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

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein–Maxwell-scalar field system with a cosmological constant Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}, with the data on the outgoing initial null hypersurface given by a subextremal Reissner–Nordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a “suitably regular” Lorentzian manifold. In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon. In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when Λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda >0$$\end{document}.


Introduction
The Einstein equations are a covariant system of equations relating the geometry of spacetime to its energy and matter content. They are written in appropriate units as where g μν is the spacetime metric, R μν is the corresponding Ricci tensor, R is the scalar curvature, is the cosmological constant and T μν is the energy-momentum tensor. In vacuum, T μν = 0 and (1) is then a closed system of partial differential equations for the metric g μν . In general, however, T μν depends on nonvanishing matter fields, and so (1) must be coupled to other equations governing the matter dynamics.
In suitable coordinates, the Einstein equations become a system of quasi-linear wave equations, naturally leading to an initial value problem. Nonetheless, unlike in other evolution equations of Mathematical Physics, the spacetime geometry is not known a priori. This enables the occurrence of unexpected phenomena, in particular the failure of uniqueness of global solutions without loss of regularity.
An example is provided by the Reissner-Nordström family of solutions 1 , corresponding to static charged black holes (see for instance [16]).
Regarding these solutions as arising from appropriate initial value problems, we encounter the surprising phenomenon that the maximal globally hyperbolic developments (informally, the largest Lorentzian manifolds determined from the initial data, via the Einstein equations) are smoothly extendible, as solutions, in a highly nonunique way. Therefore, global uniqueness fails for the Einstein equations. This puts into question the deterministic character of General Relativity, since what happens in such extensions is not uniquely determined by initial data.
The boundary of the maximal globally hyperbolic development in any given extension is known as the Cauchy horizon, and signals the breakdown of global uniqueness; in the Reissner-Nordström solutions this horizon occurs in the black hole interior. In [24], Penrose and Simpson proposed a heuristic mechanism, known as the blueshift effect, by which arbitrarily small perturbations of the black hole exterior can be infinitely amplified along the Cauchy horizon, turning it into a "singularity" beyond which no "suitably regular" extensions exist. Therefore, the previously discussed pathological features of the Reissner-Nordström family would be artifacts of those particular solutions, unstable under perturbations, and therefore devoid of physical meaning. This reinstated the belief in global uniqueness as a generic property of reasonable initial value problems for the Einstein equations, an idea substantiated in Penrose's strong cosmic censorship conjecture (SCCC), see [4,6,20].
Making the notion of "suitably regular" precise is remarkably subtle, and has evolved considerably during the last decades: the original expectation was that the nonlinearities of the Einstein equations would suffice to turn the Cauchy horizon of the Reissner-Nordström solution, under arbitrarily small perturbations, into a Schwarzschild-like singularity, across which not even continuous extensions of the metric are possible (see [23]). This would completely settle the question of the SCCC in the affirmative. On the other hand, the existence of generic C 2 extensions as solutions of the Einstein equations would completely falsify this conjecture. Nonetheless, C 2 -inextendibility does not necessarily provide a compelling argument in favor of determinism, since there are relevant solutions of the Einstein equations whose regularity is well below this threshold. Thus a formulation of the SCCC in terms of the generic blow up of the Kretschmann scalar (tidal forces), favoured by many authors, is manifestly insufficient, as it only rules out C 2 extensions.
In a seminal paper, Poisson and Israel [21] identified the blow up of a renormalized version of the Hawking mass at the Cauchy horizon as a consequence of the blueshift mechanism (see also [17,19]), implying the blow up of the Kretchmann scalar. This scenario, which was confirmed by Dafermos in his celebrated non-linear analysis of the spherically symmetric Einstein-Maxwell-scalar field system with = 0 (see [9,10]), became known as mass inflation. As a consequence, the current expectation is that, for generic initial data in the context of black hole spacetimes, the metric extends beyond the Cauchy horizon in C 0 but not in C 2 . To accomodate these developments, Christodoulou [5] proposed a formulation of the SCCC that excludes the generic existence of extensions of the metric with square integrable connection coefficients. As already suggested by Chruściel, this guarantees that (generically) the potential extensions will not be regular enough to solve the Einstein equations, even in a weak sense. By now, there is strong evidence that this formulation of the SCCC, which we will refer to as the Christodoulou-Chruściel criterion, holds for asymptotically flat black holes (see [11,18]).
It turns out that for cosmological black holes, i.e. black hole solutions of the Einstein equations with a positive cosmological constant , the instability mechanism is expected to be weaker. Although the introduction of this term has a negligible impact on the causal structure of the black hole interior, where the blueshift occurs, it has dramatic consequences for the structure of the exterior. In particular, a new horizon, known as the cosmological horizon, is formed. This generates an extra redshift, which counteracts the blueshift effect. Related to this is Price's law, which predicts that, in Eddington-Finkelstein coordinates, the decay of the perturbations along the event horizon is polynomial in the asymptotically flat case, but exponential in the cosmological case. From the start, it was clear that these facts could have a strong influence on the issue of stability of the Cauchy horizon. This was enough to generate a considerable amount of activity concerning the SCCC in the positive cosmological constant setting, raising the possibility that mass inflation might not occur (see [1][2][3]). After intense debate, the consensus was reached that the SCCC would probably prevail, at least in its weaker C 2 formulation. Unfortunately, most of the reasoning leading to this conclusion was based on heuristic arguments or perturbative calculations, with, for instance, the back-reaction of the metric being "put in by hand". This calls for a more detailed analysis, that takes into account the entire non-linear structure of the Einstein equations and is able to capture the fine regularity properties of potential extensions, especially in view of the growing