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Roads to topological censorship

In this note we review some aspects of topological censorship. We present several (actually five) alternative sets of hypotheses which allow the proof of a topological censorship theorem for spacetimes with conformal completions at infinity and vanishing cosmological constant.

The setting for the original FSW result is that of asymptotically flat spacetimes in the sense of Penrose. It has been noted in the past (cf. e.g. [15,16]), that some of the arguments in [11] require somewhat stronger assumptions. Indeed, the following issues need to be addressed in the proof: 1. The question of "i 0 avoidance", namely that the future of a compact subset of the physical spacetime does not contain all of future null infinity I + . This problem was already discussed in [16] (see in particular the second paragraph on p. L2 there, where it is proposed to extend global hyperbolicity to I + ).
1. M is the interior of M , 2. on M there exists a function Ω with the property that the metric g, defined as Ω 2 g on M , extends by continuity to the boundary of M , with the extended metric maintaining its signature on the boundary, 3. Ω is (strictly) positive on M , differentiable on M , vanishes precisely on I := ∂ M , with dΩ nowhere vanishing on I .
In the case where I is a null hypersurface, Geroch and Horowitz [17] propose to add the conditions that 1) one can find a gauge such that the Hessian of Ω vanishes on I , and that 2) the integral curves of ∇Ω on I are complete in this gauge. While this is certainly useful for some purposes, these conditions do not seem to play any obvious role in our treatment of the problem at hand. Unless explicitly specified otherwise we use the terminology and notation of [6].
For the reader's convenience we start by recalling [6,Theorem 5.5] with the Kaluza-Klein group G taken to be trivial, as rewritten in the context of Penrose's definition of asymptotic flatness. For this some terminology will be needed.
We say that I satisfies asymptotic estimates uniform to order one if there exists 1. a "radial function" r and a constant R 0 such that the set {r ≥ R 0 } forms a neighborhood of I , with the level sets of r timelike there, and 2. a time function t on M such that for all r ≥ R 0 and for all τ ∈ R the sets S r,τ := {r = R, t = τ } are smooth past and future inner trapped compact spacelike submanifolds of M .
The terminology is related to the formula for the expansion scalar of null hypersurfaces,which guarantees the above in spacetimes which are asymptotically flat in a coordinate sense (as in [6]), provided that the metric and its first derivatives satisfy bounds which are uniform in time in the asymptotic region {r ≥ R 0 }.
We have the following rewording of Theorem 5.5 in [6]: Theorem 2.2 Let (M , g) be a spacetime satisfying the null energy condition, i.e. R µν X µ X ν ≥ 0 for all null vector fields X, and admitting a conformal completion with null boundary I satisfying asymptotic estimates uniform to order one. If the domain of outer communications I is globally hyperbolic and if there exists R 1 ≥ R 0 such that the set {r ≥ R 1 } is simply connected, then I is simply connected.
We continue with the following non-visibility result, closely related to the problem at hand: where I + is a connected null hypersurface, and suppose that is globally hyperbolic. If the null energy condition holds on D, then there are no compact weakly future trapped spacelike submanifolds of codimension two within D. Proof: Let p ∈ J + (K, D)∩I + . By global hyperbolicity J − (p, , D)∩J + (K, D) is compact. Suppose that J + (K, D) contains all of I + , then the generator of I + through p is contained in the compact set J − (p, , D) ∩ J + (K, D), which is impossible in a globally hyperbolic, hence strongly causal, spacetime.
For our purposes below we will need a variation of Theorem 2.3. Consider a compact codimension-two spacelike submanifold of M , say S, and let us suppose that S is two-sided, in the sense of existence of a well-defined field of spacelike normals. Let us choose to denote one of those sides "inner". Instead of supposing, as in Theorem 2.3, that both families of null future directed geodesics are weakly trapped at S, we will only assume that the inner family is weakly trapped. The proof of Theorem 2.3 applies as is to this setting to give: Theorem 2.5 Under the hypotheses of Theorem 2.3, let S be a future weakly inner trapped submanifold contained in D. Then there are no null geodesics oṅ J + (S) connecting I + with the inner side of S. Remark 2.6 As such, we will only need Theorem 2.5 for surfaces which are inner trapped rather than weakly inner-trapped. In such a case the proof of [

