypersurface and N is a timelike vector field. Therefore, we write∫

RI

K N (φ)dVol +∫


v=v∗dVolv=v∗

≤∫


v=v0dVolv=v0 +

∫


H+dVolH+ . (50)

By Theorem 3.1 we have∫


H+dVolH+ ≤ C0max {v∗ − v0, 1} v−2−2δ0 .

8 cf. Sect. 2.3.9 Refer to “Appendix A” for further discussion of the volume elements.

618 A. T. Franzen

Using that the energy current associated to the timelike vector field N is controlled bythe deformation K N as shown in (49) and substituting

E(φ; v) =∫


v=vdVolv=v (51)

into (50) as well as using the coarea formula

∫

RI

J Nμ (φ)nμ

v=vdVol ∼

v∗∫

v0

∫


v=vdVolv=vdv, (52)

for the bulk term,10 we obtain for all v0 ≥ 1 and v∗ > v0, the relation

E(φ; v∗) + b1

v∗∫

v0

E(φ; v)dv ≤ E(φ; v0) + C0max {v∗ − v0, 1} v−2−2δ0 . (53)

Note by Proposition 3.2 with k = 0, applied to u� defined through the relation rred =r(u�, 1), we have

E(φ; 1) ≤ C D0(u�, 1), (54)

since the vector field N is regular atH+ and thus E(φ; 1) is comparable to the left handside of (42). In order to obtain estimates from (53) we appeal to the following lemma.

�Lemma 4.3. Let f be a continuous function, f : [1,∞) → R

+,

f (t) + b

t∫

t

f (t)dt ≤ f (t) + C0(t − t + 1)t− p, (55)

for all t ≥ 1, where C0, p are positive constants. Then for any t ≥ 1 we have

f (t) ≤ Ct− p, (56)

where C depends only on f (1), b and C0.

Proof. For t > t0, we will show (56) by a continuity (bootstrap) argument. It suffices toshow that

f (t) ≤ 2C t− p, for t ≤ t, (57)

leads to

⇒ f (t) ≤ C t− p, for t ≤ t, (58)

for some large enough constant C .

10 where f ∼ g means that there exist constants 0 < b < B with b f < g < B f


We note first that given any t0, from (55) we obtain, ∀ 1 ≤ t ≤ t0

f (t) ≤ f (1) + C0 t ≤ [ f (1)t0p + C0 t0

p+1]t− p. (59)

Given t ≥ t0, choose t = t − L for an L to be determined later. Moreover, t0 will haveto be chosen large enough so that ∀ t ≥ t0,

(t − L)− p = t− p < 2t− p. (60)

Given a t satisfying (57) applying (55) yields

f (t) + b

t∫

t

f (t)dt ≤ [2C + C0(L + 1)]t− p (60)≤ [4C + 2C0(L + 1)]t− p. (61)

Further, by the pigeonhole principle, there exists tin ∈ [t, t] such that

f (tin) ≤ 1

L

t∫

t

f (t)dt . (62)

Since f (t) is a positive function (61) also leads to

b

t∫

t

f (t)dt ≤ [4C + 2C0(L + 1)]t− p. (63)

Thus, (62) and (63) yield

f (tin) ≤ 1

bL[4C + 2C0(L + 1)]t− p. (64)

Now let t = tin and use (64) in (55), then

f (t) ≤ f (t) + b

t∫

tin

f (t)dt ≤ 1

bL[4C + 2C0(L + 1)]t− p + C0(L + 1)t− p

in ,

(60)≤[4C

bL+2C0(L + 1)

bL+ 2C0(L + 1)

]t− p. (65)

If 1 − 4bL > 0 and

C ≥(1 − 4

bL

)−1 [2C0(L + 1)

bL+ 2C0(L + 1)

](66)

then (58) follows.Thus picking first L such that 1 − 4

bL > 0, and then t0 such that t0 ≥ L + 1 and

satisfying (60), and finally choosing C as C = max{[

f (1) + C0t0−1−2δ],(1 − 4

bL

)−1

[2C0(L+1)

bL + 2C0(L + 1)]}

(58) and thus (56) follows by continuity. �

620 A. T. Franzen

By Lemma 4.3 we obtain from (53) together with (54)

E(φ; v∗) =∫


v=v∗dVolv=v∗ ≤ ˜Cv∗−2−2δ, (67)

with ˜C depending on b1 and D0(u�, 1).Finally, in order to close the proof of Proposition 4.2 we perform again the divergence

theorem but for region RI = {rred ≤ r ≤ r+} ∩ {v∗ ≤ v ≤ v∗ + 1}:∫

RI

K N (φ)dVol +∫


r=rreddVolr=rred

+∫


v=v∗+1dVolv=v∗+1 =∫


v=v∗dVolv=v∗

+∫


H+dVolH+ . (68)

In view of the signs we obtain

⇒∫


r=rreddVolr=rred ≤∫


v=v∗dVolv=v∗

+∫


H+dVolH+ .

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.

Corollary 4.4. Let φ be as in Theorem 3.1 and for rred as in Proposition 4.1. Then, forall v∗ ≥ 1, v∗ + 1 ≤ vred(u) and for all u such that r(u, v∗ + 1) ∈ [rred, r+), we have

∫


u=udV olu=u ≤ Cv−2−2δ∗ , (69)

with C depending on C0 of Theorem 3.1 and D0(u�, 1) of Proposition 3.2, where u� isdefined by rred = r(u�, 1) and vred(u) as in (40).

Proof. The conclusion of the statement follows by applying again the divergence theo-rem and using the results of the proof of Proposition 4.2. �


4.1.2. Propagating through N from r = rred to r = rblue. Now that we have obtained adecay bound along the r = rred hypersurface in the previous section, we propagate theestimate further inside the black hole through the noshift region N up to the r = rbluehypersurface. In order to do that we will use the future directed timelike vector field

− ∂r = 1

2 (∂u + ∂v). (70)

Using (70) in (B2) of “Appendix B” we obtain

K −∂r = 4

4

[∂u

(∂vφ)2 +

∂v

(∂uφ)2

]− 4

r2 (∂uφ∂vφ), (71)

for the bulk current. It has the property that it can be estimated by

|K −∂r (φ)| ≤ B1 J−∂rμ (φ)nμ

r=r , (72)

where B1 is independent of v∗. Validity of the estimate can in fact be seen withoutcomputation from the fact that timelike currents, such as J−∂r

μ (φ)nμr=r contain all deriv-

atives. The uniformity of B1 is given by the fact that K −∂r and J−∂r are invariant undertranslations along ∂t , cf. Sect. 2.3.1 for definition of the t coordinate. Therefore, wecan just look at the maximal deformation on a compact {t = const} ∩ {rblue ≤ r ≤ rred}hypersurface and get an estimate for the deformation everywhere.

Proposition 4.5. Letφ be as in Theorem 3.1, rblue as in (37)andrred as in Proposition 4.1.Then, for all v∗ > 1 and r ∈ [rblue, rred), we have∫

{v∗≤v≤v∗+1}J−∂rμ (φ)nμ

r=r dVolr=r ≤ Cv∗−2−2δ,

with C depending on the initial data or more precisely depending on C0 of Theorem 3.1and D0(u�, 1) of Proposition 3.2, where u� is defined by rred = r(u�, 1).

Proof. Given v∗, we define regions RI I and RI I as in Fig. 9, where we use (40) and

v(r , v∗) is determined by r(ublue(v∗), v(r , v∗)) = r . (73)

Thus the depicted regions are given byRI I ∪RI I = D+({v1 ≤ v ≤ v∗ + 1}∩{r = rred})∩ N , where region RI I is given byRI I = D+({v1 ≤ v ≤ v∗} ∩ {r = rred}).

In the following we will apply the divergence theorem in regionRI I ∪ RI I to obtaindecay on an arbitrary r = r hypersurface, dash-dotted line, for r ∈ [rblue, rred), fromthe derived decay on the r = rred hypersurface.∫

RI I ∪RI I

K −∂r (φ)dVol +∫

{rblue≤r≤rred}J−∂rμ (φ)nμ

u=ublue(v∗)dVolu=ublue(v∗)

+∫


r=rbluedVolr=rblue +∫

{rblue≤r≤rred}J−∂rμ (φ)nμ

v=v∗+1dVolv=v∗+1

=∫

{v1≤v≤v∗+1}J−∂rμ (φ)nμ

r=rreddVolr=rred .

