The formation of trapped surfaces in the gravitational collapse of spherically symmetric scalar fields with a positive cosmological constant

Given spherically symmetric characteristic initial data for the Einstein-scalar field system with a positive cosmological constant, we provide a criterion, in terms of the dimensionless size and dimensionless renormalized mass content of an annular region of the data, for the formation of a future trapped surface. This corresponds to an extension of Christodoulou's classical criterion by the inclusion of the cosmological term.


Introduction and main result
The existence of a (future) trapped surface -a compact codimension 2 spacelike submanifold with negative inward and outward (future) null expansions -has profound implication for the global structure of spacetime: most notably, under quite general causal assumptions and fairly weak energy conditions, it guarantees the existence of a non-empty black hole region [11,12] and, by Penrose's Singularity/Incompleteness Theorem, it also implies future causal geodesic incompleteness.
In [7], Christodoulou established a criterion for the dynamical formation of trapped spheres in the context of a characteristic initial value problem for the spherically symmetric Einstein-scalar field system; this was an essential step in his seminal proof of both the Weak and Strong Cosmic censorship conjectures, under appropriate regularity conditions for such system, that was completed in [9]. Later, in celebrated work [10], Christodoulou extended his criterion to the context of the Einstein vacuum equations without symmetry assumptions; in the mean time, important developments have been achieved in that setting [1,2,25,26]. Returning to the context of spherically symmetric spacetimes we would like to mention that criteria for the formation of trapped surfaces were also established for solutions of the Einstein-Vlasov equations, in [5], and the Einstein-Euler system, in [6]. Concerning the impact that the formation of a trapped surface has in the global structure of asymptotically flat spherically symmetric spacetimes we refer the reader to the qualitative results in [21,27] and the quantitative results obtained in [3,4,28].
In view of the recent developments concerning the global analysis of black hole spacetimes with a positive cosmological constant -from the remarkable proofs of the non-linear stability of the local/exterior regions of Kerr-Newman de Sitter [23,24], to the new results concerning the structure of black hole interiors [13][14][15][16], as well as the improved understanding of the geometry of cosmological regions [19,22,29] -it seems relevant to study the formation of trapped surfaces in the presence of Λ > 0. In this paper we start this study by revisiting Christodoulou's original criterion and extending it to the context of solutions to the Einstein-scalar field system with a positive cosmological constant. Another motivation for this paper comes form the study of gravitational collapse in the "hard" phase of Christodoulou's two-phase model [8], where the Einstein-Euler system reduces to the Einstein-scalar field system with cosmological constant Λ = 1.
Here we will follow the general strategy developed in [10] closely, but in order to do so one needs to overcome some new challenges created by the introduction of a positive cosmological constant. Most notably the new difficulties emanate from: the well known inadequacy of the Hawking mass (13) in the context of solutions with a positive cosmological constant and, most importantly, the loss of the basic monotonicity properties of the gradient of the radial function 1 . The first issue is resolved by the standard replacing of the Hawking mass with its renormalized version (14), in terms of which our criterion is formulated (3). The second difficulty is dealt with by obtaining "weak monotonicity" estimates, see for example (60); unfortunately their "weakness" propagates to the remaining estimates and makes the subsequent analysis more involved; for instance, as a consequence, we are no longer able to rely on some elementary integration formulas that are available for the Λ = 0 case. Other specificities of the positive cosmological constant setting are discussed in Remarks 1.1 and Section 2. We end this discussion by observing that instead of relying on "a geometric Bondi coordinate together with a null frame" 2 , which provide the basic framework in [10], we rely solely on a global double null coordinate setup.
The main result of this paper is the following: be a smooth spherically symmetric solution of the Einstein-scalar field system (6)-(7) with a cosmological constant Λ > 0. Assume there exist (double null) coordinates (u, v) : Q → R 2 , with ∂ u and ∂ v future oriented and whereg is the metric of the round 2-sphere. Assume also that C + 0 = {u = 0 , v ≥ 0} is a future complete null line emanating from the timelike curve Γ := {r = 0} ⊂ Q and that Q + := J + (C + 0 ) ∩ J + (Γ) coincides with the future maximal globally hyperbolic development of the data induced on C + 0 . Let 0 < v 1 < v 2 be such that: and consider the dimensionless size and the dimensionless renormalized mass content where ̟ : Q → R is the renormalized Hawking mass (see (14)).
2. We observe that in terms of the original Hawking mass (13) we have and recall that the last quantity is the one used for Λ = 0 in [7].
3. Note that, in particular, our theorem shows that a trapped surface can form from the evolution of data, posed on C + 0 , which is "arbitrarily far from being trapped". By this we mean that, for any ǫ > 0, we can choose initial data satisfying the assumptions of the theorem and such that 4. Condition (iii) above has no parallel in the Λ = 0 case and is related to the fact that, under general conditions [17,Section 3], an apparent cosmological horizon -a causal hypersurface which is the union of marginally anti-trapped spheres, i.e., where ∂ u r = 0 and ∂ v r > 0 -must intersect the initial cone C + 0 in the region 1/Λ < r ≤ 3/Λ. 5. Note that, by invoking the results in [18], condition (i) above can be replaced by an appropriate smallest condition at the level of the scalar field. 6. Lower bounds for the mass and radius of the marginally trapped sphere can be obtained from (45), (90) and (59).

