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Weakly trapped surfaces in asymptotically de Sitter spacetimes

It is a standard fact that trapped or marginally trapped surfaces are not visible from conformal infinity, under the usual set of conditions on matter fields and the conformal completion, provided that the cosmological constant is non-positive. In this note we show that the situation is more delicate in the presence of a positive cosmological constant: we present examples of visible marginally trapped surfaces, and we provide a set of natural conditions which guarantee non-visibility.


Introduction
A classical result in the theory of black holes asserts that trapped surfaces are, in a suitable sense, 'externally invisible'. Somewhat more precisely, for spacetimes (M , g) which are asymptotically flat (in the sense of admitting a suitably regular future null infinity I + ) and which satisfy appropriate energy and causality conditions, no (future) trapped surface (θ ± < 0) can be contained in I − (I , + M ), where M = M ∪I + . In fact, this result also extends to (future) weakly trapped surfaces (θ ± ≤ 0). Indeed, using a suitable notion of global hyperbolicity, the following has been shown in [5]: If the null energy condition holds on D then there are no weakly trapped surfaces within D.
Since spacetimes with a positive cosmological constant are of considerable current interest (cf., e.g., [1,11,13]), it is useful to have an analogue of Theorem 1.1 for asymptotically de Sitter spacetimes. We provide such a result in Theorem 1.2 below.
To put our result in proper context, it should be kept in mind that weakly trapped surfaces occur in the past of I + in asymptotically de Sitter spacetimes. We review some well known and not so well known examples in Section 3. One sees from these examples that the visibility of future trapped surfaces in this case is due to the failure of a suitable analogue of the i 0 -avoidance condition in Theorem 1.1. When this is taken into account, an analogue of Theorem 1.1 can be obtained, and under a causality condition imposed only on the physical spacetime M .
Indeed, Theorem 2.1 and Lemma 2.2 below immediately yield our main result (see below for definitions and terminology): 2 Invisibility of trapped surfaces in the asymptotically de Sitter setting Let (M , g) be a spacetime and consider a compact co-dimension two spacelike submanifold S ⊂ M . Under appropriate orientability assumptions, there exists a smooth unit spacelike vector field ν normal to S. From this one obtains unique future directed null vector fields l ± normal to S such that g(ν, l ± ) = ±1. These induce the null second fundamental forms χ ± : T p S × T p S → R given by χ ± (X, Y ) = g(∇ X l ± , Y ). Tracing χ ± , we obtain the null expansion scalars (or null mean curvatures) θ ± = tr S χ ± = div S l ± . S is future trapped if both θ ± < 0 and weakly future trapped if both θ ± ≤ 0. If θ ± = 0, then S is marginally future trapped. A spacetime (M , g) will be said to admit a future conformal completion ( M ,g) provided ( M ,g) is a spacetime with boundary such that the following properties hold (c) There is a smooth function Ω on M such that on M the metricg is related to g viag = Ω 2 g where Ω = 0 on I + and dΩ is nowhere vanishing on I + .