popularity of the Christodoulou-Chruściel criterion. This analysis has became even more pertinent since, in the meantime, it was discovered that supernovae observations are best fitted by models with > 0. This paper is the third part of a trilogy dedicated to the full non-linear evolution, inside a black hole, of the Einstein equations (1) with nonvanishing cosmological constant . The matter model consists of a massless scalar field φ and an electromagnetic field F, satisfying the Maxwell and wave equations where is the Hodge star operator and is the d'Alembertian (both depending on g). These equations couple to (1) through the energy-momentum tensor We choose this matter model because we wish to consider spherically symmetric perturbations of the Cauchy horizon of the Reissner-Nordström spacetime; since Birkhoff's theorem implies that this is the only spherically symmetric electrovacuum solution, we also introduce a self-gravitating real massless scalar field to provide dynamical degrees of freedom with the same hyperbolic character. More precisely, we study the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system (1)−(4), with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordström black hole event horizon, and the remaining data otherwise free, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold. Strictly speaking, this problem does not address the strong cosmic censorship conjecture directly, because the data considered on the event horizon does not arise from the gravitational collapse of generic Cauchy initial data. However, since the Price law for > 0 is widely expected to yield exponential decay of the scalar field along the event horizon (see for instance the linear analysis of [14,15]), we believe that our conclusions will provide valuable insights for this cosmological case.
In Part 1 [7] of this trilogy, we showed the equivalence (under appropriate regularity conditions for the initial data) between the Einstein-Maxwell-scalar field equations (1)−(4) under spherical symmetry and the system of first order PDEs (19)−(28). We established existence, uniqueness and identified a breakdown criterion for solutions of this system. In Part 2 [8], we analyzed the properties of the solution up to the Cauchy horizon, proving, in particular, the stability of the radius function. See Section 2 for a summary of our previous results, as well as for the definitions and notation that will be used henceforth. We refer the reader to Parts 1 and 2 for the complete details.
In this paper we examine the behavior of the renormalized Hawking mass (see (13)) and the scalar field at the Cauchy horizon. Depending on the control that we have on these quantities, we are able to construct extensions of the metric beyond the Cauchy horizon with different degrees of regularity. The quotient of the surface gravities (see (41)) of the Cauchy and event horizons r = r − and r = r + in the reference Reissner-Nordström black hole plays an important role in our analysis.
We start by briefly recalling the strategy of Dafermos [9,10] to establish mass inflation (that is, blow-up of at the Cauchy horizon), which naturally generalizes to the case of a non-vanishing cosmological constant. This requires the initial field ζ 0 (see (16) and (29)) to satisfy (for constants c > 0, U > 0 and u ∈ [0, U ]) ζ 0 (u) ≥ cu s for some 0 < s < ρ 2 − 1, see Theorem 3.1. Since the mass is a scalar invariant involving first derivatives of the metric, its blow up excludes the existence of spherically symmetric C 1 extensions. 2 Moreover, using the techniques of [11], one can hope to prove that in this case the Christodoulou-Chruściel inextendibility criterion holds, that is, there is no extension of the metric beyond the Cauchy horizon with Christoffel symbols in L 2 loc . The previous approach only allows us to explore a particular subregion of parameter space, corresponding to sufficiently subextremal reference solutions (see the Figure  below). We proceed by extending the analysis to the full parameter range. First we prove that if the field ζ 0 satisfies the weaker hypothesis ζ 0 (u) ≥ cu s for some 0 < s < ρ − 1, then either the renormalized mass or the field θ λ (see (12) and (15)) blow up at the Cauchy horizon (Theorem 3.2). As a consequence, the Kretschmann curvature scalar also blows up (Remark 6.9).
On the other hand, when the initial field ζ 0 satisfies we show that the mass remains bounded (Theorem 4.1). This behavior is in contrast with the standard picture of spherically symmetric gravitational collapse. The case where no mass inflation occurs is then analyzed in further detail. We construct C 0 spherically symmetric extensions of the metric beyond the Cauchy horizon with the second mixed derivatives of r continuous. There are two natural coordinate choices for the extension, corresponding to either λ = −1 or κ = 1 (see (12) and (17)) on the outgoing null ray u = U . Interestingly, these lead to inequivalent C 2 structures for the extended manifolds, a fact that is reflected on the behavior of the Christoffel symbols: when the initial data satisfies we prove that one of the Christoffel symbols blows up on u = U at the Cauchy horizon in the λ = −1 coordinates, but not in the κ = 1 coordinates. Moreover, for both coordinate systems this Christoffel symbol blows up along almost all outgoing null rays, which excludes the existence of C 0,1 extensions of the metric. Nonetheless, in the κ = 1 coordinates the Christoffel symbols are in L 2 loc and the field φ is in H 1 loc for the whole range of initial data where there is no mass inflation (Corollary 5.11). Therefore, the Christodoulou-Chruściel inextendibility criterion for strong cosmic censorship does not hold in this setting.
Finally, assuming that we can bound the field θ λ at the Cauchy horizon. This allows us to prove that the solution of the first order system extends, non-uniquely, to a classical solution beyond the Cauchy horizon (Theorem 6.5). We then show that this solution corresponds to a classical solution of the Einstein equations extending beyond the Cauchy horizon (Theorem 6.7). The metric for this solution is C 1 and such that r ∈ C 2 and ∂ u ∂ v (see (5)) exists and is continuous (Remark 6.8). However, we emphasize that the metric does not have to be C 2 , in spite of the Kretschmann curvature scalar being bounded. To the best of our knowledge, these are the first results where the generic existence of extensions as solutions of the Einstein equations is established.
It should be noted that these results, while valid for all signs of the cosmological constant , only provide evidence against the strong cosmic censorship conjecture in the case > 0, since, as discussed above, only in this case does one expect an exponential decay of perturbations along the event horizon, even in the absence of symmetry assumptions.
In summary, for a given ρ and cu s ≤ ζ 0 (u) ≤ Cu s , the behavior of the solution at the Cauchy horizon depends on the value of s as described in the following figure.
The lines represent the limits of the various strict inequalities above.
In Appendix A we explain how ρ depends on the physical parameters r + , r − and . In particular, ρ is a function of r + r − and r 2 − 3 .

