On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant

This paper is the second 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 paper of this sequence (Costa et al., Class Quantum Gravity 32:015017, 2015), we established well posedness of the characteristic problem with general initial data. In this second paper, we generalize the results of Dafermos (Ann Math 158:875–928, 2003) on the stability of the radius function at the Cauchy horizon by including a cosmological constant. This requires a considerable deviation from the strategy followed in Dafermos (Ann Math 158:875–928, 2003), focusing on the level sets of the radius function instead of the red-shift and blue-shift regions. We also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin. In the third and final paper (Costa et al., On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant. Part 3. Mass inflation and extendibility of the solutions. arXiv:1406.7261, 2015), we will consider the issue of mass inflation and extendibility of solutions beyond the Cauchy horizon.


Introduction
This paper is the second 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 , 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. We are motivated by the strong cosmic censorship conjecture and the question of determinism in general relativity. As explained in detail in the Introduction of Part 1, strong cosmic censorship is one of the most fundamental open problems in general relativity (see the classic monographs [3,10] and the discussions in [1,6,9] for the general context of this problem). Although significant developments have been achieved in the last five decades (from the initial heuristic works [14,15] to rigorous mathematical results [6][7][8]), including some recent encouraging progress (see [9,11,13] and references therein), a complete resolution of the conjecture at hand still seems out of reach. Nonetheless, the spherically symmetric self-gravitating scalar field model has provided considerable insight into the harder problem of vacuum collapse without symmetries [2]; this was explored in [12] to obtain the first promising steps towards understanding the stability of Cauchy horizons without symmetry assumptions.
In Part 1, we established the equivalence (under appropriate regularity conditions for the initial data) between the Einstein Eqs. (1)-(5) and the system of first order PDE (14)-(23). We proved existence, uniqueness and identified a breakdown criterion for solutions of this system (see Sect. 2).
In the current paper we are concerned with the structure of the solutions of the characteristic problem, and wish to address the question of existence and stability of the Cauchy horizon when the initial data is as above. This is intimately related to the issue of global uniqueness for the Einstein equations: it is the possibility of extension of solutions across this horizon that leads to the breakdown of global uniqueness and, in case the phenomenon persists for generic initial data, to the failure of the strong cosmic censorship conjecture.
As in [6], we introduce a certain generic element in the formulation of our problem by perturbing a subextremal Reissner-Nordström black hole (whose Cauchy horizon formation is archetypal) by arbitrary characteristic data along the ingoing null direction. The study of the conditions under which the solutions can be extended across the Cauchy horizon is left to Part 3.
We take many ideas from [6,7] and build on these works. In particular, we borrow the following three very important techniques. Nonetheless, the introduction of a cosmological constant causes a significant difference that requires deviation from the original strategies developed in [6,7]. Moreover, we introduce some technical simplifications and obtain sharper and more detailed estimates. These improvements will be crucial for our arguments in Part 3.
Our approach therefore has three main departures from the one of Dafermos: (i) First, due to the presence of the cosmological constant , the curves of constant shift, which are used in [6,7], are no longer necessarily spacelike for > 0 large. This forces us to find an alternative approach; we have chosen to work with curves of constant r coordinate instead of working with curves of constant shift, which turns out to be a simpler approach. Furthermore, it allows us to treat the cases < 0, = 0 and > 0 in a unified framework. (ii) Second, we show that the Bondi coordinates (r, v) are the ones most adapted to estimating the growth of the fields as we progress away from the event horizon. Our approach starts by controlling the field ζ ν using (54). Although this is similar to (53), there is one distinction which makes all the difference. It consists of the fact that in the double integral in (53) the field ζ ν is multiplied by the function ν. When we pass to Bondi coordinates this function disappears, making a simple application of Gronwall's inequality, such as the one we present, possible. This would not work in the double null coordinate system (u, v). (iii) Third, our estimates are not subordinate to the division of the solution spacetime into red shift, no shift and blue shift regions. Instead, we consider the regions {r ≥ř + }, {ř − ≤ r ≤ř + } and {r ≤ř − }, whereř + is smaller than but sufficiently close to the radius r + of the Reissner-Nordström event horizon, andř − is bigger than but sufficiently close to the radius r − of the Reissner-Nordström Cauchy horizon. These may be loosely thought of as red shift, no shift and blue shift regions of the background Reissner-Nordström solution, even though the shift factor is not small and indeed changes significantly from red to blue in the intermediate region.
Our first objective is to obtain good upper bounds for −λ in the different regions of spacetime. These will enable us to show that the radius function r is bounded below by a positive constant. However, good estimates for −ν and the fields θ and ζ will also be essential in Part 3.
The main result of this paper is therefore So, under the hypotheses of Theorem 1.1, the argument in [7,Section 11], shows that, as in the case when = 0, the spacetime is extendible across the Cauchy horizon with a C 0 metric. We also prove that only in the case of the Reissner-Nordström solution does the curve {r = r − } coincide with the Cauchy horizon. As soon as the initial data field is not identically zero, the curve {r = r − } is contained in P (Theorem 8.1). This is an interesting geometrical condition and it is conceptually relevant given the importance that we confer to the curves of constant r . We also prove that, in contrast with what happens with the Reissner-Nordström solution, the presence of any nonzero field immediately causes the integral ∞ 0 κ(u, v) dv to be finite for any u > 0 (Lemma 8.2). As a consequence, the affine parameter of any outgoing null geodesic inside the event horizon is finite at the Cauchy horizon (Corollary 8.3).