622 A. T. Franzen

Fig. 9. RegionRI I ∪ RI I represented as the hatched area

The second integral of the left hand side represents the current through the u = ublue(v∗)hypersurface, defined by (40). As u = ublue(v∗) is a null hypersurface and −∂r istimelike, the positivity of that second term is immediate. Similarly, the fourth term ofthe left hand side of our equation is positive and we obtain

∫


r=rbluedVolr=rblue ≤∫

RI I ∪RI I

|K −∂r (φ)|dVol

+∫


r=rreddVolr=rred .

Further, we use that the deformation K −∂r is controlled by the energy associated tothe timelike vector field −∂r as stated in (72). Thus we obtain

∫


r=rbluedVolr=rblue ≤ B1

∫

RI I ∪RI I

J−∂rμ (φ)nμ

r=rdVol

+∫


r=rreddVolr=rred .

(74)

By the coarea formula we obtain

∫


r=rbluedVolr=rblue ≤ B1

rred∫

rblue

∫

{v(r ,v∗)≤v≤v∗+1}J−∂rμ (φ)nμ

r=rdVolr=rdr

+∫


r=rreddVolr=rred . (75)


Now let

E(φ; r , v) =∫

{v≤v≤v∗+1}J−∂rμ (φ)nμ

r=rdVolr=r , (76)

with r ∈ [rblue, rred). Replacing rblue with r in the above, considering the future domainof dependence of {v1 ≤ v ≤ v∗ + 1}∩ {r = rred} up to the r = r hypersurface we obtainsimilarly to (75)

E(φ; r , v(r , v∗)) ≤ B1

rred∫

r

E(φ; r , v(r , v∗))dr + E(φ; rred, v1). (77)

Using Grönwall’s inequality in (77) yields

E(φ; r , v(r , v∗)) ≤ E(φ; rred, v1)[1 + B1(rred − r)eB1(rred−r)]⇒ E(φ; r , v(r , v∗)) ≤ C E(φ; rred, v1). (78)

Finally, note that

[v∗ + 1] − v(rred, v∗) = [v∗ + 1] − v1 = k < ∞, (79)

where k = 2[r�(rblue) − r�(rred)] + 1. This can be seen since (24) and (28) yields

∂vr

2 = −∂vr� = −1

2

⇒ r�(ublue(v∗), v∗) − r�(ublue(v∗), v1) = r�(rblue) − r�(rred)(27)= 1

2(v∗ − v1).

Further, by using the conclusion of Proposition 4.2 and (79) we have

E(φ; rred, v1) =∫


r=rreddVolr=rred

=∫

{v1≤v≤v1+k}J−∂rμ (φ)nμ

r=rreddVolr=rred

≤ Cmax {k, 1} v1−2−2δ ∼ Cv∗−2−2δ. (80)

We thus infer

∫


r=rdVolr=r ≤ C∫


r=rred dVolr=rred ≤ CCv∗−2−2δ.

�The above now also implies the following statement.

624 A. T. Franzen

Corollary 4.6. Let φ be as in Theorem 3.1, rblue as in (37) and rred as in Proposition 4.1.Then, for all v∗ > 1 and all u such that r(u, v∗) ∈ [rblue, rred)∫

{vred(u)≤v≤vblue(u)}J Nμ (φ)nμ

u=udV olu=u ≤ Cv−2−2δ∗ , (81)

with C depending on C0 of Theorem 3.1 and D0(u�, 1) of Proposition 3.2, where u� isdefined by rred = r(u�, 1) and vred(u), vblue(u) are as in (40).

Proof. The conclusion of the statement follows by considering the divergence theoremfor a triangular region J−(x) ∩ N with x = (u, vblue(u)), x ∈ J−(r = rblue) and usingthe results of Proposition 4.5. Note that v∗ ∼ vblue(u) ∼ vred(u). �

By the previous proposition we have successfully propagated the energy estimatefurther inside the black hole, up to r = rblue. To go even further will be more difficultand we will address this in the next section.

4.1.3. Propagating through B from r = rblue to the hypersurface γ . In the followingwe want to propagate the estimates from the r = rblue hypersurface further into theblueshift region to a hypersurface γ which is located a logarithmic distance in v fromthe r = rblue hypersurface, cf. Fig. 10. We will define the hypersurface γ and its mostbasic properties in Sect. 4.1.3.1 and propagate the decay bound to γ in Sect. 4.1.3.2.

4.1.3.1. The hypersurface γ . The idea of the hypersurface γ was already entertained in[13] by Dafermos and basically locates γ a logarithmic distance in v from a constant rhypersurface living in the blueshift region.

Let α be a fixed constant satisfying

α >p + 1

β, (82)

with β as in (39) and (39). [The significance of the bound (82) will become clear later].Let us for convenience also assume that

α > 1, (83)

and

α(2 − log 2α) > 2r�blue + 1. (84)

Fig. 10. Logarithmic distance of hypersurface r = rblue and hypersurface γ depicted in a Penrose diagram


We define the function H(u, v) by

H(u, v) = u + v − α log v − 2r�(rblue) = u + v − α log v − 2r�blue, (85)

were r�(rblue) = r�blue is the r� value evaluated at rblue according to (28), and r�

blue > 0according to the choice (38). We then define the hypersurface γ as the levelset

γ = {H(u, v) = 0} ∩ {v > 2α}. (86)

Since

∂ H

∂u= 1,

∂ H

∂v= 1 − α

v, (87)

we see that γ is a spacelike hypersurface and terminates at i+, cf. “Appendix A”. (In thenotation (40), uγ (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

r�(rblue) = vblue(u) + u

2, (88)

with vblue(u) as in (40). Plugging this into (86) recalling vγ (u) defined in (40), we obtainthe relation

vγ (u) − vblue(u) = α log vγ (u). (89)

Aswe shall see in Sect. 4.1.4.1 the above properties of γ will allow us to derive pointwiseestimates of 2 in J+(γ ) ∩ B. We first turn however to the region J−(γ ) ∩ B.

4.1.3.2. Energy estimates from r = rblue to the hypersurface γ . Now we are ready topropagate the energy estimates further into the blueshift region B up to the hypersurfaceγ . We will in this part of the proof use the vector field

S0 = rq∂r� = rq(∂u + ∂v),

which we have defined in (44). Let us now consider the bulk term and derive positivityproperties which are needed later on. Plugging (44) in (B2) of “Appendix B” yields

K S0 = qrq−1[(∂vφ)2 + (∂uφ)2

]

−[

qrq−1

2[∂vr + ∂ur ] + rq

(∂u

+

∂v

)]|∇/φ|2 − 4rq−1(∂uφ∂vφ). (90)

Our aim is to show that K S0 is positive. All terms multiplied by the angular derivativesare manifestly positive in B, cf. (39), (39) together with (24) to (25). Therefore, it is onlyleft to show that the first term on the right hand side dominates the last term. Since bythe Cauchy-Schwarz inequality

2qrq−1(∂uφ∂vφ) ≤ qrq−1[(∂vφ)2 + (∂uφ)2], (91)

K S0 is positive in B for all q ≥ 2.We show now that at the expense of one polynomial power, we can extend the local

energy estimate on r = rblue to an energy estimate along γ which is valid for a dyadiclength.

626 A. T. Franzen

Proposition 4.7. Let φ be as in Theorem 3.1. Then, for all v∗ > 2α∫

{v∗≤v≤2v∗}J S0μ (φ)nμ

γ dVolγ ≤ Cv∗−1−2δ, (92)

on the hypersurface γ , with C depending on C0 of Theorem 3.1 and D0(u�, 1) of Propo-sition 3.2, where u� is defined by rred = r(u�, 1).

Remark. nμγ denotes the normal vector on the hypersurface γ which is a levelset

γ = {H(u, v) = 0} of the function H(u, v) defined in (85). For calculation of nμγ and

J S0μ (φ)nμ

γ refer to (A1) and (A5) of “Appendix A”.