Framework and Some Results from Parts 1 and The Spherically Symmetric Einstein-Maxwell-scalar Field System with a Cosmological Constant
Consider a spherically symmetric spacetime with metric where σ S 2 is the round metric on the 2-sphere. In this case, the Maxwell equations decouple from the system, since they can be immediately solved to yield Here e is a real constant, corresponding to a total electric charge 4π e, and we have assumed zero magnetic charge without loss of generality. The remaining equations can then be written as follows: a wave equation for r , a wave equation for the massless scalar field φ, the Raychaudhuri equation in the u direction, the Raychaudhuri equation in the v direction, and a wave equation for ln ,

The First Order System
Given r , φ and , solutions of the Einstein equations, we define the following quantities: and Notice that we may rewrite (13) as The Einstein equations imply the first order system for (r, ν, λ, , θ, ζ, κ) with the restriction Here by ∂ r (1 − μ) we mean Under appropriate regularity conditions for the initial data, the system of first order PDE (19)−(28) also implies the spherically symmetric Einstein equations (6)−(10) (Part 1, Propositions 6.3 and 6.4).

Initial Data
We take the initial data on the v axis to be the data on the event horizon of a subextremal Reissner-Nordström solution with mass M, electric charge e and cosmological constant . Therefore we choose initial data as follows: where 0 < U < r + , and Here r + > 0 is the radius of the event horizon. We assume that ζ 0 is continuous with ζ 0 (0) = 0, and denote M by 0 .
Here r − > 0 is the radius of the Cauchy horizon of the Reissner-Nordström reference solution and (which exists and is decreasing). Similarly, we also define Following the same argument as in [10, Section 11], Theorem 2.3 implies that the spacetime is extendible across the Cauchy horizon with a C 0 metric.

Two Effects of Any Nonzero Field
This lemma implies that the affine parameter of an outgoing null geodesic is finite at the Cauchy horizon.

Well Posedness for the Backwards Problem
In Section 6, we will extend the solutions of Einstein's equations beyond the Cauchy horizon. For this we will need to solve a backwards problem, already discussed in Part 1. The initial conditions will be prescribed as follows: We letr We assume the regularity conditions: the functions ν 0 , ζ 0 , λ 0 , θ 0 and κ 0 are continuous, and the functions r 0 and 0 are continuously differentiable.
We assume the sign conditions: We assume the three compatibility conditions:

Retrieving the Einstein Equations from the First Order System
We consider the additional regularity condition (h4) ν 0 , κ 0 and λ 0 are continuously differentiable.