Framework and Some Results from Part 1
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. The Einstein-Maxwell-scalar field system with a cosmological constant and total electric charge 4π e reduces to the following system of equations: the wave equation for r , the wave equation for φ, the Raychaudhuri equation in the u direction, the Raychaudhuri equation in the v direction, and the wave equation for ln , The first order system. Given r , φ and , solutions of the Einstein equations, let and Notice that we may rewrite (8) as The Einstein equations imply the first order system for (r, ν, λ, , θ, ζ, κ) with the restriction Under appropriate regularity conditions for the initial data, the system of first order PDE (14)-(23) also implies the Einstein Eqs. (1)- (5).
Initial data. In Part 1 we study well posedness of the first order system for general initial data. In this paper we take the initial data on the outgoing null direction v to be the data on the event horizon of a subextremal Reissner-Nordström solution with mass M. The initial data on the ingoing null direction u is free. More precisely, we choose Here r + > 0 is the radius of the event horizon. We assume ζ 0 is continuous and ζ 0 (0) = 0.
Well posedness of the first order system. Theorem 4.4 of Part 1, for the initial data above, reads: Reissner-Nordström solution. For comparison purposes, we notice that the Reissner-Nordström solution (with a cosmological constant), obtained from the initial data ζ 0 (u) = 0, corresponds to

Preliminaries on the Analysis of the Solution
We now 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 > 0. So, we choose initial data as in (24)-(25) with ζ 0 (0) = 0. Moreover, we assume ζ 0 to be continuous. Since in this case the function 0 is constant equal to M, we also denote M by 0 . In particular, when < 0, which corresponds to the Reissner-Nordström anti-de Sitter solution, and when = 0, which corresponds to the Reissner-Nordström solution, we assume that has two zeros r − ( 0 ) = r − < r + = r + ( 0 ). When > 0, which corresponds to the Reissner-Nordström de Sitter solution, we assume that r → (1 − μ)(r, 0 ) has three zeros r − ( 0 ) = r − < r + = r + ( 0 ) < r c = r c ( 0 ).
We define η to be the function The functions (r, ) → η(r, ) and (r, ) → (1 − μ)(r, ) are related by We define the function η 0 : R + → R by We will repeatedly use the fact that η(r, ) ≤ η 0 (r ) (see Lemma 3.1). If ≤ 0, then η 0 < 0. So η 0 is strictly decreasing and has precisely one zero. The zero is located between r − and r + . If > 0, then η 0 is positive, so η 0 is strictly convex and has precisely two zeros: one zero is located between r − and r + and the other zero is located between r + and r c . We denote by r 0 the zero of η 0 between r − and r + in both cases.
Using the above, we can thus particularize the result of Part 1 on signs and monotonicities to the case where the initial data is (24) and (25) as follows. Lemma 3.1 (Sign and monotonicity). Suppose that (r, ν, λ, , θ, ζ, κ) is the maximal solution of the characteristic initial value problem (14)-(23), with initial conditions (24) and (25). Then: is nondecreasing with both u and v.
To analyze the solution we partition the domain into four regions (see figure below). We start by choosingř − andř + such that r − <ř − < r 0 <ř + < r + . In Sect. 4 we treat the regionř + ≤ r ≤ r + . In Sect. 5 we consider the regionř − ≤ r ≤ř + . In Sect. 6 we treat the region where (u, v) is such that with β > 0 appropriately chosen (we will denote the curve v = (1 + β) vř − (u) by γ ). Finally, in Sect. 7 we consider the region where (u, v) is such that The reader should regardř − ,ř + and β as fixed. Later, they will have to be carefully chosen for our arguments to go through.
The crucial step consists in estimating the fields θ λ and ζ ν . Once this is done, the other estimates follow easily. By integrating (32) and (33), we obtain Formula (34) is valid provided that uř (v) is defined and uř (v) ≤ u, since the domain P is a past set; it also holds if we replace uř (v) by 0. Similarly, formula (35) is valid provided that vř (u) is defined and vř (u) ≤ v; again it holds if we replace vř (u) by 0.