If in addition to these we also have that I + is spacelike, then we say (M , g) is future asymptotically de Sitter. (We will sometimes omit the adjective "future" for conciseness.) In this case (b) above implies that I + is acausal. We say that (M , g) is future causally simple provided J + (K, M ) is closed for all compact K ⊂ M . Note that we do not assume any further causality conditions on (M , g). This agrees with [9] but differs from [2] and [10]. In addition we say that the conformal completion ( M ,g) is future causally simple with respect to M provided J + (K, M ) is closed for all compact K ⊂ M . As discussed in [7] (cf. Lemma 2.3), a spacetime with boundary ( M ,g) admits an extension to a spacetime (M , g ) without boundary such that I + separates M . The existence of such an extension is assumed in what follows. We first prove an invisibility theorem which uniformly covers the cases when I + is either a timelike, null, or spacelike hypersurface. In this theorem we assume ( M ,g) is future causally simple with respect to M . (The main use of the assumption of global hyperbolicity in the proof of Theorem 6.1 in [5] was to ensure this.) Introduce a Riemannian metric h on I + , and let U be a convex normal neighborhood about q 0 in I + . Let q 1 be a point in U \ ∂J + (S, M ) (which is open in U ), chosen so that some points of J + (S, M ) lie within the injectivity radius of q 1 . Let q ∈ U ∩ ∂J + (S, M ) minimize the h-distance between ∂J + (S, M ) and q 1 in U . Let r > 0 be this minimizing distance. Let S + be the geodesic ball with radius r centered at q 1 . Then S + is a smooth hypersurface in I + which contains q and does not meet Thus there exists a null geodesic γ : [a, b] → M such that γ(a) ∈ S, γ(b) = q, and emanating orthogonally from S. Since I + is either timelike, null, or spacelike, γ must intersect I + transversally, 1 and hence, γ [a, b) ⊂ M . Moreover, since γ does not enter the timelike future of S, there are no null focal points to S along γ. Then [12,Theorem B.7] implies that for all points along γ the null exponential map has full rank. This allows one to generate a smooth null hypersurface N 1 ⊂ J + (S, M ) containing the segment of γ [a,b− ] for > 0 arbitrarily small. Since S is future weakly trapped, the null future inwards and outwards mean curvatures θ ± of S are nonpositive. Let θ 1 = θ 1 (s) be the null mean curvature of N 1 along γ. Hence θ 1 (a) ≤ 0, then by the Raychaudhuri equation and the null energy condition, There exists a future directed outward pointing null vectorK at q orthogonal to S + in the unphysical metricg = Ω 2 g. Since Ω → 0 as one approaches I + , we see that -irregardless of the causal character of∇Ω at q -we can normalizeK so that along γ near q, we haveK (Ω) =g(K,∇Ω) = −1 .
Since S + does not meet I + (S, M ), we see that γ intersects S + orthogonally. Let N 2 be a smooth null hypersurface which is a subset of ∂J − (S + , M ) and contains γ [a, b) near q.K extends to a smooth null vector field on S 2 near q. The null mean curvature θ 2 of N 2 with respect to the physical metric g is related to the null mean curvaturẽ θ 2 in the unphysical metricg by θ 2 = −(n − 1)K(Ω) + Ωθ 2 (2.1) Our construction of N 2 from S + shows thatθ 2 is bounded on N 2 . Therefore close to q on N 2 , we have θ 2 > 0. However, since S + does not meet I + (S, M ), we know that N 1 lies to the future side of N 2 on points of γ near q. This contradicts the maximum principle for smooth null hypersurfaces.
In the asymptotically de Sitter setting, the causal simplicity of (M , g) implies ( M ,g) is causally simple with respect to M .  3. It is easy to give an example of a two-dimensional space-time with a null I + which is causally simple but the conformal completion is not. On the other hand, it is rather clear that there are no two-dimensional examples with "causally simple" replaced by "future causally simple" in the last sentence. But a three dimensional example can be given, as follows: Let p be the point in three dimensional 1 Without some control on the causal character of I + , this in general may fail.    Proof of Lemma 2.2. Let K ⊂ M be a compact set. Let q be a limit point of J + (K, M ). Then there is a sequence of points {q n } ⊂ J + (K, M ) such that q n → q, and so there is a sequence of future causal curves γ n : [0, 1] → M with γ n (0) = p n and γ n (1) = q n . By restricting to a subsequence, we can assume that p n → p ∈ K. Using a normal neighborhood centered at q in I + , we construct a compact spacelike hypersurface (with boundary) Σ ⊂ M by pushing this neighborhood a small amount to the past along the past directed normal geodesics to I + . Take Σ large enough so that, by passing to another subsequence if necessary, there exist unique points r n where γ n and Σ intersect. Since Σ is compact, there is a point r ∈ Σ such that r is a limit point of r n . So, since (M , g) is causally simple, there is a future causal curve λ from p to r. By choosing Σ sufficiently close to I + , we may assume r is in D(I + ), the domain of dependence of I + . Then by applying [4, Proposition 2.8.1] in D(I + ), there is a future causal curve σ from r to q. Concatenating λ and σ yields a future causal curve from p to q.