The Partition of Spacetime into Four Regions
In Part 2, we divide [0, U ] × [0, ∞[ into four disjoint regions, separated by three curves, ř + , ř − and γ , where different estimates can be obtained (see Appendix C). Next we explain how these curves are constructed.

The Curves ř
We denote by ř := r −1 (ř ) the level sets of the radius function. These are spacelike curves and consequently may be parameterized by We chooseř + andř − sufficiently close to r + and r − with r − <ř − <ř + < r + .

The Curve γ = γř − ,β
Givenř − as before and β > 0, we define γ to be the curve parametrized by This curve probes the region near the Cauchy horizon. The parameter β measures the deviation of γ with respect to ř − .

The Choice of β so that r and are Controlled
Choose any β such that Let 0 < ε < ε 0 . Then, by Lemma 6.1 and Corollary 6.2 of Part 2, there exists U ε > 0 such that provided that the parametersř + , ε 0 and δ are chosen so that Hereδ is a bound for ζ ν in J − (ř + ). Suppose that there exist positive constants C and s such that |ζ 0 (u)| ≤ Cu s . Then, instead of choosing β according to (36) we may choose 0 < β < 1 In this case, (38) should be replaced by

A Note on the Choice of Parameters
The estimates obtained in Part 2, some of which are listed in Appendix C, depend on the choice of a number of parameters, namely β − , β + ,ř − ,ř + , ε 0 and U . As a rule, these estimates hold if β − < β and β + > β are chosen sufficiently close to β,ř − > r − is chosen sufficiently close to r − ,ř + < r + is chosen sufficiently close to r + and ε 0 > 0, U > 0 are chosen sufficiently small. The deviations of these parameters from β, r − , r + and 0, respectively, are controlled by a generic small parameter, typically denoted by δ or ε, where we will absorb all small quantities (so that δ or ε may change from line to line).

Mass Inflation
We denote the surface gravities of the Cauchy and black hole horizons in the reference subextremal Reissner-Nordström black hole by and define (see Appendix A) This parameter measures how close the black hole is to being extremal, which corresponds to ρ = 1.
We start by presenting a sufficient condition for the renormalized mass to blow up identically on the Cauchy horizon.

Theorem 3.1 (Mass inflation) Suppose that ρ > 2 and
where c > 0 and u ∈ [0, U ]. Then In Appendix A we see how the condition ρ > 2 translates into a relationship between r − , r + and .
The proof of Theorem 3.1 generally follows the argument on pages 493-497 of [10], where the = 0 case was studied. Nonetheless, the introduction of a cosmological constant requires a different technical approach, in particular the use of a foliation by the level sets of the radius function; moreover, since later on we will need some of the estimates derived in the proof, we present the relevant details in Appendix B.
The previous techniques only allow us to explore the subregion of parameter space determined by (43). The rest of this paper will be dedicated to the analysis of the full parameter range. The first result in that direction is Proof Suppose that there exists U > 0 such that (U, ∞) < ∞. Going through the proof of Theorem 3.1, we see that Case 3.2 must occur. So −λ must be bounded above in J + (γ ) as in (117). Furthermore, the lower estimate on ζ 0 guarantees the lower bound (120) for θ in J + (γ ). Combining (117) with (120), we obtain for (u, v) ∈ J + (γ ). We chooseř + sufficiently close to r + ,ř − sufficiently close to r − , β + and β − sufficiently close to β, and U sufficiently small so that δ < ρ − s − 1 (see the note in the end of Section 2).
In Remark 6.9 we will see that under the hypothesis of the previous theorem the Kretschmann scalar always blows up at the Cauchy horizon, as a consequence of either or θ λ blowing up.