The Region
Recall that r 0 <ř + < r + . In this section, we treat the regionř + ≤ r ≤ r + , that is, J − ( ř + ). 1 Our first goal is to estimate (42) for ζ ν . This will allow us to obtain the lower bound (43) for κ, which will then be used to improve estimate (42)-(46). Finally, 1 Throughout this paper we follow the usual notations for the causal structure of the quotient Lorentzian manifold with coordinates (u, v) and time orientation such that ∂ ∂u and ∂ ∂v are future-pointing.
we successively bound θ λ , θ , , and use this to prove that the domain of vř In this region, the solution with general ζ 0 can then be considered as a small perturbation of the Reissner-Nordström solution (26)-(30): is close to 0 , κ is close to 1 and ζ, θ are close to 0. Besides, the smaller U is, the closer the approximation.
Substituting (34) in (35) [with both uř (v) and vř (u) replaced by 0], we get We make the change of coordinates The coordinates (r, v) are called Bondi coordinates. We denote by ζ ν the function ζ ν written in these new coordinates, so that The same notation will be used for other functions. In the new coordinates, Eq. (36) may be written it is easy to show that the function θ λ can be extended as a continuous function to We can rewrite this in the new coordinates as A key point is to bound the exponentials that appear in (38) and (39). As we go on, this will be done several times in different ways.

Lemma 4.1.
Assume that there exists α ≥ 0 such that, for 0 ≤ṽ ≤ v, the following bounds hold: Proof. Combining (38) with θ λ (r + , v) ≡ 0 and the bounds on the exponentials, we have Here the maximum is taken over the projection of In the same way one can show that Using Gronwall's inequality, we get This establishes (40).
We recall from Part 1 that Eqs. (15), (17), (19) and (23) imply which is the Raychaudhuri equation in the v direction. We also recall that the integrated form of (18) is These will be used in the proof of the following result.
where we have used (43) (withř + instead of r 0 ). Thus, we can improve the bounds on the exponentials in (38) that involve integrals in v as follows: as before, we have We apply Lemma 4.1 again, this time with a positive α, to get (46). Now we may use (39) and (46) to obtain In order to bound in J − ( ř + ), we note that From (45), we conclude that Inequality (49) follows from e x ≤ 1 + 2x, for small x, since ζ 0 is continuous and Moreover, since is bounded in the region J − ( ř + ), 1−μ is bounded from below, and so λ is also bounded from below. Hence (47) implies (48).
Let 0 < u ≤ min{r + −ř + , U }. We claim that To see this, first note that (17) Combining the previous inequalities with (43), we get Finally, if (51) did not hold for a given u, we would have as v → ∞, which is a contradiction. This establishes the claim.

The Region
In this section, we treat the regionř − ≤ r ≤ř + . Recall that we assume that r − <ř − < r 0 <ř + < r + . By decreasingř − , if necessary, we will also assume that In Sect. (5.1), we obtain estimates (55) and (56) for ζ ν and θ λ , which will allow us to obtain the lower bound (66) for κ, the upper bound (67) for , and to prove that the domain of vř − ( · ) is ]0, min{r + −ř − , U }]. In Sect. (5.2), we obtain upper and lower bounds for λ and ν, as well as more information about the regionř − ≤ r ≤ř + . In Sect. (5.3), we use the results from the previous subsection to improve the estimates on ζ ν and θ λ to (86) and (91). We also obtain the bound (92) for θ . As in the previous section, the solution with general ζ 0 is qualitatively still a small perturbation of the Reissner-Nordström solution (26)-(30): , κ, ζ and θ remain close to 0 , 1 and 0, respectively. Moreover, λ is bounded from below by a negative constant, and away from zero by a constant depending onř + andř − , as is also the case in the Reissner-Nordström solution (see Eq. (26)). Likewise, ν has a similar behavior to its Reissner-Nordström counterpart (see Eq. (27)): when multiplied by u, ν behaves essentially like λ.

