Higher Order Linear Stability and Instability of Reissner-Nordström's Cauchy Horizon
João L. Costa Pedro M. Girão


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Higher Order Linear Stability and Instability of  Reissner-Nordström's Cauchy Horizon

 

João L. Costa^†
 and Pedro M. Girão^





^†ISCTE - Instituto Universitário de Lisboa, Lisboa, Portugal.


^Center for Mathematical Analysis, Geometry and Dynamical Systems,

Instituto Superior Técnico, Universidade de Lisboa,

Av. Rovisco Pais, 1049-001 Lisbon, Portugal.




[2010]Primary: 35L05; Secondary: 35R01, 58J45, 83C57









We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner-Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime, whose restrictions to the event horizon have compact support. We provide criteria, in terms of surface gravities, for the waves to remain in C^l, l≥ 1, up to and including the Cauchy horizon.
We also provide sufficient conditions for the blow up of 
solutions in C^1 and H^1. 



    
    December 30, 2023
=====================







§ INTRODUCTION



Cauchy horizons are the spacetime boundary of the maximal Cauchy development of initial value problems for the Einstein field equations. Whenever non-empty, their existence and stability puts into question global uniqueness,
and consequently challenges the deterministic character of General Relativity. 
To understand how perturbations of a static charged black hole behave at the Cauchy horizon that lies 
in its interior, we will study solutions of the wave equation on the black hole region of fixed subextremal Reissner-Nordström 
(asymptotically flat, de Sitter or anti-de Sitter) spacetimes. 
In this framework, it is natural to consider that Cauchy horizons that allow solutions
with higher regularity are 
more stable than the ones that do not.


The stability of Cauchy horizons is a classical problem in General Relativity and, in recent years,  considerable progress has been made 
in its understanding through the mathematical analysis of wave equations.  Stability results can be found in <cit.> and instability results 
in <cit.>, and the references therein. 
For developments concerning the analysis of the full Einstein equations 
we refer to <cit.>.

Most of the literature about the linear problem focuses on stability-regularity at the C^0 and H^1 levels, in line with the modern formulations of the Strong Cosmic Censorship Conjecture. There are however some notable exceptions. In <cit.>, Gajic  provides criteria for the C^1 and C^2 extendibility of spherically symmetric waves on (asymptotically flat) extremal black holes. In the subextremal de Sitter setting, Hintz and Vasy <cit.> have shown that solutions of the wave equation arising from smooth Cauchy data have H^1/2+α/κ_–ϵ regularity up to the Cauchy horizon, with the degree of regularity being dictated by α, the spectral gap 
of the operator _g (which also controls the decay rate of solutions along the event horizon), and 
κ_-, the Cauchy horizon's surface gravity. 
However, recent numerical computations of the spectral gap <cit.> suggest that the regularity 
never exceeds H^3/2-ϵ.   

Here we present criteria for higher order linear stability
of the Cauchy horizon, meaning C^l with l≥ 1, in a subextremal Reissner-Nordström spacetime, as well as criteria for linear instability, in both C^1 and H^1. We will achieve this by considering waves, without symmetry assumptions, whose restrictions to the event horizon have compact support. 
Although, in view of the results in <cit.>, this behavior on the event horizon 
cannot arise from generic Cauchy data, it provides a class of bona fide characteristic initial value problems for the wave equation. We will 
show that an arbitrarily high regularity at the Cauchy horizon can be obtained by increasing the order to which the wave vanishes in a direction transverse to the event horizon. 
Moreover, for this initial value problem, the role of the  
surface gravities
in determining the degree of stability of the Cauchy horizon becomes particularly transparent. 
For instance, we will prove that, as a consequence of a well known relation between surface gravities, 
if the wave only vanishes to zeroth order at the event horizon then,
in spite of having compact support on the event horizon,
it cannot be extended in C^1 to
any neighborhood of any point on the Cauchy horizon.
In particular, this shows that we cannot expect to obtain arbitrarily high regularity for waves up to and
including the Cauchy horizon by simply increasing their decay rate along the event horizon. 





 §.§ Statement of the main results


Let us set some basic terminology and notation.
Let ( M,g) be a connected component of the black hole region of a subextremal Reissner-Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime.
Denote by κ_+ and κ_- the surface gravities of the future event horizon H^+
and the future Cauchy horizon CH^+, respectively, and let H_A^+ and  CH_A^+ denote the “right side” components of these horizons  (see Figure <ref>).
Let v be a future increasing affine parameter of the generators of H_A^+, constant on each symmetry sphere, and let _v_0 denote an ingoing null hypersurface that intersects H_A^+, at v=v_0.
Letting X be a smooth vector field which is tangent to _v_0 and transverse to H_A^+, we will say that
ϕ|__v_0 vanishes to order ∈ℤ_0^+ at H^+
 if 

    ϕ|__v_0∩  H^+=(Xϕ)|__v_0∩  H^+= ⋯ = (X^ϕ)|__v_0∩  H^+=0.

We are interested in properties of functions that belong to the space

    _   :=   {ϕ∈ C^∞( M∪ H^+):_gϕ=0,
            ϕ|_H^+∩{v≥ v_0}=0, ϕ|__v_0    H^+},

for a fixed ∈ℤ^+_0 and some v_0∈ℝ.


We may now state our four main theorems. In all of them s
belongs to ℤ^+_0.



 If ϕ∈_ and 
(s+1)κ_+>κ_-, then
 ϕ belongs to C^1( M∪CH_A^+). Moreover, the
second mixed null derivatives of ϕ belong to C^0( M∪CH_A^+),
the restriction of ϕ to symmetry spheres is C^2,
and ϕ satisfies the wave equation on the Cauchy horizon.


 
Let l≥ 1.
If ϕ∈_ and (s+1)κ_+>lκ_-, then ϕ belongs to C^l( M∪CH_A^+).


 If the spherical mean of ϕ (see (<ref>)) belongs to
_∖_+1 and 
(s+1)κ_+<κ_-, then ϕ does not belong to C^1(( M∪CH_A^+)∩ U), for any open set U∩CH_A^+≠∅.


Since the inequality κ_+<κ_- is valid in the entire subextremal range of Reissner-Nordström we conclude that, if ϕ∈_0∖_1, then
it cannot be extended in C^1 to
any neighborhood of any point on the Cauchy horizon.

It is an easy consequence of <cit.> that if ϕ∈_  with
2(s+1)κ_+>κ_-, then ϕ belongs to H^1_ loc(M∪CH_A^+). We prove that this result is essentially sharp.

If the spherical mean of ϕ belongs to
_∖_+1 with
2(s+1)κ_+<κ_-, then ϕ does not belong to H^1(( M∪CH_A^+)∩ U), for any open set U∩CH_A^+≠∅.



The organization of this paper is as follows. 
In Section <ref> we explain the basic setup of our problem.
In Section <ref> we recall
three energy estimates due to Sbierski. In Section <ref> we upgrade the previous to
pointwise estimates. In Section <ref> we prove Theorem <ref> which establishes
the existence of a classical solution up to and including the Cauchy horizon.
In Section <ref> we prove Theorem <ref> concerning solutions with
higher regularity. Finally, in Section <ref> we prove
Theorems <ref> and <ref> about blow up in C^1 and in H^1.