No Mass Inflation
In this section we will prove that mass inflation does not occur if ζ 0 decays sufficiently fast as u tends to zero. We also control the field ζ up to the Cauchy horizon.
Given ε 1 > 0, define The set D is connected and contains γ . Our goal is to prove that, for U small enough, D = J + (γ ). This is a consequence of Indeed, for ε 1 and U small enough, Proof of Lemma 4.2 Our goal is to improve the upper estimate (146) for −λ in D, to obtain a lower estimate for −λ in D, and to obtain an upper estimate for |θ | in D.
These will allow us to prove that θ 2 −λ (u, v) decays exponentially in v, from which the conclusion of the lemma will easily follow. Note that the estimates used in this proof will be sharper than needed here, for use in Section 6.
Integrating (24) as a linear first order ODE for , starting from γ , leads to Letε > 0. If ε 1 and U are sufficiently small, we have from (37) and (47) | On the other hand, we have r <ř − in J + (γ ) and, using (31), we know with C 1 > 1. The value of C 1 can be chosen as close to one as desired by decreasing ε 1 ,ř − − r − and U . Henceforth, C 1 will denote a constant greater than one, which can be made arbitrarily close to one by a convenient choice of parameters. C will denote a positive constant. We start by recalling some estimates over γ . Collecting (126), (139), (145) and (46), we get According to (142) and (143), Combining (49) with (50), Finally, according to (144) and (145), Recall that β − < β < β + can be chosen arbitrarily close to β. We now improve the upper estimate (146) for −λ in D. Taking into account (140), Arguing as in (114), In the last inequality we introduced a parameter 0 < q ≤ 1 whose importance will become apparent below. Equation (116) The parameter q makes the second exponential grow slower as u 0 (at the cost of making the first exponential decay slower). Note that q = 0 corresponds to (146). This is our improved estimate for −λ from above.
We now obtain a lower estimate for −λ in D. Arguing as above, we have Using (116) together with (50) once more, for (u, v) ∈ D, we obtain This is our estimate for −λ from below.
We will now control θ in D. Integrating (25) and (26) from γ leads to It follows that In the next two paragraphs we bound I and II . Collecting (126), (137), (145) and (46), we obtain Using (147), (53) and (58), we have Here p = ρβ − 1 − δ. Using (52), the integral above can be estimated as and if the parameters are chosen so that δ is sufficiently small. Therefore, it is possible to bound I as follows: For v ≥ v γ (u), we define We emphasize that the constants C will not depend on u. Using (147), (53) and (62), we see that where we used the fact that v γ (u) ≤ v γ (ũ). Again, p = ρβ − 1 − δ. Using (52), the first integral above can be estimated as and if δ is sufficiently small. Therefore it is possible to bound II as follows: For all v ≥ v γ (u), we estimate T u (v) using (51), (57), (61) and (65): We claim that and small δ.
We impose (68); it can be checked that considering also the opposite inequality will not lead to an improvement of the statement of Theorem 4.1 (for the choice of parameters that we make below).
Proof of the claim Inequality (66), with u ρ(β−q)−δ bounded by a constant, is of the form Applying Gronwall's inequality, we get To estimate e −v γ (u) we used (144). We also used A 1+β 2k Using (55) and (67), we obtain The exponent in (72) can be made negative if and the second exponent in (73) can be made negative if In what follows we will not exploit the smallness of u 2b . Below we will characterize a choice of parameters for which we have for (u, v) ∈ D, with > 0. However, before we do that, we note that estimate (76) wraps up the bootstrap argument. Indeed, as lim u 0 v γ (u) = +∞, we can choose U such that We now bring together the conditions that we must satisfy in order for the above argument to work, and we choose our parameters. The number β is bounded above by (39) and bounded below by (64) and (75); in addition, s is bounded below by (60), (68) and (74). In fact, the restrictions on s can be stated in a simpler form: inequality (68) is stricter than (60); inequality (75) implies that (68) is stricter than (74). So, all the restrictions on s amount to saying that s is bounded below by (68).
We now select the parameters q and β. The minimum of the maximum of the lower bounds for β in (64) and (75) is obtained for q = 1 3 . This is our choice of q. Inequality (39) can be satisfied when s > 2ρ 9 − 1 because For (68) to be satisfied we impose 7ρ 9 − 1 < s because Obviously, 2ρ 9 − 1 < 7ρ 9 − 1. Therefore, if s > 7ρ 9 − 1 and we choose β = 1 3 + ε, with ε > 0 sufficiently small, both (39) and (68) are satisfied. Therefore, our parameters will be chosen in the following way. Suppose that we are given initial data ζ 0 satisfying (46). We choose β > 1 3 (so that (64) and (75) hold with q = 1 3 ) and such that (39) and (68) hold. When the parameters δ above all converge to 0 (at the cost of increasing the constants C). So, we may chooseř + sufficiently close to r + ,ř − sufficiently close to r − , β + and β − sufficiently close to β, and ε 0 , ε 1 and U sufficiently small so that (40) holds, the exponent in (72) and the second exponent in (73) are negative, the integrals (59) and (63)  We finish this section by controlling the field ζ in the following result.
Then there exists a constant C > 0 such that for (u, v) ∈ J + (γ ), where δ > 0 can be chosen arbitrarily close to zero, provided that U is sufficiently small.