First estimates. By reducing
We make the change of coordinates (37). Then, Eq. (53) may be written Proof. From (52) we have (For the second inequality, see the graph of η 0 in Sect. 3.) Each of the five exponentials in (54) is bounded by and Again consider r ≤ s ≤ř + and letṽ In the same way one can show that Using Gronwall's inequality, we get To bound Z (r,v) and T (r,v) , it is convenient at this point to use (42) and (valid forř + ≤ r < r + ), in spite of having the better estimates (46) and (47). Indeed, if these better estimates are used, the improvement is just e −αvř + (u r (v)) (that is, an exponential factor computed over ř + for the same value of u); to turn this into an exponential decay in v we must first obtain a more accurate control of the various quantities in the regionř − ≤ r ≤ř + . Applying first the definition (60) and then (42), we have because uř + (ṽ) ≤ u r (v). Applying first the definition (61) and then (63), we have We use (64) and (65) in (62). This yields (55). Finally, writing (34) in the (r, v) coordinates (withř =ř + ) gives The exponentials are bounded by the constant C in (58). We use the estimates (63) and (55) to obtain |ζ 0 |(u) which is (56).
Proof. The proof of (67) is identical to the proof of (49). Because is bounded, the function 1 − μ is bounded below in J − ( ř − ). Also, by (57), the function 1 − μ is bounded above in The proof is similar to the proof of (51): since κ is bounded below by a positive constant and 1 − μ is bounded above by a negative constant, λ is bounded above by a negative constant in J − ( ř − ) ∩ J + ( ř + ), say −c λ . Then, as long as (u, v) belongs to J − ( ř − ), we have the upper bound for r (u, v) given by Finally, if (68) did not hold for a given u, we would have r (u, v) → −∞ as v → ∞, which is a contradiction. This proves the claim.

Lemma 5.4. Let ε > 0. If δ is sufficiently small, then for any point
For any point Proof. Obviously, we have Since r is C 1 and ν does not vanish, v → u r + −δ (v) is C 1 . Differentiating both sides of the last equality with respect to v we obtain Using (71) and (72), we have Integrating the last inequalities between 0 and v, as u r + −δ (0) = δ, we have This proves (84).
. Integrating (70) between u r + −δ (v) and u, we get Combining v r + −δ (u) ≤ v with the first inequality in (84) applied at the point (u, v r + −δ (u)), and combining u ≤ e r+ c u r + −δ (v) with the second inequality in (84) applied at the point
Remark 5.7. For use in Part 3, we observe that (47) and (91) imply In this section, we define a curve γ to the future of ř − . Our first aim is to obtain the bounds in Corollary 6.2, In the process, we will bound (102)). Then we will obtain a lower bound on κ, as well as upper and lower bounds on λ and ν. Therefore this region, where r may already be below r − , is still a small perturbation of the Reissner-Nordström solution.
We choose a positive number 2 and define γ = γř − ,β to be the curve parametrized by for u ∈ [0, U ]. Since the curve ř − is spacelike, so is γ [u → vř − (u) is strictly decreasing].
(105) This is a consequence of (66) and (102), since r is bounded away from zero.
Consider the reference subextremal Reissner-Nordström black hole with renormalized mass 0 , charge parameter e and cosmological constant . The next remark will turn out to be crucial in Part 3.
Thus, the exponent in the upper bound for in (103) may be replaced by This is positive for Given β satisfying (106), we can guarantee that it satisfies the condition above by choosing (ř + , ε 0 , δ) sufficiently close to (r + , 0, 0). and Proof. Let us first outline the proof. According to (16) and (17), In this region we cannot proceed as was done in the previous section because we cannot guarantee 1 − μ is bounded away from zero. The idea now is to use these two equations to estimate λ and ν. For this we need to obtain lower and upper bounds for and The estimates for (115), and thus for ν, are easy to obtain. We estimate (114) by comparing it with Using (73), we see that (114) is bounded above by (116). We can also bound (114) from below by (116), divided by 1 + ε, once we show that The estimates for θ λ are obtained via (34) and via upper estimates for (114). To bound (116) we use the fact that the integrals of ν and λ along ř − coincide.
We start the proof by differentiating the equation By making the change of variablesṽ = vř − (ũ), this last equation can be rewritten as We may bound the integral of λ along ř − in terms of the integral of κ along ř − in the following way: Analogously, we may bound the integral of ν along ř − in terms of the integral of ν 1−μ along ř − in the following way: Using successively (73), (121), (118) and (120), we get We can now bound the field θ λ for (u, v) ∈ J − (γ ) ∩ J + ( ř − ). Using (34), We can bound the exponentials in (125) by Combining this inequality with (91), (98) and (100), leads to We consider the two possible cases. Suppose first that the exponent in (126) is nonpositive. Then, from (100) we get Suppose now the exponent in (126) is positive. Using (126) and (100) Therefore, in either case, given ε > 0 we may choose U sufficiently small so that if On the other hand, using (124), We continue assuming (u, v) ∈ γ . Taking into account (69), estimate (131) allows us to obtain an upper bound for −λ(u, v), and estimate (132) allows us to obtain a lower bound for −λ(u, v), Next, we turn to the estimates on ν. Let, again, (u, v) ∈ γ . Using (105), These two inequalities imply We note that according to (85) we have Recalling (113) and (70), and using (133) and (135), whereas using (134) and (136), we can make our choice of β and other parameters (ř − , ε 0 , U ) so that Having done so, for (u, v) on the curve γ , we obtain with 0 < s 1 < s 2 .