Proof. In the following we will again make use of notation (40).Let v∗ > 2α, such that γ is spacelike for v > v∗, cf. Sect. 4.1.3.1. Define u3 by

(u3, v∗) ∈ γ , i.e. uγ (v∗) = u3 and define vblue as the intersection of u3 with rblue,i.e. vblue(u3) = vblue. And similarly the hypersurfaces u = u1 and u = u2 as shown inFig. 10 are given by ublue(2v∗) = u1 and uγ (2v∗) = u2. Having defined the relationsbetween all these quantities we can now carry out the divergence theorem for regionRI I I :∫

RI I I

K S0(φ)dVol +∫

{vblue≤v≤v∗}J S0μ (φ)nμ

u=udVolu=u +∫


γ dVolγ

+∫

{u1≤u≤u2}J S0μ (φ)nμ

v=2v∗dVolv=2v∗ =∫

{vblue≤v≤2v∗}J S0μ (φ)nμ

r=rbluedVolr=rblue . (93)

Positivity of the flux along the u = u3 segment and the flux along the v = 2v∗segment, as well as positivity of K S0 for the choice q ≥ 2, which was derived in (90)and (91), leads to∫


γ dVolγ ≤∫

{vblue≤v≤2v∗}J S0μ (φ)nμ

r=rbluedVolr=rblue

≤ Cmax {2v∗ − vblue, 1} v−2−2δblue

(89)≤ C (v∗ + α log v∗) v−2−2δblue ≤CCv−1−2δ∗ , (94)

where the second step is implied by Proposition 4.5 and the last step follows from theinequality v∗ ≤ Cvblue 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 weightedenergy estimate along the hypersurface γ .

Corollary 4.8. Let φ be as in Theorem 3.1. Then, for all v∗ > 2α∫

{v∗≤v<∞}v p J S0

μ (φ)nμγ dVolγ ≤ Cv

−1−2δ+p∗ , (95)

on the hypersurface γ , with C depending on C0 of Theorem 3.1 and D0(u�, 1) of Propo-sition 3.2, where u� is defined by rred = r(u�, 1) and p as in (46).


Proof. This follows by weighting (92) with vp∗ and summing dyadically. �

Further, we can state the following.

Corollary 4.9. Let φ be as in Theorem 3.1, rblue as in (37) and γ as in (86). Then, forall v∗ > 2α and for all u ∈ [ublue(v∗), uγ (v∗))

∫

{vblue(u)≤v≤vγ (u)}v p J S0

μ (φ)nμ

u=udV olu=u ≤ Cv−1−2δ+p∗ , (96)

with C depending on C0 of Theorem 3.1 and D0(u�, 1) of Proposition 3.2, where u� isdefined by rred = r(u�, 1) and vγ (u), vblue(u) as in (40).

Proof. The proof is similar to the proof of Corollary 4.6 by considering the divergencetheorem for a triangular region J−(x) ∩ B with x = (u, vγ (u)), x ∈ J−(γ ) and usingthe results of the proof of Proposition 4.7. �

4.1.4. Propagating through B from the hypersurface γ to CH+ in the neighbourhood ofi+. In order to prove our Theorem 3.3 and close our estimates up to the Cauchy horizonin the neighbourhood of i+ we are interested in considering a region RI V within thetrapped region whose boundaries are made up of the hypersurface γ , a constant u and aconstant v segment, which can reach up to the Cauchy horizon, cf. Fig. 11.

Let v∗ > 2α and let v > v∗. We may write RI V = J+(γ ) ∩ J−(x) withx = (uγ (v∗), v), x ∈ J+(γ ) ∩ B. Note that RI V lies entirely in the blueshift region,

which was characterized by the fact that the quantity M − e2r 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 hyper-

surface γ . With this estimate, the bulk term will be bounded in terms of the currentsthrough the null hypersurfaces. Consequently, we will be able to absorb the bulk termand to show that the currents through the null hypersurfaces can be bounded by thecurrent along the hypersurface γ , cf. Sect. 4.1.4.3.

4.1.4.1. Pointwise estimates on 2 in J+(γ ). In the following we will derive pointwiseestimates on 2 in J+(γ ). We note that these will imply that the spacetime volume tothe future of the hypersurface γ is finite, Vol(J+(γ )) < C .

Fig. 11. Blueshift region of Reissner-Nordström spacetime from hypersurface γ onwards

628 A. T. Franzen

We first derive a future decay bound along a constant u hypersurface for the function2(u, v) for (u, v) ∈ B. Let x = (ufix, vfix), x ∈ B, then, from (25) we can immediatelysee that

log(2(ufix, v)

)∣∣∣vvfix

=v∫

vfix

1

2r2

(M − e2

r

)dv

(39)≤ −β[v − vfix]. (97)

It then immediately follows that

2(u, v)≤2(u, vfix)e−β[v−vfix], for all (u, vfix) ∈ B and v > vfix. (98)

Analogously, we obtain

2(u, vfix)≤2(ufix, vfix)e−β[u−ufix], for all (u, vfix) ∈ J+(x), (99)

and plugging (98) into (99) it yields

2(u, v)≤2(ufix, vfix)e−β[u−ufix+v−vfix], for all (u, v) ∈ J+(x). (100)

From (98) and (89) we obtain a relation for2(u, v) on the hypersurface γ as follows

2(u, v)≤2(u, vblue(u))e−βα log vγ (u) = 2(u, vblue(u))vγ (u)−βα, for (u, v) ∈ γ .

(101)

For J+(γ ), using (98) we further get

2(u, v) ≤ Cvγ (u)−βαe−β[v−vγ (u)], for (u, v) ∈ J+(γ ), (102)

where we have used 2(u, vblue(u)) ≤ C . Moreover, we may think of a parameter v

which determines the associated u value via intersection with γ , we denote this valueby the evaluation the function uγ (v) which was introduced in (40), cf. Fig. 6b).

Moreover, by (25) we can also state

2(u, v) ≤ C |uγ (v)|−βαeβ[uγ (v)−u] for (u, v) ∈ J+(γ ). (103)

Note that the choice (82) of α implies that βα > 1. From (103), the fact that |uγ (v)| ∼ v,and the extra exponential factor, finiteness of the spacetime volume to the future of γ

follows,

Vol(J+(γ )) < C. (104)

See also [27].


4.1.4.2. Bounding the bulk term K S. To derive energy estimates in RI V we use thetimelike vector field multiplier

S = |u|p∂u + v p∂v,

which we have given before in (45). The weights of S are chosen such that they willallow 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 scalarcurrent K S , in terms of J S

μ(φ)nμv=v and J S

μ(φ)nμu=u . 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

K S = −2

r

[v p + |u|p] (∂uφ∂vφ) −

[∂u

|u|p +

∂v

v p +

p

2(v p−1 + |u|p−1)

]|∇/φ|2.

(105)

Recall (39) and (39). For large absolute values of v and u the first two terms mul-tiplying the angular derivatives of φ dominate the last two terms, so in total the termmultiplying the angular derivatives is always positive in D+(γ ). Consequently we willbe able to use this property to derive an inequality by using the divergence theorem inthe proof of Proposition 4.11. Let us therefore define

K S = − 2

r

[v p + |u|p] (∂uφ∂vφ), (106)

and state

− K S ≤ |K S| for v >p2β and |u| >

p2β , cf. (39). (107)

(Note that K S coincides with the bulk term for spherically symmetric φ.) We have thefollowing

Lemma 4.10. Let φ be an arbitrary function. Then, for all v∗ > 2α and all v > v∗, theintegral over region RI V = J+(γ ) ∩ J−(x) with x = (uγ (v∗), v), x ∈ B, cf. Fig. 11,of the current K S, defined by (106), can be estimated by

∫

RI V

|K S|dVol ≤ δ1 supuγ (v)≤u≤uγ (v∗)

∫

{vγ (u)≤v≤v}J Sμ(φ)nμ

u=udVolu=u

+ δ2 supv∗≤v≤v

∫

{uγ (v)≤u≤uγ (v)}J Sμ(φ)nμ

v=vdVolv=v ,

where δ1 and δ2 are positive constants, with δ1 → 0 and δ2 → 0 as v∗ → ∞.

Remark. In the proof of Proposition 4.11 we will see that the above proposition deter-mines u✂ of Theorem 3.3, depicted in Fig. 7b). We have to choose u✂ = uγ (v∗), withv∗ such that δ1 is small.