Extensions of the Metric Beyond the Cauchy Horizon
In this section we assume that the field ζ 0 satisfies |ζ 0 (u)| ≤ cu s for some nonnegative s > 7ρ 9 − 1, so that there is no mass inflation, and we examine the possibility of extending the metric beyond the Cauchy horizon. We regard the (u, v) plane, the domain of our first order system, as a C 2 manifold. Since the Cauchy horizon corresponds to v = ∞, we must change this coordinate to one with a finite range. There are two natural choices to do so: either resorting to the radius function along the outgoing null ray u = U for the new coordinate (i.e. choosing λ = −1 on u = U ), or setting κ = 1 on the null ray u = U (as was done for the initial data along the event horizon).
In the first coordinate system, v is then replaced byṽ = r (U, 0) − r (U, v). This is the coordinate system that we will later use in Section 6; it transforms the domain [0, ∞[ of v into a bounded interval forṽ, even when the field ζ 0 is identically zero. In the second coordinate system, which has finite range only when ζ 0 is not identically zero, v is replaced byv :=  (93)). By Remark 5.5, the two coordinate systems (u,ṽ) and (u,v) are not equivalent (as C 2 coordinate systems) when θ λ is unbounded along u = U . In both coordinate systems we can extend the metric continuously to the Cauchy horizon, and consequently beyond the Cauchy horizon, with the second mixed derivatives of r continuous. In the coordinate system (u,v) this can be done so that the Christoffel symbols are in L 2 loc and the field φ is in H 1 loc . Therefore, the Christodoulou-Chruściel inextendibility criterion for strong cosmic censorship does not hold.

Coordinates with v Replaced byṽ = r(U, 0) − r(U, v)
If there exists a positive sequence (u n ) converging to 0 such that ζ 0 (u n ) = 0 then we choose U such that (1 − μ)(U, ∞) < 0. In the proof of Lemma 2.5 we showed that such a U exists; in Proposition 5.2 we will see that under the present assumptions (1 − μ)(U, ∞) < 0 for any U > 0, so that actually any choice of U will do. If ζ 0 vanishes in a right neighborhood of the origin then the solution is simply Reissner-Nordström and we can choose any U . We define f : so that and set We will change the v coordinate toṽ The functions ν 0 , κ 0 and λ 0 for the original characteristc initial value problem (equal to −1, 1 and 0, respectively, see (29) and (30)) satisfy hypothesis (h4) (see Section 2). By Lemma 2.7, the function r is C 2 . Moreover, λ(U, · ) < 0. Therefore, the change of coordinates of the previous paragraph is admissible (that is, C 2 ).
We denote byr the function r written in the new coordinates, i.e.
On the other hand, to obtain the equations that involve the derivative with respect to v, we write these equations in integrated form, say from 0 toṽ n , and letṽ n V . From the (trivial) continuity of the indefinite integral of a continuous function and the Fundamental Theorem of Calculus, we deduce that the equations are valid at V .
Obviously, (28) is satisfied on the segment [δ, U ] × {V }. Finally, taking into account (122)) and thatν is negative on [δ, U ] × {V } (see (83)), we conclude that The metric and the field. Recall that the reason to study our first order system is that its solutions allow the construction of spherically symmetric Lorentzian manifolds This allows for C 0 extensions of the metric beyond the Cauchy horizon, by a similar construction as the one that will be used below for the coordinate system (u,v). The fieldφ is determined, after prescribingφ(0, 0), by integrating (15) and (16). According to [10,Proposition 13.2] (with the choice u 1 (see [12, Appendix A]).