§ SETUP




 §.§ Some useful coordinate systems.

We will study solutions of the wave equation on a fixed background consisting
of  the black hole region of a subextremal Reissner-Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime.
This spacetime has a
metric given in a local coordinate system by

    g   =   -D dt^2+1/D dr^2+r^2σ_𝕊^2,

where
σ_𝕊^2=dθ^2+sin^2θ dφ^2 is the round metric on the 2-sphere, and

    D=D(r)=1- 2m/r+e^2/r^2- Λ/3r^2.

Here m>0 is the mass, e≠ 0 is the charge parameter and Λ∈ℝ is the
cosmological constant. We will assume that
the function D has at least two positive roots, the smallest of which are

    0<r_-<r_+.

The values r_- and r_+ correspond to the values of r at the Cauchy horizon CH^+ and at the event horizon H^+, respectively.
The Penrose diagram of this spacetime for positive Λ is
given in Figure <ref>.





The surface gravities of the Cauchy and event horizons, defined by

    κ_-=1/2|D'(r_-)|   and  κ_+=1/2|D'(r_+)|,

are of fundamental importance to us here. Throughout we will assume that the surface gravities do not vanish, which restricts the scope of our analysis to the subextremal setting.

For r∈[r_-,r_+], we have

    D(r)= - e^O(1)(r-r_-)(r_+-r).

Moreover, any tortoise coordinate

    dr^*/dr=1/D

satisfies, for r∈ ]r_-,r_+[,

    r^*(r)=-1/2κ_-ln|r-r_-|
    +1/2κ_+ln|r-r_+|+O(1).


 The black hole region corresponds to

    (t,r,ω)∈ M:= ]-∞,+∞[×]r_-,r_+[×,

a region where the function D is negative, and where
r^* varies in ]-∞,+∞[.


We will often rely on the double null coordinates (,)∈ ]-∞,+∞[^2 given in
terms of t and r by

    =t-r^*(r),
    =t+r^*(r).

In these coordinates the metric takes the form

    g
       =   -D d d+r^2σ_𝕊^2.

Clearly, we have

    r^*(r)=-/2.

Note that the event horizon corresponds
to =+∞ and the Cauchy horizon corresponds to =+∞.
Since our double null
coordinates are singular at these horizons,
at the event horizon we change from (,,ω) coordinates to (v,r,ω) coordinates
using

    =v-2r^*(r),
    =v,
        ∂_=- D/2∂_r,
    ∂_=∂_v+D/2∂_r.

In these coordinates the metric becomes

    g=-D dv^2+2 dvdr+r^2σ_𝕊^2.

At the Cauchy horizon we change from (,,ω) coordinates to (u,r̃,ω)
coordinates using

    =u,
    =u+2r^*(r̃),
        ∂_=∂_u- D/2∂_r̃,
    ∂_=D/2∂_r̃.

In these coordinates the metric is written as

    g=-D du^2-2 dudr̃+r^2σ_𝕊^2.

Note that to change from (v,r,ω) coordinates to (u,r̃,ω) coordinates we can use

    u=v-2r^*(r),
    r̃=r,
        ∂_u=∂_v,
    ∂_r̃=2/D∂_v+∂_r.

By abuse of notation, we will write ϕ(,,ω)=ϕ(v,r,ω)=
ϕ(u,r,ω).

It is important to note that the vector field ∂_t=∂_v
=∂_u is Killing. We also denote by Ω_i, for i=1,2,3, the generators of spherical symmetry, and
just by Ω any one of the three.
The vector fields Ω
are also Killing.



 §.§ The wave equation

Define

    G_n=1/rD+n/2D',


    S=- 1/r∂_v-1/2Δ        S̃=1/r∂_u-1/2Δ,

with Δϕ denoting the spherical laplacian of ϕ,

    Δϕ=1/r^2∂_θ^2ϕ+1/r^2θ ∂_θϕ
    +1/r^2sin^2θ∂_φ^2ϕ.

The wave equation,

    _gϕ=∇^μ∇_μϕ=0,

is equivalent to both

    ∂_∂_rϕ+G_1∂_rϕ=S(ϕ)

and

    ∂_∂_r̃ϕ-G_1∂_r̃ϕ=-S̃(ϕ).




 §.§ The energy-momentum tensor

Recall that to a scalar function ϕ we may associate the energy-momentum tensor

    T_μν=∂_μϕ∂_νϕ-1/2
    g_μν(∂_αϕ ∂^αϕ),

whose relevance for the study of solutions of the wave equation
stems from the fact that its divergence satisfies

    ∇^μ T_μν=(_gϕ) ∂_νϕ.

Our energy estimates for ϕ will be obtained by applying the Divergence Theorem
to certain currents, which are contractions of the energy-momentum tensor
with appropriate vector fields. It will be useful to have the
expression of the energy-momentum tensor in coordinates.
One readily checks that

    T(∂_,∂_)=(∂_ϕ)^2,
    
    T(∂_,∂_)=D/4|∇ϕ|^2,
    
    T(∂_,∂_)=(∂_ϕ)^2.

Again, ∇ϕ denotes the spherical gradient of ϕ,

    ∇ϕ=1/r^2(∂_θϕ) ∂_θ+1/r^2sin^2θ
    (∂_φϕ) ∂_φ,

and

    |∇ϕ|^2=r^2σ_𝕊^2(∇ϕ,∇ϕ).





 §.§ Energy identities and the Divergence Theorem

We will apply the Divergence Theorem in regions bounded
by hypersurfaces _v, where  is constant equal to v, hypersurfaces C_u,
where  is constant equal to u, and hypersurfaces Σ_r, where
the geometric variable r is constant
equal to r.
Denoting by n__v, n_ C_u and n_Σ_r the corresponding
normals, with
n_Σ_r unit and all three future directed, and denoting by
dV__v, dV_ C_u and dV_Σ_r the corresponding volume elements,
we have

    [              n__v                 =             -∂_r,                               dV__v                 =         r^2 drdω,;                  ;            n_ C_u                 =            -∂_r̃,                             dV_ C_u                 =        r^2 dr̃dω,;                  ;             n_Σ_r                 = -∂_u+D∂_r̃/√(-D),                              dV_Σ_r                 =   √(-D) r^2 dudω, ]

where dω is the volume form associated to σ_𝕊^2. Note that along the null hypersurfaces there is no natural choice of normal or volume form, so one can just
choose a convenient normal and then
let the Divergence Theorem determine the volume form.


Our currents will be vector fields of the form

    J^N_ν=J^N_ν(ϕ):=T_μνN^μ=T_μν(ϕ)N^μ,

with N timelike and future pointing, so that if ϕ is a solution of
the wave equation, then

    ∇^ν J^N_ν   =   
    (∇^ν T_μν)N^μ+T_μν∇^ν N^μ
       =   
    (_gϕ) N·ϕ+T_μν∇^ν N^μ=T_μν∇^ν N^μ.

Our choices of N will be such that T_μν∇^ν N^μ is nonnegative.
We denote by

    C_u(r_2,r_1)= C_u∩{r_2≤ r≤ r_1},   
       _v(r_2,r_1)=_v∩{r_2≤ r≤ r_1}.