The Einstein-scalar field system with a positive cosmological constant, in spherical symmetry
We will consider the Einstein-scalar field system in the presence of a positive cosmological constant Λ: where a Lorentzian metric g µν is coupled to a scalar field φ via the Einstein field equations (6), with energy-momentum tensor (7). We will work in spherical symmetry by assuming that M = Q × S 2 and by requiring the existence of double-null coordinates (u, v), on Q, such that the spacetime metric takes the form (1). Then the system (6)- (7) becomes Since we will only probe the future of the null cone u = 0, truncated at v = v 2 , from now on we will redefine Q + , as defined in Theorem 1, to be the set We define the Hawking mass m = m(u, v) by It turns out to be convenient, both by its physical relevance and by its good monotonicity properties, to also introduce the renormalized Hawking mass ̟ = ̟(u, v) defined by Then we have and by interpreting 1 − µ as a function of (r, ̟) we write In this paper we will mainly use a first order formulation of the Einsteinscalar field system, obtained by introducing the quantities: Note, for instance, that in terms of these new quantities (15) gives Consequently, in the non-trapped region we are allowed to define both and It is then well known (see for instance [13]) that the Einstein-scalar field system with a cosmological constant satisifies the following overdetermined system of PDEs and algebraic equations: It will also be useful to note that in terms of the (original) Hawking mass (28) reads One of the assumptions of Theorem 1 is the non-existence of anti-trapped spheres along the initial null cone {u = 0}, up to v ≤ v 2 , i.e, that ν(0, v) < 0, for all v ≤ v 2 (recall point 4 of Remarks 1.1). It is then a well known consequence of the Raychaudhuri equation (11) that this sign is preserved by evolution along the ingoing direction, that is As an immediate consequence we get the global upper bound Most importantly, the following extension criteria then follows from [13, Corollary 5.5]: Proposition 2.1. Under the previous conditions (including the sign condition (36)), let p ∈ Q + be such that there exists q ∈ J − (p) for which If there exists c > 0 such that inf D r > c , It is also useful to consider the marginally-trapped region and the trapped region We then have the following:

Proposition 2.3. If non-empty,
A is a C 1 curve such that: and Moreover and, consequently, Proof. Assume (u * , v * ) ∈ A. Then (22) implies that 1 − µ(u * , v * ) = 0 which in turn gives rise to the identities and The last identity together with (37) implies that We will also need the fact that which follows from (24) and (36). We then use (8), (22) and (24) to write and conclude that ∂ u λ(u * , v * ) < 0; in particular, this shows that A is a continuously differentiable curve. The same reasoning shows that once λ < 0, we get 1 − µ < 0, which implies that ∂ r (1 − µ) > 0 and consequently ∂ u λ < 0. We can then conclude that Just as in the discussion leading to (36) the fact that is an immediate consequence of the Raychaudhuri equation (12). Consequently both (40) and (41) follow. Now assume there existes (u * , v * ) ∈ A ∩ Γ, i.e., r(u * , v * ) = λ(u * , v * ) = 0: then, in view of (49) and the fact that r ≥ 0, we get r(u * , v) = 0, for v ≥ v * , which is in contradiction with the causal character of the center of symmetry Γ. The remanning conclusions then follow immediately.