Coordinates with v Replaced byv
Assume there exists a positive sequence (u n ) converging to 0 such that ζ 0 (u n ) = 0. We change the v coordinate tov According to (32),V := ∞ 0 κ(U,v) dv < ∞. From Lemma 2.7, κ is C 1 ; since κ is also positive, this change of coordinates is admissible (C 2 ).
We denote byr ,ν,λ,ˆ ,θ,ζ andκ the functions written in the coordinates (u,v). In particular,κ Fromκ(U,v) ≡ 1 and (27), we get Relationship between theṽ and thev coordinates. We now show that when θ λ is unbounded the change of coordinates fromṽ tov is not C 2 at the Cauchy horizon. From (81) and (91), we write Using the chain rule, (14) and (24), we obtain According to Proposition 5.2, (1 − μ)(U, · ) is bounded away from zero (note that this quantity does not depend on the choice of coordinate system). So indeed, we have Proof This is a consequence of Propostion 5.2 and the fact that the mapv →ṽ extends to a C 1 map from [0,V ] to [0, V ]. For example, to check (20) at the Cauchy horizon, note that from ∂ṽr (u,ṽ) =λ(u,ṽ) we conclude that The spherically symmetric Lorentzian manifold M is nowQ×S 2 , whereQ admits the global null coordinate system (u,v) defined on [0, U ] × [0,V ] \ {(0,V )}, and the metric is The fieldφ(u,v) equals φ(u, v) and soφ(u,ṽ). The nonvanishing Christoffel symbols of the metric on M are written as the ones above, with tildes replaced by hats. For example, instead of (85), we havê and so d 2ṽ dv 2 must blow up at the Cauchy horizon (as was already shown in (94) by direct computation). This again shows that the two coordinate systems (u,ṽ) and (u,v) are not C 2 compatible. More generally, the same reasoning can be applied to show the C 2 incompatibility of any two coordinate systems whose Christoffel symbols v vv have different asymptotic behavior at the Cauchy horizon. In particular, different choices of U yield incompatible (u,v) coordinates (when θ λ is unbounded).
It turns out that, although unbounded, the Christoffel symbols of the (u,v) coordinates are in L 2 . we then know that V 0 |θ | 2 (u,v) dv is bounded for u ∈ [δ, U ]. Differentiating both sides of (92) with respect tov, and using (22) and (26), we get ). The previous equality, (22) and (95) then imply that there exists a C > 0 such that , and so, using Hölder's inequality, Finally, note that the square of the L 2 norm of a functionĥ on M δ is given by Since the functionsr andˆ 2 = −4νκ are bounded in [δ, U ] × [0,V ], we conclude that the Christoffel symbols andθ are in L 2 (M δ ).
So, in our framework the Christodoulou-Chruściel formulation of strong cosmic censorship (see [5]) does not hold: a C 0 metric onM. HereQ has a global null coordinate system (u,v) defined on Proof The extensionsˇ 2 ,φ andř ofˆ 2 ,φ andr are continuous. For u > 0 andv >V , we get Clearly, ∂ uˇ 2 ,λ andν are also continuous. Therefore,ˇ C AB ,ˇ u AB ,ˇ v AB ,ˇ A Bû ,ˇ A Bv , andˇ u uu are continuous, and so is the fieldζ . Finally,ˇ v vv andθ are zero forv >V . It would be easy to construct other extensions of (M, g) andφ satisfying (96).
Note that there is no guarantee that the extensions above satisfy the Einstein equations. Moreover, the functionθ may not admit a continuous extension to the Cauchy horizon.

Remark 5.12 Since in the previous extension
forv >V , we constructed a C 0 extension of the metric such that (ˇ ∈ L 2 loc ,φ ∈ H 1 loc and) the second mixed derivatives ofř are continuous. This would not be possible ifˆ ( · ,V ) were +∞ (see (21) and (22)). In [10, Theorem 11.1] M. Dafermos constructs C 0 extensions of the metric without assuming any restriction on the continuous function ζ 0 , so without any control onˆ ( · ,V ).