Applying the Divergence Theorem to the current J^N_ν in the region

    R={u≥ u_0∧ v≤ v_0∧ r_2≤ r≤ r_1}


(see Figure <ref>) we get the energy identity

    ∫_Σ_r_1∩{u≥ u_0 ∧ v≤ v_0}J^N_ν n^ν_Σ_r_1 dV_Σ_r_1
    -∫__v_0(r_2,r_1)J^N_ν n^ν__v_0 dV__v_0              
    
    -∫_C_u_0(r_2,r_1)J^N_ν n^ν_C_u_0 dV_C_u_0
    -∫_Σ_r_2∩{u≥ u_0 ∧ v≤ v_0}J^N_ν n^ν_Σ_r_2 dV_Σ_r_2              
    
    =∬_R∇^ν J^N_ν dV_M.


For a hypersurface S, the integral

    ∫_SJ^N_ν n^ν_S dV_S=∫_ST(N,n_S) dV_S

controls first order derivatives of ϕ. Let us give an example by defining, near the Cauchy horizon, N=Ñ_b=-∂_-∂_r̃. This choice leads to

    T(Ñ_b,n__v) dV__v   =   (- 2/DT(∂_,∂_)- 4/D^2T(∂_,∂_))r^2 drdω
       =   [- 1/D(2(∂_ϕ)^2+|∇ϕ|^2
    )]r^2 drdω,


    T(Ñ_b,n_ C_u) dV_ C_u   =   (2/DT(∂_,∂_)+4/D^2
    T(∂_,∂_))r^2 dr̃dω
       =   [(∂_r̃ϕ)^2+1/2|∇ϕ|^2
    ]r^2 dr̃dω,


    T(Ñ_b,n_Σ_r) dV_Σ_r
                  =(T(∂_,∂_)
    +(2/D-1)T(∂_,∂_)
    - 2/DT(∂_,∂_))r^2 dudω
                  =[(∂_ϕ)^2- D/2(∂_r̃ϕ)^2+1/2(1- D/2)|∇ϕ|^2
    ]r^2 dudω.

Note that the expressions inside the square parentheses above are
nonnegative as required by the fact that
energy-momentum tensor satisfies the Dominant Energy Condition.

For -1≤ D≤ 0 we have

    T(Ñ_b,n_Σ_r)   ≤   (2(∂_uϕ)^2-D(∂_r̃ϕ)^2+3/4|∇ϕ|^2
    )1/√(-D)

and

    T(Ñ_b,n_Σ_r)   ≥   (1/2(∂_uϕ)^2- D/4(∂_r̃ϕ)^2+1/2|∇ϕ|^2
    )1/√(-D),

so that for r sufficiently close to r_- we have

    T(Ñ_b,n__v)≥ 0,


    T(Ñ_b,n_ C_u)∼ (∂_r̃ϕ)^2+|∇ϕ|^2,


    T(Ñ_b,n_Σ_r)∼((∂_uϕ)^2-D(∂_r̃ϕ)^2+|∇ϕ|^2
    )1/√(-D).

The notation f∼ h means that there exist positive constants c and C such that
cf≤ h≤ Cf.
The blue-shift vector field, which will play a relevant role below, satisfies

    N_b=-(1+O(r-r_-))(∂_+∂_r̃)
 and so expressions (<ref>),
(<ref>) and (<ref>) are also valid if we replace Ñ_b by N_b.




§ BASIC ENERGY ESTIMATES


We denote by

    Σ_r(u_0)=Σ_r∩{u≥ u_0},	  Σ_r(v_0)=Σ_r∩{v≥ v_0}.

We will now recall some basic energy estimates. The first one applies to the red-shift region.
According to <cit.> we have

For every δ>0,
there exists a future directed timelike time invariant vector field N_r in
M∩ H^+, a r_-<r_0<r_+ and
a constant C>0 such that we have

    ∫_Σ_r_0(u)J^N_r(ϕ)· n_Σ_r_0≤ Ce^-(2+2-δ)κ_+u,

for all ϕ belonging to _.

At the event horizon N_r=∂_-∂_r and N_r satisfies
T_μν∇^ν N_r^μ≥ 0 for r_0≤ r≤ r_+.

The second energy estimate applies to the no-shift region.
According to the proof of <cit.> we have

	Given r_-<r_1<r_0<r_+ and a future directed timelike time invariant
	vector field N in M, there exists a constant C>0 such that

    ∫_ C_u(r_1,r_0)J^N(ϕ)· n_ C_u+
    ∫_Σ_r_1(u)J^N(ϕ)· n_Σ_r_1≤ C
    ∫_Σ_r_0(u)J^N(ϕ)· n_Σ_r_0,
	
for all ϕ∈ C^∞( M) satisfying _gϕ=0.

Note that, in the previous case, T_μν∇^ν N^μ might be negative but we can apply the Divergence
Theorem with vector field Ñ=e^C(r_0,r_1)rN and C sufficiently large so that
T_μν∇^νÑ^μ is nonnegative.


The third energy estimate applies to the blue-shift region. According to the proof of <cit.> we have

Assume 2(+1)κ_+>κ_->0.
For every sufficiently small δ, >0,
there exists a future directed timelike time invariant vector field N_b in
M∪ C  H^+, a r_1>r_- and a constant C>0
such that, for all r_1>r_2>r_-,
we have (see <cit.>)

    ∫_Σ_r_1(u_0)e^(2+2-δ)κ_+uJ^N_b(ϕ)· n_Σ_r_1≤ C

and (see <cit.> together with the previous inequality)

    e^κ_-(1+)u_0∫_ C_u_0(r_2,r_1)J^N_b(ϕ)· n_ C_u_0+
    ∫_Σ_r_2(u_0)
    e^κ_-(1+)uJ^N_b(ϕ)· n_Σ_r_2
                      ≤∫_Σ_r_1(u_0)e^κ_-(1+)uJ^N_b(ϕ)· n_Σ_r_1
                      ≤ Ce^-[(2+2-δ)κ_+-κ_-(1+)]u_0,

for all ϕ belonging to _.

Moreover, the function ϕ belongs to
H^1_ loc(M∪CH_A^+).

At the Cauchy horizon N_b=-∂_-∂_r̃ and N_b satisfies
T_μν∇^ν N_b^μ≥ 0 for r_-≤ r≤ r_1.




§ POINTWISE ESTIMATES FOR ∂_UΦ, ∇Φ AND ΔΦ



Let ϕ∈_. Throughout this section we will assume that 2(s+1)κ_+>κ_->0.
We will fix κ satisfying κ<(+1)κ_+.
The objective of the next three subsections is to prove that the three estimates

    |∂_uϕ|(u,r, · )   ≤    Ce^-κ u,
    
    |∇ϕ|(u,r, · )   ≤    Ce^-κ u,
    
    |Δϕ|(u,r, · )   ≤    Ce^-κ u

hold for r_-≤ r≤ r_0<r_+ and u∈ℝ.
We choose a δ>0 satisfying κ≤ (+1-δ)κ_+.