630 A. T. Franzen

Proof. Using the Cauchy-Schwarz inequality twice for the remaining part of the bulkterm we obtain

|K S| ≤ 1

r

[(1 +

|u|p

v p

)v p(∂vφ)2 +

(1 +

v p

|u|p

)|u|p(∂uφ)2

], (108)

with the related volume element

dVol = r22

2dudvdσ 2

S. (109)

Note that the currents related to the vector field S with their related volume elements aregiven by

J Sμ(φ)nμ

v=v = 2

2

[|u|p(∂uφ)2 +

2

4v p|∇/φ|2

], dVolv=v = r2

2

2dσS2du, (110)

J Sμ(φ)nμ

u=u = 2

2

[v p(∂vφ)2 +

2

4|u|p|∇/φ|2

], dVolu=u = r2

2

2dσS2dv, (111)

cf. “Appendix A”. Taking the integral over the spacetime region yields

∫

RI V

|K S(φ)|dVol ≤uγ (v∗)∫

uγ (v)

∫

{vγ (u)≤v≤v}

2(u, v)

2r

(1 +

|u|p

v p

)J Sμ(φ)nμ

u=udVolu=udu

+

v∫

v∗

∫

{uγ (v)≤u≤uγ (v∗)}

2(u, v)

2r

(1 +

v p

|u|p

)J Sμ(φ)nμ

v=vdVolv=vdv,

(112)

with uγ (v) and vγ (u) in the integration limits as defined in (40).Note the following general relation for positive functions f (u, v) and g(u, v)

uγ (v∗)∫

uγ (v)

v∫

vγ (u)

f (u, v)g(u, v)dvdu ≤uγ (v∗)∫

uγ (v)

v∫

vγ (u)

[sup

vγ (u)≤v≤v

f (u, v)

]g(u, v)dvdu

≤uγ (v∗)∫

uγ (v)

[sup

vγ (u)≤v≤v

f (u, v)

] v∫

v

g(u, v)dvdu

≤uγ (v∗)∫

uγ (v)

supvγ (u)≤v≤v

f (u, v)du supuγ (v)≤u≤uγ (v∗)

v∫

v

g(u, v)dv

≤uγ (v∗)∫

uγ (v)

supvγ (u)≤v≤v

f (u, v)du supuγ (v)≤u≤uγ (v∗)

v∫

v∗

g(u, v)dv.

(113)


Similarly, it immediately follows that

v∫

v∗

uγ (v∗)∫

uγ (v)

f (u, v)g(u, v)dudv ≤v∫

v∗

supuγ (v)≤u≤uγ (v∗)

f (u, v)dv supv∗≤v≤v

uγ (v∗)∫

uγ (v)

g(u, v)du.

(114)

Using (113) and (114) in (112) we obtain

∫

RI V

|K S(φ)|dVol ≤uγ (v∗)∫

uγ (v)

supvγ (u)≤v≤v

[2(u, v)

2r

(1 +

|u|p

v p

)]du sup

uγ (v)≤u≤uγ (v∗)

×∫

{v∗≤v≤v}J Sμ(φ)nμ

u=udVolu=u

+

v∫

v∗


[2(u, v)

2r

(1 +

v p

|u|p

)]dv sup

v∗≤v≤v

×∫

{uγ (v)≤u≤uγ (v∗)}J Sμ(φ)nμ

v=vdVolv=v . (115)

It remains to showfiniteness and smallness ofuγ (v∗)∫uγ (v)

supvγ (u)≤v≤v

[2(u,v)

2r

(1 + |u|p

v p

)]du

andv∫

vγ (u)

supuγ (v)≤u≤uγ (v∗)[

2(u,v)2r

(1 + v p

|u|p

)]dv. Earlier we obtained the relation

(103) for 2 in region RI V . Therefore, we can write

uγ (v∗)∫

uγ (v)

supvγ (u)≤v≤v

[2(u, v)

2r

(1 +

|u|p

v p

)]du

≤ C

uγ (v∗)∫

uγ (v)

supvγ (u)≤v≤v

[|uγ (v)|−βαeβ[uγ (v)−u]

(1 +

|u|p

v p

)]du

≤ C

uγ (v∗)∫

uγ (v)

|u|−βα

(1 +

|u|p

vp∗

)du ≤ ˜C

uγ (v∗)∫

uγ (v)

|u|−βα+pdu

≤˜C

| − βα + p + 1|[|u|−βα+p+1

]uγ (v∗)

uγ (v)≤ δ1, (116)

where δ1 → 0 for |uγ (v∗)| → −∞ and thus for v∗ → ∞. Note that we have here used(82).

632 A. T. Franzen

For finiteness of the second term in (115) we follow the same strategy and use (102)for the second term to obtain

v∫

v∗


[2(u, v)

2r

(1 +

v p

|u|p

)]dv

≤ C

v∫

v∗


[v−βα

(1 +

v p

|u|p

)]dv

≤ C

v∫

v∗

v−βα

(1 +

v p

|uγ (v∗)|p

)dv ≤ ˜C

[|v|−βα+p+1]vv∗

| − βα + p + 1| ≤ δ2, (117)

where δ2 → 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 cometo the actual proof of weighted energy boundedness up to the Cauchy horizon.

Proposition 4.11. Let φ be as in Theorem 3.1 and p as in (46). Then, for u✂ sufficientlyclose to −∞, for all v∗ ≥ vγ (u✂) and v > v∗

∫


v=vdVolv=v +

∫


u=uγ (v∗)dVolu=uγ (v∗) ≤ Cv∗−1−2δ+p,

(118)

where C is a positive constant depending on C0 of Theorem 3.1 and D0(u�, 1) of Propo-sition 3.2, where u� is defined by rred = r(u�, 1).

Remark.Refer to (40) for the definition of uγ (v) and see Fig. 6b) for further clarification.

Proof. In Sect. 4.1.3, Corollary 4.8, we have obtained the global estimate (95) for theweighted S0 current which follows from Proposition 4.7. Recall that in D+(γ ) we have|u|p ≤ v p, cf. Sect. 4.1.3.1, which immediately leads to

∫


γ dVolγ ≤ C∫

{v∗≤v≤v}v p J S0

μ (φ)nμγ dVolγ , (119)

cf. “Appendix B” for explicit expressions of J S0μ (φ)nμ

γ and J Sμ(φ)nμ

γ . From (119) wesee that

∫


γ dVolγ ≤ Cv∗−1−2δ+p for all v∗ > α and p as in (46), (120)

is implied by Corollary 4.8.


Let v∗ > 2α and v > v∗. In order to obtain (118) we consider a regionRI V = J+(γ ) ∩ J−(x)with x = (uγ (v∗), v), x ∈ B, as shown in Fig. 11. Applying thedivergence theorem we obtain

∫

RI V

K S(φ)dVol +∫


v=vdVolv=v

+∫


u=uγ (v∗)dVolu=uγ (v∗)

=∫


γ dVolγ . (121)

In Sect. 4.1.4.2 we found that the angular part of K S(φ) is positive inRI V and we calledthe remaining part K S(φ) given in (106). Using (107) we can therefore write

∫


v=vdVolv=v +

∫



≤∫

RI V

|K S(φ)|dVol +∫


γ dVolγ . (122)

Using Lemma 4.10 we obtain∫


v=vdVolv=v +

∫



≤ δ1 supuγ (v)≤u≤uγ (v∗)

∫


u=udVolu=u


∫


v=vdVolv=v +∫


γ dVolγ .

(123)

Repeating estimate (123) with u, v in place of uγ (v∗), v and taking the supremum wehave


∫


u=udVolu=u

+ supv∗≤v≤v

∫


v=vdVolv=v

≤ δ1 supuγ (v)≤u≤uγ (v∗)

∫


u=udVolu=u

634 A. T. Franzen


∫


v=vdVolv=v

+∫


γ dVolγ . (124)

Recalling δ1 → 0, δ2 → 0 as v∗ → ∞, choose u✂ sufficiently close to −∞, suchthat for v∗ > vγ (u✂), say

δ1, δ2 ≤ 1

2(125)

holds. The conclusion of Proposition 4.11 then follows by absorbing the first two termsof the right hand side of (124) by the two terms on the left and estimating the third from(120). �

4.1.4.4. Energy estimates globally in the rectangle � up to CH+ in the neighbourhoodof i+. In the previous Sects. 4.1.1 to 4.1.4.3 we have proven energy estimates for eachregion with specific properties separately. Putting all results together we can state thefollowing proposition.