Proof of Theorem 1
Proof. The proof follows by obtaining several estimates valid in the region R, as defined in (23), that will allow us to conclude that, under the conditions of Theorem 1, the ingoing line v = v 2 has to exit R before leaving Q + . So from now one, and unless otherwise stated, assume that we are in R.
We start by noting that (13) implies that and then, after recalling (47), (35) shows that ∂ v m ≥ 0 and since Q + ⊂ J + (Γ) we obtain m ≥ 0 . Consequently with the first inequality, valid in R, as a consequence of (22). The previous together with (31), the monotonicity of the radial function in v and (28) gives (recall (34)) which we save for later use as Now notice that ̟ ≥ 0 (55) since ̟| Γ = (m − Λr 3 /6)| Γ = 0 and, in view of (28), ∂ v ̟ ≥ 0. We then obtain the estimate (recall (37)) Now in the region so that by imposing the condition r(0, v 2 ) < 1 √ Λ (which we recall is one of the hypothesis of Theorem 1) we have first in R + and then in the entire region R, since the left hand side is negative in R \ R + and the right hand side is positive.
We can now integrate (26) to obtain, for 0 ≤ u 1 ≤ u 2 , which we save for future use in the form Integrating the last inequality, in v, readily gives Another consequence of (60) and condition (i) of Theorem 1 is the following: In fact, under such conditions we can consider a parameterization of Γ of the form (u, v Γ (u)), with v Γ (u) → v 1 , as u →ū 1 . Then by integrating (26) from Γ, while recalling (44), we get as u → u 1 , and (62) then follows from the monotonicity of r.
Relying once again on (26) and (58) gives which implies We now turn to the scalar field's derivative and using, in sequence, (30), (34), Hölder's inequality, (28), (33), (64) and the sign in (31) which, after introducing the quantity To obtain our next estimate we introduce the notations We also define with the understanding that u η = +∞ if the defining set is empty. Note that, in view of (4), u η > 0.
The following estimates will be restricted to the set R ∩ {u < u η }; later we will see that this is in fact the entire R, i.e., that u η = +∞. That being said, using (54) and recalling that κ < 0, we have In conclusion, in R ∩ {u < u η }, we have Noting that we can rewrite (53) as we have, using (27), (68), (66), Exactly as in [7] we introduce a new variable which, after noting that δ 0 = δ(0) and recalling (61), we use to obtain In the new variable we have and then, using (69), the elementary inequality e η − 1 ≥ η and (71), we arrive at dη dx Let which will be made smaller than unity by future restrictions on δ 0 . Then we can easily integrate (72) and obtain, for x ∈ (x 0 , 1], where and Now let and note that dH We can then conclude that is the only critical point of H and that moreover and It is now essential to note that α 0 decreases with δ 0 (just recall (59) and that r(0, v 2 ) = (1 + δ 0 )r(0, v 1 )); then it becomes clear that there exists δ 1 = δ 1 (Λ) such that In conclusion, for δ 0 < δ 1 , x 1 is the global minimum of H in (x 0 , 1]. We then define A direct computation gives rise to the elementary formula from which we immediately see that, for δ 0 < δ 1 (after decreassing δ 1 if necessary), we have, for appropriate choices of 0 < c 1 < C 1 , However we are unaware of the existence of an elementary formula for F ; note, in contrast, that in the Λ = 0 case we obtain e α0 = 1 and then such a formula can be easily obtained [7]. Nonetheless we have the following estimate: There are positive constants c 1 , C 1 and δ 1 , such that for δ 0 < δ 1 Proof. Using (81) we get y − e α 0 δ0 δ0+1 e α 0 +1 dy If we set n = min{m ∈ Z : m ≥ e α0 } ≥ 2 and write a = e α 0 δ0 δ0+1 we see that, since y y−a > 1 then We then conclude that we also have In particular, from (4) we see that η 0 > E(δ 0 ) > F (x), for all x, so that (75) implies that u η = +∞ and the derived estimates hold in the entire region R.
Moreover, again by Proposition 2.3 we also have [0, u * ) × {v 2 } ⊂ R and we are allowed to apply (86) to obtain, for all u < u * ≤ u 1 , By taking the limit when u → u * we see that which, in view of (62), allows us to conclude that the marginally trapped surface forms, along v = v 2 , before (as measured by u) v = v 1 reachesΓ.