 §.§ Estimates for r=r_0



In a parallel manner to (<ref>), there exists r_0<r_+
such that

    J^N_r_μ(ϕ)n_Σ_r^μ dV_Σ_r∼((∂_vϕ)^2-D(∂_rϕ)^2+
    |∇ϕ|^2)r^2 dvdω

holds for r_0≤ r<r_+.

To obtain a uniform bound on ∂_uϕ we use the following five ingredients:

(i)
From (<ref>) and the fact that u=v-2r^*(r) (or from <cit.>),
for v≥ v_0,
we have that

    ∫_Σ_r_0(v)
    J^N_r(ϕ)· n_Σ_r_0≤
    Ce^-(2+2-δ)κ_+v,

for a red-shift vector field  that satisfies
N_r=(1+O(r_+-r))(∂_v-∂_r).
Since ∂_v and the vector fields Ω are Killing, and ϕ∈_ implies that
∂^l_vΩ^Iϕ∈_, for every l∈ℤ^+_0 and every multi-index I
(see Remark <ref>),
we see that estimate (<ref>) holds with ϕ replaced by ∂^l_vΩ^Iϕ.

(ii) We now apply Sobolev's inequality in symmetry spheres and (<ref>) to obtain

    c∫_v_1^v_2sup_ω∈(∂_vϕ)^2(v,r_0,ω) dv≤∫_v_1^v_2∂_v^2ϕ(v,r_0, · )^2_H^2() dv
        
                  ≤ C∫_Σ_r_0(v_1)
    	J^N_r_μ(∂_vϕ)n_Σ_r_0^μ dV_Σ_r_0
                     +C∑_i=1^3∫_Σ_r_0(v_1)
    	J^N_r_μ(Ω_i∂_vϕ)n_Σ_r_0^μ dV_Σ_r_0
                     +C∑_i=1^3∫_Σ_r_0(v_1)
    	J^N_r_μ(Ω_i^2∂_vϕ)n_Σ_r_0^μ dV_Σ_r_0

and then we use (<ref>) to conclude that

    ∫_v_1^v_2sup_ω∈(∂_vϕ)^2(v,r_0,ω) dv≤ Ce^-(2+2-δ)κ_+v_1.

(iii) We recall <cit.>.


	Let f:[t_0,∞[→ and assume that for some α_1, C>0, and for all
	t≥ t_0,
	
    ∫_t^∞ f(s) ds≤ Ce^-α_1 t.

	Then, for all 0<α_2<α_1,
	
    ∫_t^∞ e^α_2s f(s) ds≤ Ce^-(α_1-α_2)t.



Below we will take α_1=(2+2-δ)κ_+ and α_2=(2+2-2δ)κ_+.

(iv) If we take squares of both sides of

    ∂_vϕ(v_2,r_0,ω)=∫_v_1^v_2∂^2_vϕ(v,r_0,ω)dv + ∂_vϕ(v_1,r_0,ω)

and then apply Hölder's inequality we get

    (∂_vϕ)^2(v_2,r_0,ω)
       ≤   
    2 (∫_v_1^v_2∂^2_vϕ(v,r_0,ω)dv)^2 + 2(∂_vϕ)^2(v_1,r_0,ω)
    
       ≤   
    C e^-(2+2-2δ)κ_+ v_1∫_v_1^v_2e^(2+2-2δ)κ_+ v(∂^2_vϕ)^2(v,r_0,ω)dv
    
           +
    2(∂_vϕ)^2(v_1,r_0,ω).


We can then use (<ref>) and Lemma <ref> to conclude that

    sup_ω∈(∂_vϕ)^2(v_2,r_0,ω)
    
                  ≤ C
    e^-(2+2-2δ)κ_+v_1∫_v_1^v_2e^(2+2-2δ)κ_+v∂_v^2ϕ(v,r_0, · )^2_H^2() dv
    
                      +
    2sup_ω∈(∂_vϕ)^2(v_1,r_0,ω)
    
                  ≤ C
    e^-(2+2-2δ)κ_+v_1+2sup_ω∈(∂_vϕ)^2(v_1,r_0,ω).



(v) From <cit.>, we know that

Let t_0≥ 0, and b, B, M, Δ>0. Let f:[t_0,∞[→^+ be continuous
and such that

    f(t_2)+b∫_t_1^t_2f(t) dt≤ Mf(t_1)+Be^-Δ t_1,

for all t_0≤ t_1≤ t_2. Then, for α<min{b/M,Δ},
we have

    f(t)≤ C_α(1+f(t_0))e^-α t,

for all t≥ t_0.


If we consider the function f(v)=sup_ω∈(∂_vϕ)^2(v,r_0,ω) and add a large multiple of ∫_v_1^v_2 f(v)dv (which satisfies (<ref>)) to both sides of (<ref>),
we can apply the previous lemma
to obtain the pointwise estimate

    |∂_vϕ|(v,r_0, · )≤ Ce^-κ v,

for κ<(+1-δ)κ_+ and v≥ v_0.

Note that the constant C, in the last estimate, is uniform in ω because
∂_vϕ(v_0,r_0, · ) is a bounded function of ω.
For r=r_0 we also have (<ref>)
since ∂_u=∂_v and v=u+2r^*(r).


Since estimate (<ref>) also holds with ϕ replaced by Ω^Iϕ, for any multi-index I,
we can repeat the
proceeding of the previous paragraphs
to obtain (<ref>) and (<ref>), at r=r_0,
with constants C, once again,  uniform in ω.




 §.§ Estimates for r_1≤ r≤ r_0


Let N be a timelike future directed and time independent vector field. Similarly to (<ref>), we have

    J^N_μ(ϕ)n_ C_u^μ dV_ C_u∼((∂_r̃ϕ)^2+|∇ϕ|^2)r^2 drdω.

Since ∂_u=∂_v is Killing, and ϕ∈_ implies that
∂_uϕ∈_, Ω∂_uϕ∈_ and
Ω^2∂_uϕ∈_, for r_1≤ r≤ r_0, by (<ref>) we have

    ∫_r^r_0∂_r̃∂_uϕ(u,r̃, · )^2_H^2() dr̃
              ≤ C∫_ C_u(r,r_0)
    	J^N_μ(∂_uϕ)n_ C_u^μ dV_ C_u+C∑_i=1^3∫_ C_u(r,r_0)
    	J^N_μ(Ω_i∂_uϕ)n_ C_u^μ dV_ C_u
                 +C∑_i=1^3∫_ C_u(r,r_0)
    	J^N_μ(Ω_i^2∂_uϕ)n_ C_u^μ dV_ C_u
              ≤ Ce^-(2+2-δ)κ_+u.

This together with (<ref>), applied with r=r_0, implies that

    sup_ω∈(∂_uϕ)^2(u,r,ω)
                  ≤ C
    	∫_r^r_0∂_r̃∂_uϕ(u,r̃, · )^2_H^2() dr̃+
    	2sup_ω∈(∂_uϕ)^2(u,r_0,ω)
                  ≤ Ce^-(2+2-2δ)κ_+u.

Thus, we obtain (<ref>)
for r_1≤ r≤ r_0.

Applying (<ref>) to Ωϕ, Ω^2ϕ, Ω^3ϕ and Ω^4ϕ, and using (<ref>) and (<ref>) for r=r_0,
we obtain (<ref>) and (<ref>) for r_1≤ r≤ r_0.
The constants C do not depend on r∈[r_1,r_0] or ω.