Proposition 4.12. Let φ be as in Theorem 3.1 and p as in (46). Then, for u✂ sufficientlyclose to −∞, for all v∗ > 1, v > v∗ and u ∈ (−∞, u✂).

∫


u=udVolu=u ≤ Cv∗−1−2δ+p, (126)

where C is a positive constant depending on C0 of Theorem 3.1 and D0(u�, 1) of Propo-sition 3.2, where u� is defined by rred = r(u�, 1).

Proof. First of all we partition the integral of the statement into a sum of integrals of thedifferent regions. That is to say

∫


u=udVolu=u =∫

{v∗≤v≤v}∩RJ Sμ(φ)nμ

u=udVolu=u

+∫

{v∗≤v≤v}∩NJ Sμ(φ)nμ

u=udVolu=u

+∫

{v∗≤v≤v}∩J−(γ )∩BJ Sμ(φ)nμ

u=udVolu=u

+∫

{v∗≤v≤v}∩J+(γ )∩BJ Sμ(φ)nμ

u=udVolu=u .

For the integral inR and the integral inN we use Corollaries 4.4 and 4.6. (Note that theformer has to be summed resulting in the loss of one polynomial power). Further, forthe 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 conclusionof 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✂ sufficientlyclose to −∞, for all vfix ≥ 1, and u ∈ (−∞, u✂),

∫

S2

∞∫

vfix

[v p(∂vφ)2(u, v) + 2|∇/φ|2(u, v)

]r2dvdσS2 ≤ C, (127)

where C is a positive constant dependent on C0 of Theorem 3.1 and D0(u�, 1) of Propo-sition 3.2, where u� is defined by rred = r(u�, 1).

Proof. The conclusion of the proposition follows immediately examining the weightsin Proposition 4.12. �

4.2. Pointwise estimates from higher order energies.

4.2.1. The � notation and Sobolev inequality on spheres. Recall that we had stated theexpressions for the generators of spherical symmetry �i , i = 1, 2, 3, in Sect. 2.3.2.They were explicitly given in (32). Further, having the expressions of (33) in mind weintroduce the following notation

(�kφ)2 =

∑(�i1 · · · �ik φ)2

,(∂v�kφ

)2

=∑(

∂v�i1 · · · �ik φ)2

,

∣∣∣∇/�kφ

∣∣∣2 =∑∣∣∇/�i1 · · · �ik φ

∣∣2 , (128)

with i j = 1, 2 or 3. By Sobolev embedding on the standard spheres we have in thisnotation

sup{θ,ϕ}∈S2

|φ(u, v, θ, ϕ)|2 ≤ C2∑

k=0

∫

S2

(�kφ)2

(u, v, θ, ϕ)dσS2 , (129)

which means that we can derive a pointwise estimate from an estimate of the integralson the spheres, see e.g. [16].

4.2.2. Higher order energy estimates in the neighbourhood of i+. We will need thefollowing 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 vfix ≥ 1 andufix > −∞

∫

S2

∞∫

vfix

[v p(∂v�kφ)2(ufix, v, θ, ϕ) + 2|∇/�kφ|2(ufix, v, θ, ϕ)

]r2dvdσS2 ≤ Ek,

for all k ∈ N0. (130)

636 A. T. Franzen

Proof. Statement (130) with k = 0 was already derived in Corollary 4.13. Recall that�iφ,�i� jφ etc. also satisfy the massless scalar wave equation, cf. Sect. 2.1. Summingover all angular momentum operators, keeping in mind notation (128), we thereforeobtain (130) for all k ∈ N

0. �

4.2.3. Pointwise boundedness in the neighbourhood of i+. We turn the discussion tothe derivation of pointwise boundedness from energy estimates. In particular we proveTheorem 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∗ andu ∈ (−∞, u✂) that

φ(u, v, θ, ϕ) =v∫

v∗

(∂vφ) (u, v, θ, ϕ)dv + φ(u, v∗, θ, ϕ)

≤v∫

v∗

(∂vφ)(u, v, θ, ϕ)vp2 v− p

2 dv + φ(u, v∗, θ, ϕ)

≤⎛⎜⎝

v∫

v∗

v p(∂vφ)2(u, v, θ, ϕ)dv

⎞⎟⎠

12⎛⎜⎝

v∫

v∗

v−pdv

⎞⎟⎠

12

+ φ(u, v∗, θ, ϕ),

(131)

where we have used the Cauchy-Schwarz inequality in the last step. Squaring the entireexpression, using Cauchy-Schwarz again and integrating over S2 we obtain the expres-sion that we had sketched in Sect. 3.2 already

∫

S2

φ2(u, v)dσS2 ≤ C

⎡⎢⎣∫

S2

⎛⎜⎝

v∫

v∗

v p(∂vφ)2(u, v)dv

v∫

v∗

v−pdv

⎞⎟⎠ r2dσS2 +

∫

S2

φ2(u, v∗)dσS2

⎤⎥⎦ ,

(132)

with p as in (46) and the first term on the right hand side controlled by the flux forwhich we derived boundedness in Sect. 4.1.4. Therefore, by using Theorem 4.14 andapplying all our estimates to �iφ, �i� jφ etc. and summing, we obtain in the notationof Sect. 4.2.1 the following:

∫

S2

(�kφ)2(u, v)dσS2 ≤ C

⎡⎢⎣Ek

∫

S2

v∫

v∗

v−pdvdσS2 +∫

S2

(�kφ)2(u, v∗)dσS2

⎤⎥⎦

≤ C

⎡⎢⎣ ˜C Ek +

∫

S2

(�kφ)2(u, v∗)dσS2

⎤⎥⎦ , (133)

for all k ∈ N0. It is here that we have used the requirement p > 1 of (46).


Fig. 12. Penrose diagram depicting regionRV

Let us now use (43) of Proposition 3.2 to estimate the right hand side of (133) withv∗ = 1. Adding all equations up we derive pointwise boundedness according to (129)

sup{θ,ϕ}∈S2

|φ(u, v, θ, ϕ)|2 ≤ C

⎡⎢⎣∫

S2

(φ)2(u, v)dσS2 +∫

S2

(�φ)2(u, v)dσS2

+∫

S2

(�2φ)2(u, v)dσS2

⎤⎥⎦ , (134)

≤ C[ ˜C (E0 + E1 + E2) + D0(u�, 1)

+ D1(u�, 1) + D2(u�, 1)]

≤ C, (135)

with C depending on the initial data on �. We therefore arrive at the statement given inTheorem 3.3. �

4.3. Energy along the future boundaries ofRV . Let u� > u✂ and v∗ ≥ vγ (u✂). DefineRV = {u✂ ≤ u ≤ u�} ∩ {v∗ ≤ v ≤ v

}, cf. Fig. 12, and note that RV ⊂ B.

We will apply the vector field

W = v p∂v + ∂u (136)

as a multiplier. The bulk can be calculated as

K W = − 2

r[v p + 1](∂vφ∂uφ) −

[1

2pv p−1 +

∂v

v p +

∂u

]|∇/φ|2. (137)

Let us define

K W = −2

r[v p + 1](∂vφ∂uφ) and K W∇/ = −

[1

2pv p−1 +

∂v

v p +

∂u

]|∇/φ|2,

(138)

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

− K W ≤ |K W | (139)

638 A. T. Franzen

in RV . We aim for estimating it via the currents along v = constant and u = constanthypersurfaces.

Lemma 4.15. Let φ be an arbitrary function. Then, for all v∗ ≥ vγ (u✂), v > v∗, foru� ≥ u2 > u1 ≥ u✂ and ε ≥ u2 − u1 > 0

∫

RV1

|K W |dVol ≤ δ1 supu1≤u≤u2

∫

{v∗≤v≤v}J Wμ (φ)nμ

u=udVolu=u

+δ2 supv∗≤v≤v

∫

{u1≤u≤u2}J Wμ (φ)nμ

v=vdVolv=v , (140)

where RV1 = {u1 ≤ u ≤ u2} ∩ RV and δ1, δ2 are positive constants, depending onlyon v∗ and ε such that δ1 → 0 for ε → 0 and δ2 → 0 as v∗ → ∞.