The Region J + (γ )
Using (112) and (113), we wish to obtain upper bounds for −λ and for −ν in the future of γ while r is greater than or equal to r − − ε. To do so, we partition this set into two regions, one where the mass is close to 0 and another one where the mass is not close to 0 . In the former case ∂ r (1 − μ) < 0 and in the latter case ∂ r (1−μ) 1−μ is bounded. This information is used to bound the exponentials that appear in (112) and (113).
Here the solution with general ζ 0 departs qualitatively from the Reissner-Nordström solution (26)-(30), but the radius function remains bounded away from zero, and approaches r − as u → 0. This shows that the existence of a Cauchy horizon is a stable property when ζ 0 is perturbed away from zero.
Proof. We recall that on γ the function r is bounded above byř − and that then clearly η > 0.
On the other hand, if ≥ 0 + min then, for r ∈ [r − − ε,ř − ], where we used In case (139) we have (recall (31)) In case (141), the absolute value of is bounded, say by C. Indeed, this is a consequence of two facts: (i) the denominators 1 − μ and r are bounded away from zero (we recall η also has a denominator equal to r ); (ii) the equality We define and (u, v) ≥ 0 + min and (u, v) < 0 + min In order to estimate λ, we observe that Similarly, to estimate ν we note that In conclusion, let (u, v) ∈ {r > r − − ε} ∩ J + ( ř − ). Using (112) and (110), we have Similarly, using (113) and (111), we have Proof. We denote by ε the value of ε that is provided in Lemma 7.1. Let δ > 0. Without loss of generality, we assume that δ is less than or equal to ε. Choose the value of ε in Corollary 6.2 equal to δ. This determines an U ε as in the statement of that corollary. Let (u, v) ∈ J + (γ ) with u ≤ U ε . Then are parameterizations of the spacelike curve γ . Integrating (138), we obtain for a positive p. This estimate is valid for (u, v) ∈ {r > r − − ε} ∩ J + (γ ). It yields provided u < min U ε , p δp 2C =:Ũ δ . Since δ is less than or equal to ε and γ ⊂ {r > r − −ε}, if (u, v) ∈ J + (γ ) and u <Ũ δ , then (u, v) ∈ {r > r − −ε} and the estimate (145) does indeed apply.
Alternatively, we can obtain (146) integrating (137): Ce −qs ds, for a positive q. This yields for a positiveq, according to (135). For u < min U ε ,˜q δ 2C we obtain, once more, Due to the monotonicity of r (u, · ) for each fixed u, we may define r (u, ∞) = lim v→∞ r (u, v).
The previous two corollaries prove Theorem 1.1. The argument in [7,Section 11], shows that, as in the case when = 0, the spacetime is then extendible across the Cauchy horizon with C 0 metric.

Two Effects of Any Nonzero Field
This section contains two results concerning the structure of the solutions with general ζ 0 . Theorem 8.1 asserts that only in the case of the Reissner-Nordström solution does the curve r − coincide with the Cauchy horizon: if the field ζ 0 is not identically zero, then the curve r − is contained in P.
Lemma 8.2 states that, in contrast with what happens with the Reissner-Nordström solution, and perhaps unexpectedly, the presence of a nonzero field immediately causes the integral ∞ 0 κ(u, v) dv to be finite for any u > 0. This implies that the affine parameter of any outgoing null geodesic inside the event horizon is finite at the Cauchy horizon.
For any fixed index n, there exists a v n such that for v ≥ v n . It follows that κ(u n , v) ≤ c n (−λ(u n , v)), for v ≥ v n .
where the Christoffel symbol v vv is given by v vv = ∂ v ln 2 .
We integrate both sides of this equation to obtain lnv + ln c = − ln 2 (u, v), with c > 0, or dt dv = c 2 (u, v).
On the other hand, suppose that there exists a positive sequence (u n ) converging to 0 such that ζ 0 (u n ) = 0. Then, since ν is continuous, it satisfies the bound (138) for large v, and (149) holds. So we also have ∞ 0 2 (u,v) dv < ∞.