 §.§ Estimates for r_-<r≤ r_1



In this region, according to (<ref>), we have

    ∫_ C_u(r_1,r_-)J^N_b(ϕ)· n_ C_u≤ Ce^-(2+2-δ)κ_+u.

Applying (<ref>) to ϕ,
Ωϕ and Ω^2ϕ implies that ϕ extends continuously to C H^+ along segments of constant u. Moreover, ϕ( · ,r_-, · )
is the uniform  limit of ϕ( · ,r, · ) as r↘ r_- (for (u,ω)∈[u,U]×, where -∞<u<U<∞). Arguing as in <cit.>, ϕ
is continuous in M∪ C H^+.
The same reasoning can be used to show that ∂_uϕ, Ωϕ
and Ω^2ϕ
extend continuously to C H^+. A simple argument implies that
the derivative of the continuous extension of ϕ with respect to u exists and
coincides with the continuous extension of ∂_uϕ. Analogous statements apply
to Ωϕ and Ω^2ϕ.
Using (<ref>), and reasoning as we did in the region r_1≤ r≤ r_0,
we see that
(<ref>), (<ref>) and (<ref>) hold
for r_-<r≤ r_1.



§ EXISTENCE OF A CLASSICAL SOLUTION UP TO THE CAUCHY HORIZON


Henceforth, by “up to the Cauchy horizon” we mean up to and including the Cauchy horizon.
In this section, we will use the energy estimates of the previous sections, together with Lemma <ref>,
to obtain a pointwise bound for ∂_r̃ϕ, for a fixed r=r_1>r_-.
This together with the previously established pointwise bounds for other derivatives of ϕ, which are valid up to the Cauchy horizon, can then be
used to integrate (<ref>) and obtain a pointwise bound ∂_r̃ϕ,
up to the Cauchy horizon. Finally, the control of this quantity in C^1 will allow us to extend ϕ as a classical solution of the wave equation, all the way up to the Cauchy horizon.


	We proceed in four steps.
	
(i) Bounding ∂_r̃ϕ for r=r_1>r_-.	
Assume (+1)κ_+>κ_-. We now fix κ satisfying
κ_-<κ<(+1)κ_+ and, as before, choose
δ>0 satisfying κ<(+1-δ)κ_+.


We will start by showing that, for a fixed r=r_1>r_-, we have

    |∂_r̃ϕ|(u,r_1,ω)   ≤    Ce^-κ u.

Indeed, this follows by the procedure developed in Section <ref>:  we start
by realizing that for any ψ=∂^l_uΩ^Iϕ, with l∈ℤ_0^+ and I a mutli-index, we have, in view of (<ref>),

    ∫_Σ_r_1(u_1,u_2)J^N_b_μ(ψ)n^μ_Σ_r_1   ∼   ∫_u_1^u_2∫_S^2[(∂_uψ)^2+(-D)
    (∂_r̃ψ)^2+|∇ψ|^2](u,r_1,ω) dω du,
    
       ≤    Ce^-(2+2-δ)κ_+u_1,

which implies that

    ∫_u_1^u_2∂_u∂_r̃ϕ(u,r_1, · )^2_H^2(S^2) du
                      ≤
    	C∫_Σ_r_1(u_1,u_2)J^N_b_μ(∂_uϕ)n^μ_Σ_r_1 dV_Σ_r_1
                         +C∑_i=1^3∫_Σ_r_1(u_1,u_2)J^N_b_μ(Ω_i∂_uϕ)n^μ_Σ_r_1 dV_Σ_r_1
                         +C∑_i=1^3∫_Σ_r_1(u_1,u_2)J^N_b_μ(Ω_i^2∂_uϕ)n^μ_Σ_r_1 dV_Σ_r_1
                      ≤ Ce^-(2+2-δ)κ_+u_1,

and allows one to estimate

    sup_ω∈(∂_r̃ϕ)^2(u_2,r_1,ω)
                  ≤ C
    	e^-(2+2-2δ)κ_+u_1∫_u_1^u_2e^(2+2-2δ)κ_+u∂_u∂_r̃ϕ(u,r_1, · )^2_H^2() du
                      +
    	2sup_ω∈(∂_r̃ϕ)^2(u_1,r_1,ω)
                  ≤ C
    	e^-(2+2-2δ)κ_+u_1+2sup_ω∈(∂_r̃ϕ)^2(u_1,r_1,ω).


We can now apply Lemma <ref>
to u↦sup_ω∈(∂_r̃ϕ)^2(u,r_1,ω) to finish the proof of (<ref>).


(ii) Bounding ∂_r̃ϕ up to the Cauchy horizon.
Let

    u_r_1():=-2r^*(r_1)

so that r_1(u_r_1(),)=r_1.
Integrating the wave equation, in its form (<ref>), along a segment with fixed ,
from u_r_1() to <u_r_1(),
we get

    ∂_r̃ϕ(,,ω)   =   ∂_r̃ϕ(u_r_1(),,ω)
    e^∫_u_r_1()^ G_1(r(ũ,)) dũ
          -
    ∫_u_r_1()^S̃(ϕ)(ũ,,ω)e^∫_ũ^ G_1(r(s,)) ds dũ.

Choose 0<<κ-κ_- and r_1 such that G_1(r)≥-κ_–,
for r_-<r≤ r_1.
Using (<ref>), and (<ref>) and (<ref>) to estimate S̃(ϕ),
yields

    |∂_r̃ϕ(,,ω)|   ≤   Ce^-κ u_r_1()
    e^(κ_-+)(u_r_1()-)+∫^u_r_1()_Ce^-κũe^(κ_-+)(ũ-) 
    dũ
       ≤   Ce^-(κ-κ_–)u_r_1()e^-(κ_-+)+Ce^-(κ_-+) e^-(κ-κ_–)
       ≤   C e^-κ,

for ∈ℝ and ≥+2r^*(r_1).

(iii) Continuity of ∂_r̃ϕ up to the Cauchy horizon.
We define ∂_r̃ϕ(,∞,ω) by

    ∂_r̃ϕ(,∞,ω)=
    ∫^+∞_S̃(ϕ)(ũ,∞,ω)e^κ_-(ũ-) dũ

Let -∞<u<U<+∞. To prove the uniform convergence of ∂_r̃ϕ( · ,, · ) to
∂_r̃ϕ( · ,∞, · ), as →∞, for
the first variable  belonging to [u,U], we write

    ∂_r̃ϕ(,,ω)-∂_r̃ϕ(,∞,ω)
       =   ∂_r̃ϕ(u_r_1(),,ω)
    e^∫_u_r_1()^ G_1(r(ũ,)) dũ
          +∫^U_S̃(ϕ)(ũ,,ω)e^∫_ũ^ G_1(r(s,)) ds dũ
          -∫^U_S̃(ϕ)(ũ,∞,ω)e^∫_ũ^ (-κ_-) ds dũ
          +∫^u_r_1()_US̃(ϕ)(ũ,,ω)e^∫_ũ^ G_1(r(s,)) ds dũ
          -∫^+∞_US̃(ϕ)(ũ,∞,ω)e^∫_ũ^ (-κ_-) ds dũ
       =   A+B+C+D+E.