Proof. Using the Cauchy-Schwarz inequality for the first equation of (138) we obtain

|K W | ≤ 1

r

[(1 + v−p) v p(∂vφ)2 +

(1 + v p) (∂uφ)2

], (141)

with the related volume element

dVol = r22

2dudvdσ 2

S. (142)

Note that the currents related to the vector field W with their related volume elementsare given by

J Wμ (φ)nμ

v=v = 2

2

[(∂uφ)2 +

2

4v p|∇/φ|2

], dVolv=v = r2

2

2dσS2du, (143)

J Wμ (φ)nμ

u=u = 2

2

[v p(∂vφ)2 +

2

4|∇/φ|2

], dVolu=u = r2

2

2dσS2dv, (144)

cf. “Appendix A”. Taking the integral over the spacetime region therefore yields

∫

RV1

|K W (φ)|dVol ≤u2∫

u1

∫

{v∗≤v≤v}

2(u, v)

2r

(1 + v−p) J W

μ (φ)nμu=udVolu=udu

+

v∫

v∗

∫

{u1≤u≤u2}

2(u, v)

2r

(1 + v p) J W

μ (φ)nμv=vdVolv=vdv,

≤u2∫

u1

supv∗≤v≤v

[2(u, v)

2r

(1 + v−p)] du

× supu1≤u≤u2

∫


u=udVolu=u


+

v∫

v∗

supu1≤u≤u2

[2(u, v)

2r

(1 + v p)] dv

× supv∗≤v≤v

∫

{u1≤u≤u2}J Sμ(φ)nμ

v=vdVolv=v . (145)

It remains to show finiteness and smallness ofu2∫

u1supv∗≤v≤v

[2(u,v)

2r

(1 + v−p

)]du and

v∫v∗supu1≤u≤u2

[2(u,v)

2r (1 + v p)]dv. Recall the properties of the hypersurface γ shown

in Sect. 4.1.3.1. Since v∗ > vγ (u✂), (103) implies that

2(u, v) ≤ C2(u✂, v∗), for any (u, v) ∈ J+(x), with x = (u✂, v∗), x ∈ B,(146)

so that we obtainu2∫

u1

supv∗≤v≤v

[2(u, v)

2r

(1 + v−p)] du

(146)≤ C

u2∫

u1

supv∗≤v≤v

2(u✂, v∗)(1 + v−p) du

≤ C

u2∫

u1

|u✂|−βα(1 + v∗−p) du

≤ ˜C |u2 − u1| ≤ δ1, (147)

and moreover δ1 → 0 for ε → 0.Further, in Sect. 4.1.4.1 we derived that similarly

2(u, v) ≤ C v−βα, for any (u, v) ∈ J+(x), with x = (u✂, v∗), x ∈ B, (148)

where v∗ > vγ (u✂).

v∫

v∗

supu1≤u≤u2

[2(u, v)

2r

(1 + v p)] dv (148)≤ C

v∫

v∗

v−βα(1 + v p) dv

≤ C

| − βα + p + 1|[v−βα+p+1

]vv∗

≤ δ2, (149)

where δ2 → 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). For all v∗ > vγ (u✂)

sufficiently large, v ∈ (v∗,∞), for u� ≥ u2 > u1 ≥ u✂ and ε ≥ u2 − u1 > 0. Then forε sufficiently small, the following is true. If∫


u=u1dVolu=u1 ≤ C1, (150)

640 A. T. Franzen

then

∫


u=u2dVolu� +∫


v=vdVolv=v ≤ C2(C1, u�, v∗),

(151)

where C2 depends on C1, C0 of Theorem 3.1 and D0(u�, v∗) of Proposition 3.2.

Remark. Note already that the hypothesis (150) is implied by the conclusion of Propo-sition 4.12 for u1 = u✂.

Proof. By the divergence theorem and (139) we can state

∫


u=u2dVolu=u2 +∫


v=vdVolv=v

≤∫

RV1

|K W |dVol +∫


v=v∗dVolv=v∗

+∫


u=u1dVolu=u1 . (152)

We can replace the hypersurfaces u = u2 and v = v with u = u and v = v hypersurfacesand therefore obtain

supu1≤u≤u2

∫

{v∗ ≤v≤v}J Wμ (φ)nμ

u=udVolu=u + supv∗≤v≤v

∫


v=vdVolv=v

≤∫

RV1

|K W |dVol +∫

{u✂≤u≤u�}J Wμ (φ)nμ

v=v∗dVolv=v∗

+∫


u=u1dVolu=u1 ,

Lem. 4.15≤ δ1 supu1≤u≤u2

∫


u=udVolu=u


∫


v=vdVolv=v

+∫


v=v∗dVolv=v∗ +∫


u=u1dVolu=u1 . (153)


Thus, we have

⇒ supu1≤u≤u2

∫


u=udVolu=u + supv∗≤v≤v

∫


v=vdVolv=v

≤ 1

1 − max {δ1, δ2}

⎡⎢⎣

∫


v=v∗dVolv=v∗ +∫


u=u1dVolu=u1

⎤⎥⎦ ,

≤ C D0(u�, v∗) + C1, (154)

where the last step follows by statement (42) of Proposition 3.2, with k = 0, and by(150). We have chosen ε sufficiently small and v∗ sufficiently close to ∞, such that δ1and δ2 satisfy say

δ1, δ2 ≤ 1

2. (155)

The conclusion of Proposition 4.16 is obtained. �

We are now ready to make a statement for the entire region RV .

Proposition 4.17. Let φ be as in Theorem 3.1 and p as in (46). Then, for all v∗ > vγ (u✂)

sufficiently large, v > v∗, and u� > u > u✂,

∫


v=vdVolv=v +

∫


u=udVolu=u ≤ C(u�, v∗), (156)

where C depends on C0 of Theorem 3.1 and D0(u�, v∗) of Proposition 3.2.

Proof. Let ε be as in Proposition 4.16. We choose a sequence ui+1 − ui ≤ ε and i ={1, 2, .., n} such that u1 = u✂ and un = u. Denote RVi = {ui ≤ u ≤ ui+1} ∩ {v∗ ≤ v

≤ v}, cf. Fig. 13. Iterating the conclusion of Proposition 4.16 from u1 up to un then

completes the proof. Note that n depends only on the smallness condition on ε fromProposition 4.16, since n � u�−u✂

ε. �

Fig. 13. Penrose diagram depicting regions RVi

642 A. T. Franzen

Fig. 14. Penrose diagram depicting region RV I

4.4. Propagating the energy estimate up to the bifurcation sphere. All results stated sofar, correspond to the neighbourhood of i+ at u = −∞, which we call the right handside of the spacetime. Analogously, we can derive all statements for the left hand side,in the neighbourhood of i+ at v = −∞ by substituting all u parameters by v and allv parameters by u. In this section we will use both results from the right and left sideon CH+. Fix u� = v�, such that moreover Proposition 4.17 holds with v� = v∗, andsuch that its left side analog holds with u� = u∗. We will consider a region RV I ={u� ≤ u ≤ u, v� ≤ v ≤ v

}, with u ∈ (u�,∞) and v ∈ (v�,∞), cf. Fig. 14.

Recall, that in Sect. 4.1.4.2 we have defined the weighted vector field11

S = v p∂v + u p∂u, (157)

which we are going to use again to obtain an energy estimate up to the bifurcate two-sphere. Recall K S given in (105), where the terms multiplying the angular derivativesare positive sinceRV I is located in the blueshift region. We further defined K S in (106)and stated (107) which will be useful to state the following proposition.

Lemma 4.18. Let φ be an arbitrary function. Then, for all (u�, v�) ∈ J+(γ ) ∩ B andall u > u�, all v > v�, the integral over RV I , cf. Fig. 14 of the current K S, defined by(106), can be estimated by

∫

RV I

|K S|dVol ≤ δ1 supu�≤u≤u

∫

{v�≤v≤v}J Sμ(φ)nμ

u=udVolu=u

+ δ2 supv�≤v≤v

∫

{u�≤u≤u}J Sμ(φ)nμ

v=vdVolv=v , (158)

where δ1 and δ2 are positive constants, with δ1 → 0 as u� → ∞ and δ2 → 0 asv� → ∞.