Extensions of Solutions Beyond the Cauchy Horizon
It is clear that in order to improve on the results of the previous section we need to control the field θ λ . In view of Proposition 5.3, this requires a stronger restriction on the exponent s. Once the field is controlled, it turns out to be possible to construct smooth extensions of our spacetime which in fact are solutions of the Einstein equations.
More precisely, in the main part of this section we assume that where c > 0 and u ∈ [0, U ]. In Lemma 6.1, we obtain the desired bound for θ λ in J + (γ ). We then start by proving that our solution of the first order system (19) for (u, v) ∈ J + (γ ), provided that U is sufficiently small. Furthermore, Proof Integrating (123), we obtain By Theorem 4.1, we know that we have |∂ r (1−μ)+2k − | < δ in J + (γ ) for sufficiently small U . Using (49) and (54), This exponent can be made negative for Now, according to (122) and (129) due to the monotonicity of ν 1−μ . Thus, Combining this with (77), if s > ρ(β 2 + 1) − 1 and if the parameters are chosen appropriately, we get Using (100) and (103) in (99), taking into account that the right-hand side of (101) would be 13ρ 9 − 1 if β were 1 3 , and recalling that we can choose β = 1 3 + ε, we obtain (97).
To prove the last assertion, notice that for (u, v) ∈ J − (γ ) ∩ J + ( ř − ) the estimate on the right-hand side of (49) applies since u ≤ u γ (v). Also, recall (130). All this information, together with (99) and the bounds (100)  Proof of Theorem 6.2 We fix 0 < δ < U . We already did most of the work in Proposition 5.2. So, we just need to prove the assertion forθ and that (24), (25) and (26) are satisfied on [δ, U ] × [0, V ]. As before, we proceed in three steps.
Step 1 We prove thatθ(·,ṽ) converges uniformly toθ(·, V ) in [δ, U ] asṽ → V , that is, We let v ∞ in (99). Taking into account the estimate (100) for the first term on the right-hand side, and using Lebesgue's Dominated Convergence Theorem and (103) for the second term on the right-hand side, we conclude that is also well defined. We now wish to prove uniform convergence of θ λ ( · , v) to θ λ ( · , ∞), as v ∞. We write Suppose that we are given ε > 0. Notice the upper limits of the integrals in IV and V : the outer integrals have upper limitδ, while the inner integrals have upper limit u. Nevertheless, we may do computations similar to (103), using (102), to conclude that we may chooseδ > 0 so that |IV | + |V | < ε 3 , for all u ∈ [δ, U ]. We fix such aδ. By (100), there existsṼ ε > 0 such that for v ≥Ṽ ε we have |I | < ε 3 , again for all u ∈ [δ, U ]. When estimating |II + III | we replace the upper limit of integration u by U (after gathering this difference into a single integral and taking absolut values). Finally, by uniform convergence of the functions in the integral II to the functions in the integral III , in [δ, U ], there exists V ε ≥Ṽ ε such that |II + III | < ε 3 , for v ≥ V ε . So for v ≥ V ε and for all u ∈ [δ, U ], we have This establishes the desired uniform convergence.
Step 2 As in Step 2 of the proof of Proposition 5.2, we conclude thatθ is continuous in the closed rectangle [δ, U ] × [0, V ].
On the choice of initial data beyond the Cauchy horizon. Fix 0 <ε <r (U, V ), and consider the continuous extensionλ(U, · ) ≡ −1 to the interval [0, V +ε]. According to this choice, definẽ forṽ ∈ ]V, V +ε]. The upper bound onε is imposed to guarantee that Choose a continuously differentiable extension of˜ (U, · ) to the interval [0, V +ε], with ∂ṽ˜ (U, · ) ≥ 0, forṽ ∈ ]V, V +ε]. Since (1 − μ)(U, V ) < 0, by continuity, there exists 0 < ε ≤ε such that andθ Take the sign ofθ(U, V ) to be +1 ifθ(U, V ) ≥ 0, and −1 ifθ(U, V ) < 0. These choices guarantee (34) and (35). Together with the values ofr (u, V ),ν(u, V ) and ζ (u, V ), they provide initial data for the first order system (19)  In this case, we should consider the first order system with initial data on [δ, U ]×{V }∪ {δ}×[V, V +ε]. We would then obtain an extension of the solution to [δ, U ]×[V, V +ε] for some 0 <ε < ε. However, our approach above, using Theorem 2.6, guarantees that the domain of our extended solution contains a neighborhood of the whole Cauchy horizon ]0, U ]×{V }. If we had insisted on using the original existence and uniqueness theorem in Part 1, we would only have known that there existed a solution whose domain contained a neighborhood of ]δ, U ] × {V }, for δ arbitrarily small; but if δ changed, the solution might change, because we would have to change the initial data.
We now wish to see that the solution of our first order system corresponds to a solution of the Einstein equations. Using Propositions 2.8 and 2.9, we know that this is the case provided that the regularity hypothesis (h4) (see Section 2) is satisfied, which it is. Indeed, the extended solution is a solution of the backward problem wherẽ λ(U,ṽ) ≡ −1 andκ(U,ṽ) are C 1 on [0, V + ε] by our choice of initial data. On the other hand,ν(u, 0) = ν 0 (u) ≡ −1. Hence, we proved Remark 6.8 By Lemma 2.7 we conclude thatř is C 2 , andν andκ are C 1 . Therefore, 2 is C 1 , and so the metric is also C 1 . The fieldφ is also C 1 becauseθ andζ are continuous. Furthermore, ∂ u ∂ṽˇ 2 exists and is continuous in this (u,ṽ) chart. We emphasize thatˇ 2 does not have to be C 2 in this (u,ṽ) chart. Indeed, 2 (u, 0) = −4ν 0 (0)κ(u, 0) = 4e to the present situation because we only need˜ (u, V ) to be close to 0 to show that ν(u, V ) < 0). Therefore, the result follows from θ λ (u,ṽ) → +∞, asṽ V , for u > 0.
In the next figure we sketch part of the (σ, ϒ)-plane. As we just saw, the restriction r + < r c translates into (109) and this region (shaded in the figure) is the only relevant one for our purposes. We remark that the limit value of ρ on the line σ = 1 is one.

Appendix B: Proof of Theorem 3.1
We start by establishing the following useful result.

Proof
The proof proceeds in three steps.

Corollary 6.11
Under the hypotheses of Lemma 6.10, for u >ū and v >v, we have Proof This is an easy consequence of the fact that Next we use (122), (134) and (112). We may bound the integral of ν along ř − in terms of the integral of ν 1−μ on the segment u γ (v), u × {v} in the following way: Applying successively (113), (125), (131), and (138), We can now obtain a lower bound for I (u) using (117) and (120): This integral is infinite if or, equivalently, provided thatε andδ are chosen sufficiently small (which we can achieve by decreasing U and δ, if necessary). To complete the proof of Theorem 3.1 we just have to note that given s < ρ 2 − 1 we can always chooseř − so that (121) holds, contradicting for appropriate choices of the parameters β − , β + ,ř − ,ř + , ε 0 and U .