Let >0.
Again using estimates (<ref>) and (<ref>) to control S̃, we can
fix U and V_1 sufficiently big so that |D|+|E|</3, for
≥ V_1.
Now, using (<ref>), fix V_2≥ V_1 sufficiently large so that |A|</3, for ≥ V_2 and
∈[u,U].
Finally, invoking the uniform convergence, in [u,U]×, of
S̃(ϕ)( · ,, · ) to S̃(ϕ)( · ,∞, · )
and of G_1(r(,)) to -κ_-,
as →∞,
we are allowed to fix V≥ V_2 such that |B+C|</3, for v≥ V.
For v≥ V and (,ω)∈[u,U]×, we have

    |∂_r̃ϕ(,,ω)-∂_r̃ϕ(,∞,ω)|<.

This proves the stated uniform convergence. Again, arguing as in <cit.>, ∂_r̃ϕ
is continuous in M∪ C H_A^+.
A simple argument implies that
the derivative of the continuous extension of ϕ with respect to r̃ exists and
coincides with the continuous extension of ∂_r̃ϕ.

(iv) The wave equation is satisfied on the Cauchy horizon.
To justify that the wave equation (<ref>) is satisfied on the Cauchy horizon we
just have to 
differentiate the right-hand side of (<ref>) with respect to . Note that we are not claiming
that ϕ is C^2 up to the Cauchy horizon but merely that ∂_∂_r̃ϕ exists, is continuous and satisfies (<ref>). We can also
guarantee that ∂_r̃∂_ϕ exists and is continuous. Indeed,
define

    Ŝ=- 1/r∂_u- 1/2Δ.

The wave equation can also be written as

    ∂_r̃∂_ϕ+2/r∂_ϕ=-Ŝ(ϕ).

This can be integrated to

    ∂_ϕ(u,r,ω)=r_2^2/r^2∂_ϕ(u,r_2,ω)
    -∫_r_2^rs^2/r^2Ŝ(ϕ)(u,s,ω) ds.

Note that ∂_ϕ is continuous up to the Cauchy horizon because
it is equal to ∂_uϕ- D/2∂_r̃ϕ. Therefore,
equation (<ref>) holds with r_2=r_-. Another application of the
Fundamental Theorem of Calculus guarantees that
∂_r̃∂_ϕ exists and is continuous up to the Cauchy horizon
and that (<ref>) is satisfied also on the Cauchy horizon. Of course,
the fact that ∂_r̃∂_ϕ exists and is continuous up to the Cauchy horizon is enough to guarantee that (<ref>) is satisfied on the Cauchy horizon.




§ SOLUTIONS WITH HIGHER REGULARITY


This section is devoted to the proof of Theorem <ref>.



 §.§ Wave equations for ∂_r̃^nϕ

We start by deriving the inhomogeneous wave equations satisfied by higher order r-derivatives
of ϕ.
The commutators of _g and ∂_r, and of _g and ∂_r̃, are

    [_g,∂_r]=-D'∂_r^2+2

and

    [_g,∂_r̃]=-D'∂_r̃^2+2,

where

    =1/r^2∂_v+(
    1/r^2D-1/rD'-D”/2)∂_r
    +1/rΔ

and

    =- 1/r^2∂_u+(
    1/r^2D-1/rD'-D”/2)∂_r̃
    +1/rΔ.

Consequently, the function ∂_rϕ satisfies the inhomogeneous wave equation

    _g(∂_rϕ)=-D'∂_r^2ϕ+2(ϕ),

the function ∂_r^2ϕ satisfies the inhomogeneous wave equation

    _g(∂_r^2ϕ)   =   [_g,∂_r]∂_rϕ+∂_r[_g(∂_rϕ)]
       =   -2D'∂_r^3ϕ-D”∂_r^2ϕ+2(∂_rϕ)+2∂_r(ϕ),

and, in general, ∂_r^nϕ satisfies the inhomogeneous wave equation

    _g(∂_r^nϕ)
       =   -∑_l=1^n(n l)D^(l)∂_r^n+2-lϕ+2∑_l=0^n-1∂_r^l(∂_r^n-1-lϕ)
       =   -nD'∂_r^n+1ϕ+2H_n(ϕ),

with

    H_n(ϕ)=- 1/2∑_l=2^n(n l)D^(l)∂_r^n+2-lϕ+∑_l=0^n-1∂_r^l(∂_r^n-1-lϕ).

We also define

    H̃_n(ϕ)=- 1/2∑_l=2^n(n l)D^(l)∂_r̃^n+2-lϕ+∑_l=0^n-1∂_r̃^l(∂_r̃^n-1-lϕ).

Because T (T̃) is a differential operator of order one in r (r̃),
H_n(ϕ) (H̃_n(ϕ)) involves a sum of derivatives of ϕ whose order with respect to r (r̃) is
at most n.

Taking into account that the sequence defined by (<ref>) satisfies

    G_1+n/2D'=G_n+1,

we can write (<ref>) as

    ∂_∂_r^n+1ϕ+G_n+1∂_r^n+1ϕ=S(∂_r^nϕ)+H_n(ϕ).

Similarly, we have that

    ∂_∂_r̃^n+1ϕ-G_n+1∂_r̃^n+1ϕ=
    -S̃(∂_r̃^nϕ)-
    H̃_n(ϕ).




 §.§ Derivatives of ϕ in _

Recall the definition of _ in (<ref>).
 Let l,i_1,i_2,i_3∈ℤ_0^+. Then ϕ∈_ ⇒ ∂_v^lΩ_1^i_1Ω_2^i_2Ω_3^i_3ϕ∈_.



For l=0 the result follows immediately from the fact that the vector fields Ω are Killing and tangent to the sphere {r=r_0}∩{v=v_0}. For the remaining cases it suffices to note that:


  * Obviously, ϕ∈_0 ⇒ ∂_v^lϕ∈_0.

  * ϕ∈_1 ⇒ ∂_v^lϕ∈_1.
Recalling that ∂_=∂_v+D/2∂_r, the wave equation (<ref>) shows that if ϕ vanishes on the event horizon
and ∂_rϕ(v_0,r_+, · )=0, then
∂_v∂_rϕ(v_0,r_+, · )=0.
Differentiating both sides of (<ref>) once with respect to v we then
conclude that ∂_v^2∂_rϕ(v_0,r_+, · )=0.
If we keep differentiating both sides of (<ref>) with respect to v,
we see that
∂_v^l∂_rϕ(v_0,r_+, · )=0.

  * ϕ∈_2 ⇒ ∂_v^lϕ∈_2. Before handling the
general case, lets us also go over the case =2 in detail.
Suppose now that ϕ vanishes on the event horizon, ∂_rϕ(v_0,r_+, · )=0
and ∂_r^2ϕ(v_0,r_+, · )=0. Using (<ref>)
together with the previous paragraph
we conclude that ∂_v∂_r^2ϕ(v_0,r_+, · )=0.
One can also argue that ∂_v^l∂_r^2ϕ(v_0,r_+, · )=0, for all l>0.