Proof. The proof is similar to the proof of Lemma 4.10 of Sect. 4.1.4.2 and Lemma 4.15of Sect. 4.3.

11 Since u is always positive in the remaining region under considerationRV I , we have omitted the absolutevalue in the u-weight.


We still need to show finiteness and smallness ofu∫

u�supv�≤v≤v

[2(u,v)

2r

(1 + u p

v p

)]du

andv∫

v�supu�≤u≤u

[2(u,v)

2r

(1 + v p

u p

)]dv. In Sect. 4.1.4.1 we derived (100) which wewill

use now for all u, v ∈ J+(u�, v�) Therefore, we can write

u∫

u�

supv�≤v≤v

[2(u, v)

2r

(1 +

u p

v p

)]du

≤ C

u∫

u�

supv�≤v≤v

[2(u�, v�)e−β[v−v�+u−u�]

(1 +

u p

v p

)]du,

≤ C

u∫

u�

2(u�, v�)e−β[u−u�](1 +

u p

vp�

)du ≤ δ1, (159)

where δ1 → 0 as u� = v� → ∞ (since 2(u�, v�) → 0, cf. (23)). Similarly, forfiniteness of the second term we obtain

v∫

v�

supu�≤u≤u

[2(u, v)

2r

(1 +

v p

u p

)]dv

≤ C

v∫

v�

supu�≤u≤u

[2(u�, v�)e−β[v−v�+u−u�]

(1 +

v p

u p

)]dv,

≤ C

v∫

v�

2(u�, v�)e−β[v−v�](1 +

v p

u p�

)dv ≤ δ2, (160)

where δ2 → 0 as u� = v� → ∞.Thus we obtain the statement of Lemma 4.10 by fixing u� = v� sufficiently large.

�Proposition 4.19. Let φ be as in Theorem 3.1. Then, for u� = v� sufficiently close to ∞and u > u�, v > v�∫

{v�≤v≤v}J Sμ(φ)nμ

u=udVolu=u +∫

{u�≤u≤u}J Sμ(φ)nμ

v=vdVolv=v ≤ C(u�, v�), (161)

where C depends on C0 of Theorems 3.1 and D0(u�, v�) of Propositions 3.2 (and theirleft side analogs, which we obtain from interchanging u and v).

Proof. The proof follows from applying the divergence theorem for the current J Sμ(φ)

in the region RV I . The past boundary terms are estimated by Proposition 4.17 and itsleft side analog. Note that the weights of J S

μ(φ) are comparable to the weights of J Wμ (φ)

for fixed u�, and similarly the weights of J Sμ(φ) are comparable to the weights of the

left analog of J Wμ (φ) for fixed v�. The bulk term is absorbed by Lemma 4.18. �

644 A. T. Franzen

Now that we have shown boundedness for different subregions of the interior we canstate the following proposition for the entire interior region M|I I , cf. Sect. 2.3.1.

Corollary 4.20. Let φ be as in Theorem 3.1. Then

∫

S2

∞∫

vfix

[(|v| + 1)p(∂vφ)2(ufix, v, θ, ϕ) + 2|∇/φ|2(ufix, v, θ, ϕ)

]r2dvdσS2

≤ C, for vfix ≥ v✂, ufix > −∞ , (162)∫

S2

∞∫

ufix

[(|u| + 1)p(∂uφ)2(u, vfix, θ, ϕ) + 2|∇/φ|2(u, vfix, θ, ϕ)

]r2dudσS2

≤ C, for ufix ≥ u✂, vfix > −∞, (163)

where p is as in (46) and C depends on C0 of Theorem 3.1 and D0(u�, v�) of Proposi-tion 3.2 (and their left side analogs) where u� = v� is as in Proposition 4.19.

Proof. This follows by examining the weights in Proposition 4.17 (and its left sideanalog) and 4.19 together with Theorem 4.14 (and its left side analog). �

4.5. Global higher order energy estimates, pointwise boundedness and continuity. Toobtain pointwise bounds in analogy to Sect. 4.2 we first have to extend Corollary 4.20to a higher order statement.

Theorem 4.21. Let φ be as in Theorem 3.1 and p as in (46). Then, for vfix ≥ v✂,ufix > −∞∫

S2

∞∫

vfix

[(|v| + 1)p(∂v�kφ)2(ufix, v, θ, ϕ)+2|∇/�kφ|2(ufix, v, θ, ϕ)

]r2dvdσS2 ≤ Ek,

(164)

and for ufix ≥ u✂, vfix > −∞∫

S2

∞∫

ufix

[(|u| + 1)p(∂u�kφ)2(u, vfix, θ, ϕ)+2|∇/�kφ|2(u, vfix, θ, ϕ)

]r2dudσS2 ≤ Ek,

(165)

(166)

for all k ∈ N0.

Proof. This follows immediately from Corollary 4.20 by commutation. �

Having proven Theorem 4.21, the pointwise boundedness of |φ| in all of M|I I followsanalogously to Sect. 4.2.3 but by integration in u-direction.


We estimate

∫

S2

(�kφ)2(u, v)dσS2 ≤ C

⎡⎢⎣∫

S2

⎛⎜⎝

u∫

u∗

(|u| + 1)p(∂u�kφ)2(u, v)du

u∫

u∗

(|u| + 1)−pdu

⎞⎟⎠ r2dσS2

+∫

S2

(�kφ)2(u∗, v)dσS2

⎤⎥⎦ ,

≤ C

⎡⎢⎣ ˜C Ek +

∫

S2

φ2(u∗, v)dσS2

⎤⎥⎦ , (167)

where u∗ ≥ u✂, u ∈ (u∗,∞) and v ∈ (1,∞) and k ∈ N0.

By using the result (135) in (167) we derive pointwise boundedness according to(129)

sup{θ,ϕ}∈S2

|φ(u, v, θ, ϕ)|2 ≤ C

⎡⎢⎣∫

S2

(φ)2(u, v)dσS2 +∫

S2

(�φ)2(u, v)dσS2

+∫

S2

(�2φ)2(u, v)dσS2

⎤⎥⎦ ,

≤ C[ ˜C (E0 + E1 + E2) + C)

]≤ C, (168)

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 interiorhas compact closure in spacetime for which (2) thus follows by Cauchy stability. Wehave thus shown (2) globally in the interior.

The continuity statement of Theorem1.1 follows easily by estimating |φ(u, v, ϕ, θ)−φ(u, v, ϕ, θ)| via the fundamental theorem of calculus and Sobolev embedding,12 andsimilarly for v, ϕ and θ in the role of of u.

5. Outlook

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

5.1. Scalar waves on subextremal Kerr interior backgound. All constructions of thispaper have direct generalizations to the subextremal Kerr case.13 The following theoremwill appear in a subsequent companion paper.

12 We estimate as in (167) and (168) but exploiting that∫ u

u (|u| + 1)−p → 0 as u, u → ∞.13 This is in contrast to the exterior region for which the analysis of the Kerr case is significantly harder

than for spherically symmetric black hole spacetimes in view of the difficulties of superradiance and the morecomplicated trapping. See the references below.

646 A. T. Franzen

Theorem 5.1. On a fixed subextremal Kerr background 0 �= |a| < M, with angularmomentum per unit mass a and mass M, let φ be a solution of the wave equation arisingfrom sufficiently regular initial data on a Cauchy hypersurface �. Then

|φ| ≤ C

globally in the black hole interior up to CH+, to which φ in fact extends continuously.

The above theorem of course depends on the fact that the analog of Theorem 3.1 hasbeen proven in the full subextremal range |a| < M on Kerr backgrounds by Dafermoset al. cf. [19,22]; see also [1,18,20,21,31,35,44] for the |a| � M case and [42,46] formode stability.

5.2. Mass inflation. In view of our stability result, what remains of the “blueshift insta-bility”?

The result of Theorem 1.1 is still compatible with the expectation that the transversederivatives (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

|∂vφ| ≥ cv−4, (169)

for some constant c > 0 and all sufficiently large v onH+, Dafermos has shown for thespherically symmetric Einstein-Maxwell-scalar field model that transverse derivativesblow up, cf. [13]. This blow up result of [14] would also apply to our setting here if theabove 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 genericdata on �; see also Sbierski [40]. One might therefore aim for proving the followingconjecture.