We can form analogous statements for trapped regions with assumptions only on the outward expansion θ + . To be precise let (M , g) be a spacetime, and let T be a compact connected spacelike hypersurface with boundary S in M . If S has null expansion θ + ≤ 0 with respect to the outward null normal vector field + along S, then we say that T is a future weakly trapped region and S is a future weakly outer trapped surface. Analogous statements for future weakly outer trapped surfaces hold for Theorems 2.1 and 1.2. For brevity we only state and prove the analogous version of Theorem 1.2.

Examples
De Sitter Space. Consider de Sitter space, where dω 2 is the usual round metric on the sphere S n . It conformally embeds into the Einstein static universe (M , g ) = (R × S n , −dt 2 + dω 2 ). The future conformal completion is M = (−π/2, π/2] × S n with I + = {π/2} × S n . Note that for any p ∈ M , J + (p, M ) does not cover all of I + . Hence, by Theorem 1.2 there are no future weakly trapped surfaces in J + (p, M ). Now consider any t-slice Σ t = {t} × S n . Let K be the second fundamental form of Σ t within M . Let S be any hypersurface in Σ t and H the mean curvature of S within Σ t . Then θ ± = tr S K ± H , (3.2) (With the notation as in the beginning of Section 2, here ± are scaled with respect to the unit normal ν tangent to the time slice.) If 0 < t < π/2, then tr S K > 0 and so either θ + or θ − is positive. Consequently there are no future weakly trapped surfaces in Σ t for t > 0. 2 On the other hand, there are many future weakly trapped surfaces in Σ 0 . For example take S = E to be the equator. Since Σ 0 is totally geodesic within M and E is a minimal surface within Σ 0 , we have K = H = 0. Thus, by (3.2), θ ± = 0, and so E is future weakly (in fact, marginally) trapped. This is consistent with Theorem 1. also recognize that Theorem 1.2 cannot be weakened by replacing 'causal future' with 'timelike future' because I + (E, M ) does not contain all of I + , since it misses the north pole and south pole associated with the equator. The above example shows that any compact embedded minimal surface in Σ 0 is a future weakly trapped surface (see [6] for some related discussion). But there are infinitely many such examples in the 3-sphere, of arbitrary genus; see e.g. [3]. (The maximum principle implies that any such example cannot be contained in an open hemisphere; somewhat amusingly, this also follows from Theorem 1.2.) An interesting example in this case, where dim M = 4, is the Clifford torus C ⊂ Σ 0 . Expressing S 3 as the sphere in R 4 , x 2 1 + x 2 2 + x 2 3 + x 2 4 = 1 , the Clifford torus is defined by the equations x 2 1 + x 2 2 = 1/2 = x 2 3 + x 2 4 . The Clifford torus is a minimal surface, so one again has θ ± = 0, and C is a weakly trapped surface in M . This is consistent with Theorem 1.2 since the causal future of C covers all of I + . In fact, the causal future of C already covers the time slice { π 4 } × S 3 ; see Figure 3.
Schwarzschild-de Sitter space. Schwarzschild-de Sitter space is the spacetime M = R × R × S 2 with metric in static form, with m > 0, Λ > 0, and 9Λm 2 < 1. The Penrose diagram for (M , g) is given in Figure 4; cf. [8]. I + has topology R × S 2 . (Here we consider a conformal compactification consisting of a single component of future null infinity.) The g tt -component of (3.3) has positive roots r 1 < r 2 , corresponding to a black hole event horizon and a cosmological horizon, respectively. Regular Kruskal-Szekeres type coordinates can be defined near r = r 1 and r = r 2 . Consider the totally geodesic time slice V = {0} × R × S 2 . Let A be the subset of V consisting of the union of all rotationally symmetric 2-spheres S r , for r 1 < r ≤ r 0 . Provided r 0 < r 2 , Theorem 1.2 implies that there are no future weakly trapped surfaces in J + (A, M ) ∩ I − (I + , M ). In particular, we note that the 2-spheres S r have positive null expansion with respect to the null normal pointing to the right. However, when we allow r 0 = r 2 , then J + (A, M ) ∩ I − (I + , M ) contains the 2-sphere at r = r 2 , which is future weakly trapped, since it is minimal. Again this is consistent with Theorem 1.2, since now J + (A) contains all of I + .