  * ϕ∈_n+1 ⇒ ∂_v^lϕ∈_n+1.
More generally, using (<ref>), if ϕ vanishes on the event horizon and its first
n+1 derivatives with respect to r vanish at (v_0,r_+, · ),
then
∂_v^l∂_r^mϕ(v_0,r_+, · )=0, for all m≤ n+1.







 §.§ Bounding higher ∂_r̃ derivatives of ϕ


 The wave equation (<ref>) can be used to bound ∂_r̃^2ϕ when r=r_1>r_-.
If we integrate the
inhomogeneous wave equation for ∂_r̃ϕ and use the bounds
for derivatives of ϕ, whose order with respect to r̃ is at most equal to one, then
we can bound ∂_r̃^2ϕ up to the Cauchy horizon if
2κ_-<(s+1)κ_+. The 2κ_- comes from the fact that
in (<ref>), with n=1, the function in front of ∂_r̃^2ϕ is G_2
and G_2(r_-)=-2κ_-. Moreover, when lκ_-<(s+1)κ_+ we will be able to generalize the previous procedure and establish boundedness of ∂_r̃^lϕ.


 We proceed in three steps.

(i) ϕ∈ C^2( M∪CH_A^+).
Since ϕ∈_ implies ∂_uϕ=∂_vϕ∈_,
using (<ref>) applied to ∂_uϕ, when κ_-<κ<(s+1)κ_+
we have

    |∂_r̃∂_uϕ|(u,r,ω)≤ Ce^-κ u,

for r_-≤ r≤ r_1. Then the wave equation (<ref>) shows that
on the hypersurface r=r_1>r_- we have that

    |∂_r̃^2ϕ|(u,r_1,ω)≤ Ce^-κ u.

Integrating the wave equation (<ref>) with n=1 we obtain

    ∂_r̃^2ϕ(,,ω)   =   ∂_r̃^2ϕ(u_r_1(),,ω)
    e^∫_u_r_1()^ G_2(r(ũ,)) dũ
          -
    ∫_u_r_1()^(S̃(∂_r̃ϕ)+H̃_1(ϕ))(ũ,,ω)e^∫_ũ^ G_2(r(s,)) ds dũ.

The derivatives of ϕ appearing inside the integral
have order at most one with respect to r̃, namely they are
∂_uϕ, Δϕ, ∂_r̃ϕ, ∂_r̃∂_uϕ and
∂_r̃Δϕ.
Moreover, 
    |∂_r̃Δϕ|(u,r,ω)≤ Ce^-κ u,

for r_-≤ r≤ r_1. Therefore, under the assumption that 2κ_-<κ<(s+1)κ_+
we obtain that

    |∂_r̃^2ϕ|(u,r,ω)≤ Ce^-κ u,

for r_-≤ r≤ r_1. As, in addition, when κ_-<κ<(s+1)κ_+, ∂_u^2ϕ,
∂_r̃Ωϕ and ∂_uΩϕ
are continuous on
M∪CH_A^+
we have that ϕ∈ C^2( M∪CH_A^+) when 2κ_-<(s+1)κ_+.

(ii) ϕ∈ C^3( M∪CH_A^+).
Let us consider another specific case, n=2, before analyzing the general situation: repeating the previous argument, using (<ref>) applied to ∂_uϕ, when
2κ_-<κ<(s+1)κ_+ we have

    |∂_r̃^2∂_uϕ|(u,r,ω)≤ Ce^-κ u,

for r_-≤ r≤ r_1. The wave equation (<ref>) shows that
on the hypersurface r=r_1>r_- we have that

    |∂_r̃^3ϕ|(u,r_1,ω)≤ Ce^-κ u.

Since the above mentioned derivatives, ∂_r̃^2∂_uϕ and
∂_r̃^2Δϕ are controlled when 2κ_-<κ<(s+1)κ_+,
integrating the wave equation (<ref>) with n=2, when 3κ_-<κ<(s+1)κ_+ we obtain
that

    |∂_r̃^3ϕ|(u,r,ω)≤ Ce^-κ u,

for r_-≤ r≤ r_1. All other third order derivatives of ϕ
are continuous when 2κ_-<κ(s+1)κ_+ and we are able to conclude that ϕ∈ C^3( M∪CH_A^+) when 3κ_-<(s+1)κ_+.

(iii) ϕ∈ C^l( M∪CH_A^+).
The general case now clearly follows by induction: in fact, when lκ_-<κ<(s+1)κ_+ we obtain that

    |∂_r̃^lϕ|(u,r,ω)≤ Ce^-κ u,

for r_-≤ r≤ r_1. In conclusion, ϕ∈ C^l( M∪CH_A^+) provided lκ_-<(s+1)κ_+.




§ BLOW UP IN C^1 AND IN H^1




 §.§ Blow up in C^1

 This subsection is devoted to the proof of Theorem <ref>. Let us start by sketching the main ideas for this proof.
Suppose that ϕ is spherically symmetric.
If for all large  we have that -∂_ϕ(,v_0, · ) is positive, then it turns out that
-∂_ϕ(,, · ) and
∂_ϕ(,, · ) are positive for (,)∈[u_r_0(v_0),∞[×[v_0,∞[.
This fact can be used to propagate a lower bound for -∂_ϕ, at v=v_0,
all the way up to the Cauchy horizon. We can then
obtain a lower bound for ∂_ϕ and a negative upper bound for
∂_r̃ϕ, 
which can be used to obtain the desired blow up result.

 We proceed in six steps.

(i) Initial data for -∂_ϕ.
Assume first that ϕ is spherically symmetric and that
the restriction of ϕ to the ingoing null hypersurface v=v_0, through the event horizon,
vanishes to order s and does not vanish to order s+1,
on the event horizon.
Then there exist constants 0<c<C such that (eventually replacing ϕ by -ϕ)

    c(r_+-r)^s≤-∂_rϕ(v_0,r, · )≤ C(r_+-r)^s,

for r_0≤ r≤ r_+.
As ∂_ϕ=- D/2∂_rϕ, using (<ref>) we have

    c(r_+-r(,v_0))^s+1≤-∂_ϕ(,v_0, · )≤ C(r_+-r(,v_0))^s+1,

for ≥ u_r_0(v_0). 
According to (<ref>) and (<ref>), for r_0≤ r≤ r_+,
there exist constants 0<c<C such that

    ce^κ_+(-)≤ r_+-r(,)≤ Ce^κ_+(-).

Thus, we get

    ce^-(s+1)κ_+≤-∂_ϕ(,v_0, · )≤ Ce^-(s+1)κ_+,

for ≥ u_r_0(v_0).

(ii) -∂_ϕ in M.
Since ϕ is spherically symmetric
the wave equation reduces to

    ∂_(r∂_ϕ)=-∂_r ∂_ϕ.

According to <cit.>,
-∂_ϕ(,, · ) and
∂_ϕ(,, · ) are positive for
(,)∈[u_r_0(v_0),∞[×[v_0,∞[.
So ↦ -r∂_ϕ( · ,, · ) is an increasing
function. This implies that

    ce^-(s+1)κ_+≤-∂_ϕ(,, · ),

for (,)∈[u_r_0(v_0),∞[×[v_0,∞[.