Conjecture 5.2. Let φ be as in Theorem 1.1 or 5.1. For generic data on �, transversederivatives of φ blow up on CH+, in fact φ is not H1

loc.

5.3. Extremal black holes. For a complete geometric understanding of black hole interi-ors one must also consider extremal black holes. Aretakis proved stability and instabilityproperties for the evolution of a massless scalar field on a fixed extremal Reissner-Nordström exterior background. For data on a spacelike hypersurface� intersecting theevent horizon and extending to infinity he has proven decay of φ up to and including thehorizon, cf. [2]. In subsequent work [3] Aretakis showed that first transverse derivativesof φ generically do not decay along the event horizon for late times. He further provedthat higher derivatives blow up along the event horizon.

The analysis of the evolution of the scalar wave in the region beyond the eventhorizon for the extreme case remains to be shown. Motivated by heuristics and numericsof Murata, Reall and Tanahashi [36], which suggest stronger stability results in theinterior than in the subextremal case, we conjecture

Conjecture 5.3. (See [36]). In extremal Reissner-Nordström spacetime (M, g), withmass M and charge e and M = |e| �= 0, let φ be a solution of the wave equation�gφ = 0 arising from sufficiently regular Cauchy data on �. Then, for V− regular atCH+

|φ| ≤ C, |∂V−φ| ≤ C,

globally in the maximal domain of dependence D+(�) of a hypersurface �.


See work of Gajic [24].Aretakis has further considered �gφ on a fixed extremal Kerr background, c.f [4].

Analogously to the extremal Reissner-Nordström case, he shows decay up to and includ-ing the event horizon for axisymmetric solutions φ and in [5] shows instability propertieswhen considering transverse derivatives. This has been further generalised in [28,29].In analogy with Conjecture 5.3, we thus also make the following conjecture:

Conjecture 5.4. In extremal Kerr spacetime (M, g), with mass M and angular momen-tum per unit mass a and M = |a| �= 0, let φ be an axisymmetric solution of the waveequation �gφ = 0 arising from sufficiently regular Cauchy data on �. Then, for V−regular at CH+

|φ| ≤ C, |∂V−φ| ≤ C

globally in the maximal domain of dependence D+(�) of the hypersurface �.

The case of non axisymmetric φ seems to be rather complicated and we thus do notventure a conjecture here.

5.4. Einstein vacuum equations. Wenow return to the problem that originallymotivatedour work, namely the dynamics of the Einstein vacuum equations. Our result and theforthcoming extension to subextremal Kerr backgrounds further support the expectationthat the spherically symmetric toy models [12–14,37–39] are indeed indicative of whathappens for the full nonlinear Einstein vacuum equationswithout symmetry. In particularour results support the following conjecture given in [14].

Conjecture 5.5. (Ori). Let (M, g) be the maximal vacuum Cauchy development of suf-ficiently small perturbations of asymptotically flat two-ended Kerr data correspondingto parameters 0 < |a| < M. Then;

(a) there exist both a future and past extension (M, g) of M with C0 metric g such that∂M is a bifurcate null cone in M and all future (past) incomplete geodesics in γ

pass into M/M.(b) Moreover, for generic such perturbations, any C0 extension M will fail to have L2

Christoffel symbols in a neighbourhood of any point of ∂M.

A proof of part (a) has recently been announced by Dafermos and Luk, given the con-jectured stability of Kerr exterior (i.e. given the analog of Theorem 3.1 for the fullnonlinear Einstein vacuum equations). Specific examples of vacuum spacetimes withnull singularities as in (b) have been constructed by Luk [32]. For a discussion of whatall this means to Strong Cosmic Censorship see [14].

Acknowledgments. I am very grateful to Mihalis Dafermos for suggesting this problem to me and guiding metowards its completion. Further, I would like to thank Gerard ’t Hooft for his continuous encouragement andcomments on the manuscript. Parts of this work were conducted at Cambridge University, ETH Zürich andPrincetonUniversity, whose hospitality is gratefully acknowledged. This work is part of the D-ITP consortium,a program of the Netherlands Organisation for Scientific Research (NWO) which is funded by the DutchMinistry of Education, Culture and Science (OCW).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Inter-national License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons license, and indicate if changes were made.

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648 A. T. Franzen

Appendix A: The J-Currents and Normal VectorsIn the following we will derive the J -currents on constant r , u and v hypersurfaces as well as the hypersurfaceγ , defined in (85).We consider an arbitrary function F(u, v) independent of the angular coordinates. Let ζ be a levelsetζ = {F(u, v) = 0}. Then, the normal vector to the hypersurface ζ is given by

nμζ = 1√

2|∂u F∂v F |(∂v F∂u + ∂u F∂v). (A1)

In particular, for the future directed normal vector of an r(u, v) = const hypersurface we obtain

nμr=const = 1√

2(∂u + ∂v), (A2)

and on constant u and v null hypersurfaces with their related volume elements we have

nμu=const = 2

2 ∂v, dVolu=const = r22

2dσ

S2dv,

nμv=const = 2

2 ∂u , dVolv=const = r22

2dσ

S2du. (A3)

For (A3), note that since vectors orthogonal to null hypersurfaces cannot be normalized, their proportionalityhas to be chosen consistent with an associated volume form in the application of the divergence theorem.Further, the volume element of the four dimensional spacetime is given by

dVol = r22

2dσ

S2dudv. (A4)

According to (7), using an arbitrary vector field X = Xu∂u + Xv∂v we then obtain:

J Xμ (φ)nμ

ζ = 1√2

[Xv

√∂u F

∂v F(∂vφ)2 + Xu

√∂v F

∂u F(∂uφ)2

]

+

√2

4

[Xv

√∂v F

∂u F+ Xu

√∂u F

∂v F

]|∇/φ|2, (A5)

J Xμ (φ)nμ

r=const = 1√2

[Xv(∂vφ)2 + Xu(∂uφ)2

]+

√2

4

[Xv + Xu] |∇/φ|2, (A6)

J Xμ (φ)nμ

v=const = 2

2 Xu(∂uφ)2 +1

2Xv |∇/φ|2,

J Xμ (φ)nμ

u=const = 2

2 Xv(∂vφ)2 +1

2Xu |∇/φ|2. (A7)

Appendix B: The K -CurrentIn order to compute all scalar currents according to (8) in (u, v) coordinates we first derive the componentsof the deformation tensor which is given by

(π X )μν = 1

2(gμλ∂λ Xν + gνσ ∂σ Xμ + gμλgνσ gλσ,δ Xδ), (B1)

where X is an arbitrary vector field, X = Xu∂u + Xv∂v without angular components.14 From this we obtain

(π X )vv = − 2

2 ∂u Xv, (π X )uu = − 2

2 ∂v Xu ,

(π X )uv = − 1

2 (∂v Xv + ∂u Xu) − 2

2

(∂v

Xv +

∂u

Xu)

,

(π X )θθ = 1

r2

(∂vr

rXv +

∂ur

rXu)

, (π X )φφ = 1

r2 sin2 θ

(∂vr

rXv +

∂ur

rXu)

.

14 Recall that all our multipliers N , −∂r , S0 and S only contain u and v components.


From (6) we calculate the components of the energy momentum tensor in (u, v) coordinates as

Tvv = (∂vφ)2, Tuu = (∂uφ)2, Tuv = Tvu = 2

4|∇/φ|2,

Tθθ = (∂θ φ)2 +2r2

2 (∂uφ∂vφ) − 1

2r2|∇/φ|2,

Tφφ = (∂φφ)2 +2r2 sin2 θ

2 (∂uφ∂vφ) − 1

2r2 sin2 θ |∇/φ|2.

Multiplying the components according to (8) and using the relations (24) we obtain

K X = − 2

2

[∂u Xv(∂vφ)2 + ∂v Xu(∂uφ)2

]− 2

r

[Xv + Xu] (∂uφ∂vφ)

−[1

2

(∂v Xv + ∂u Xu) +

(∂v

Xv +

∂u

Xu)]

|∇/φ|2. (B2)

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