(iii) -∂_ϕ for r_-≤ r≤ r_0.
Now we integrate the following (spherically symmetric) version of the wave equation

    ∂_(r∂_ϕ)=-∂_r ∂_ϕ

between +∞ and >u_r_0().
Taking into account that in this region

    -∂_r(,)=- D/2(r(,))≥
    c(r_+-r(,))≥ ce^κ_+(-),

we obtain

    (r∂_ϕ)(,, · )   =   (
    r∂_ϕ)(+∞,, · )+∫_+∞^
    (-∂_r ∂_ϕ)(ũ,, · ) 
    dũ
       =   ∫^+∞_[
    (-∂_r) (-∂_ϕ)](ũ,, · ) 
    dũ
       ≥   ∫^+∞_ce^κ_+(-ũ)
    e^-(s+1)κ_+ũ dũ
       =   ce^κ_+(-(s+2)).

On the hypersurface r=r_0 we have =-2r^*(r_0), and so

    ∂_ϕ(u_r_0(),, · )≥
    ce^-(s+1)κ_+.

As -∂_(r∂_ϕ) is positive, it follows that

    ∂_ϕ(,, · )≥
    ce^-(s+1)κ_+,

for u_r_0(v_0)≤≤ u_r_0().

(iv) ∂_r̃ϕ for r_-≤ r≤ r_0.
In the region u_r_0(v_0)≤≤ u_r_0(),
according to (<ref>) and (<ref>),
there exist constants 0<c<C such that

    ce^-κ_-(-)≤ r(,)-r_-≤ Ce^-κ_-(-).

Therefore,

    ∂_r̃ϕ(,, · )   =   (2/D∂_ϕ)(,, · )≤
    -ce^κ_-(-)e^-(s+1)κ_+v
       =   -ce^-κ_-e^(κ_–(s+1)κ_+).


(v) Blow up.
For κ_->(s+1)κ_+ the right-hand side of (<ref>) goes to -∞ as
v goes to +∞. In this case ϕ does not extend to a C^1
function up to the Cauchy horizon.

Suppose now that ϕ is not spherically symmetric. Its spherically
mean

    ψ(,):=1/4π∫_ϕ(,,ω) dω

is also a solution of the wave equation.
According to our hypotheses, ψ is not C^1 up to the Cauchy horizon.
Therefore ϕ cannot be C^1 up to the Cauchy horizon.

(vi) Uniform blow up. The previous analysis only provides blow up of the C^1 norm of ϕ along null rays
=u_1, with u_1≥ u_r_0(v_0). To extend the result to all outgoing null rays  intersecting CH_A^+,  assume that ∂_r̃ϕ is bounded along some u=u_2<u_r_0(v_0). Then we would be able to solve the spherically symmetric wave equation sideways, with characteristic initial data provided by the spherical mean ψ along
[u_2,u_1]×{v_0}∪{u_2}×[v_0,∞], for u_1≥ u_r_0(v_0). But since the spacetimes region  [u_2,u_1]×[v_0,∞] is compact, local well posedness (see for instance <cit.>) and the regularity of the initial data would imply boundedness of ∂_r̃ψ along u=u_1. This is a contradiction.




 §.§ Blow up in H^1

To prove that ϕ does not belong to H^1 it is enough to prove that its
spherically symmetric part does not belong to H^1. Moreover, the negative upper bound (<ref>)
applied to the spherical mean can be used to obtain a lower bound for the H^1 norm of ϕ.

 We proceed in three steps.

(i) The H^1 norm of ϕ.
To define a H^1 norm on M∪CH_A^+, we define a Riemannian metric h on M∪CH_A^+
in the usual way. After choosing a unit timelike vector field X, we
let

    h_μν=g_μν+2X_μ X_ν.

Our choice of X is

    X=1/√(2)(-∂_-∂_r̃)
    =1/√(2)(-∂_u-(1- D/2)∂_r̃).
	
This leads to

    h=(1+D^2/4) du^2+D dudr̃+dr̃^2+r^2σ_𝕊^2.

The square of the norm of the gradient of ϕ is

    h(∇ϕ,∇ϕ)=(∂_ϕ)^2+(∂_r̃ϕ)^2
    +|∇ϕ|^2.

The volume element on M∪CH_A^+ is

    dV_ M=r^2 dω dr̃du.
	
Now we may define the H^1 norm of ϕ to be

    ϕ_H^1(M)^2   =   ϕ_L^2(M)^2+
    	∇ϕ_L^2(M)^2
       =   ∫_Mϕ^2 dV_ M+∫_Mh(∇ϕ,∇ϕ) dV_ M.


(ii) Decomposition of the H^1 norm.
Again, let ψ be the spherical mean of ϕ. We remark that

    ∇ϕ_L^2(M)^2=∇(ϕ-ψ)_L^2(M)^2+∇ψ_L^2(M)^2.

Indeed, this follows from

    ∫_Mh(∇ϕ,∇ψ) dV_ M
              =∫_M((∂_ϕ)(∂_ψ)+
    	(∂_r̃ϕ)(∂_r̃ψ)
    	)r^2 dω dr̃du
              =1/4π∫_M[(∂_ϕ)
    	(∫_∂_ϕ dω)+
    	(∂_r̃ϕ)
    	(∫_∂_r̃ϕ dω)
    	]r^2 dω dr̃du
              =∫_M/SO(3)((∫_∂_ϕ dω)^2+
    	(∫_∂_r̃ϕ dω)^2
    	)r^2 dr̃du
              =∫_Mh(∇ψ,∇ψ) dV_ M.

For the second equality we used (<ref>) and the fact that ϕ belongs to
C^∞(M).

(iii) Blow up.
From (<ref>), to prove that ϕ does not belong to H^1_ loc it is enough to prove
that ψ does not belong to H^1_ loc. Using (<ref>),
the L^2 norm of ∂_r̃ψ is bounded below by

    ∫_u_0^u_1∫_r_-^r_2∫_(∂_r̃ψ)^2
    (u,r,ω) dV_ M
              =∫_u_0^u_1∫_r_-^r_2∫_(∂_r̃ψ)^2
    (u,r,ω)r^2 dω dr̃du
              =1/2∫_u_0^u_1∫_v_r_2(u)^+∞∫_(∂_r̃ψ)^2
    (u,v,ω)(-D(r(u,v)))r^2(u,v) dω d du
              ≥ c
    ∫_u_0^u_1∫_v_r_2(u)^+∞∫_
    e^-2κ_-e^2(κ_–(s+1)κ_+)e^-κ_-(-) 
    dω d d
              =
    ∫_u_0^u_1∫_v_r_2(u)^+∞∫_e^-κ_-
    e^(κ_–2(s+1)κ_+) 
    dω d d,

where r_2>r_- and u_0<u_1<u_r_0(v_1). This integral is infinite provided
κ_->2(s+1)κ_+. The extension of the H^1 blow up to any neighborhood of CH_A^+ follows once again by local well posedness, as in the end of Subsection <ref>.





§ ACKNOWLEDGEMENTS


We thank J. Natário and J.D. Silva for useful comments on a preliminary version of this paper.
We also thank Anne Franzen for sharing and allowing us to use Figure <ref>.
This work was partially supported by FCT/Portugal through UID/MAT/04459/2013 and grant (GPSEinstein) PTDC/MAT-­ANA/1275/2014. 


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