The theorem
We have the following alternative to Theorem 2.2: Theorem 3.1 Let (M , g) be a globally hyperbolic space-time satisfying the null energy condition. Assume that (M , g) admits a completion at infinity such that both I + and I − are connected null hypersurfaces, and that is globally hyperbolic. Suppose that there exists a simply connected neighborhood U of I and a foliation of a (perhaps smaller) neighborhood of I − with spacelike up-to-boundary and smooth up-to-boundary acausal hypersurfaces S τ ⊂ M , τ ∈ R, such that for each τ the intersections S τ,ε of S τ with the ε-level sets of Ω are non-empty, smooth, connected and compact for all ε ≥ 0 small enough.
(3.2) If, again for each τ and for all ε small enough, the null geodesics normal to S τ,ε withΩ| Sτ,ε < 0 never leave U , whereΩ denotes the derivative of Ω along the geodesic, then the domain of outer communications I + (I − ) ∩ I − (I + ) is simply connected.
The key property of the collection of smooth compact manifolds S τ,ε is that they are future-inner trapped, cover a neighborhood of I − , and satisfy (3.3). There are many ways of obtaining such a family. First, one could, e.g., assume that there exists a foliation of I − by smooth compact submanifolds, each intersecting the generators of I − exactly once, and extend this foliation to a foliation of a neighborhood of I − in M by moving along a field of spacelike directions, leading to the family S τ of the theorem.
Next, the manifolds S τ,ε can then, e.g., be obtained by moving a g-distance ε away from I − along S τ , rather than intersecting with the level sets of Ω.
Finally, one could define an alternative τ -foliation, as needed in the statement of the theorem, by extending the just-mentioned foliation of I − into M along null rather than spacelike hypersurfaces S τ , with the manifolds S τ,ε obtained by moving cuts of I − an affine-parameter-distance ε along the generators of S τ .
In each case the resulting S τ,ε 's will be future-inner trapped for ε small enough. But note that in each case the additional condition (3.3) needs to be imposed.
Remark 3.3 For simplicity we assume that both metricsg and g are smooth up to boundary. One can check, using e.g. [8] and the conformal transformation formula for the divergence of null hypersurfaces, that the proof applies if g is C 2 on M andg is C 1 up-to-boundary.
Remark 3.4 The theorem remains true, with an essentially identical proof, when the null energy condition is replaced by the following integrated null energy condition: for all half-geodesics γ that start near I − it holds that  In the case where I is a null hypersurface, Geroch and Horowitz [17] propose to add the conditions that 1) one can find a gauge such that the Hessian of ⌦ vanishes on I , and that 2) the integral curves of e r⌦ on I are complete in this gauge. While this is certainly useful for some purposes, these conditions do not seem to play any obvious role in our treatment of the problem at hand.
As usual, we set The domain of outer communications hhI ii is defined as 2. Notation as in the proof of Theorem 3.1.  Suppose hhI ii is not simply connected. Let U 1 and U 2 be as in the proof of Theorem 3.1. Let S 1 , S 0 1 , S and A 1 be the lifts of S, S 0 , S ⌧ 0 ,✏ and A, respectively, to U 1 . We may assume J + (A 1 ) meets I + 2 but, by Lemma 2.4, does not contain all of it. Then there exists a future complete null geodesic onJ(A 1 ) that extends from I + 2 to either S 1 or S 0 1 when followed to the past. If it extends to S 0 1 , it will have to cross D + ( c S 1 ) ⇢ U 1 at some point But then there will be a null generator, sayˆ of H + ( c S 1 ) from p that extends to S 1 . Now, generators of H + ( c S 1 ) are either past inextendible, or have an end point on the edge of c S 1 . We claim that the former case cannot occur. In order to see this, we first note that any point q lying on a generator of If the curve connects q with S there is nothing to prove; otherwise it connects to I S , but then we can concatenate it with a segment of generator As such, the least satisfactory condition in Theorem 3.1 appears to be (3.3), and the question arises whether it can be replaced by some other natural conditions. We list here some alternatives; we have not been able to verify that it can be gotten rid of altogether, as hinted-to in [11] (compare Lemma 1 there): Proposition 3.4 Under the remaining hypotheses of Theorem 3.1, suppose that instead of (3.3) we have either 1. for each ⌧ and for all ✏ small enough those generators ofJ + (S ",⌧ ) on which d⌦ ds | S",⌧ < 0 never leave U ; is also globally hyperbolic, and it holds that where  Proof of Theorem 3.1. The proof uses a mixture of well-known-by-now ideas from [6,11,13], with the heart of the argument stemming from [11]. For the sake of clarity we present most details, without claim to novelty. Suppose that the domain of outer communications I is non-empty and not simply-connected, otherwise there is nothing to prove.
Let M 1 be the universal cover of I ∪ I equipped with the covering metrics which we continue to denote by g andg. The covering manifold M 1 contains several copies, say U α , with α ∈ A for some index set A, of the simply connected neighborhood U of the original I . Each U α comes with its adjacent conformal boundaries I + α , I − α and I α := I + α ∪ I − α . Let us choose two distinct lifts of U , call them U 1 and U 2 , with conformal boundaries I 1 and I 2 , and suppose that there exists a causal curve, sayγ, connecting I − 1 with I + 2 . Letτ be such thatγ meets I − 1 at Sτ . We can choose a small parameter > 0 and a value of τ 0 nearτ so that 4. S is inner future trapped, in the sense that it has negative divergence on those null geodesics which initially move normally away from S to the future and in the direction of increasing Ω. This follows from the following transformation formula 2 for the future-and past-divergences θ ± , in space-time dimension n + 1, whereθ ± are the divergences calculated using the metricg,T is the field of unit future-directed normals to S τ 0 andÑ is the field of outer-pointing normals to the level sets of Ω within S τ 0 . By hypothesis ∇Ω is null and past-directed on I − , tangent to I − , while (T ±Ñ ) is null, future-directed and transverse to I − . This shows that θ ± is positive, bounded away from zero on S τ 0 ∩ I − , hence positive for all small enough.
Let S 1 denote the lift of S to U 1 . We claim that S 1 is a future outer trapped surface as seen from I + 2 . For this, note that the achronal boundaryJ + (S 1 ) forms near S 1 a union of two smooth null hypersurfaces ruled by null geodesic segments normal to S 1 . Let us denote by N L 1 the subset ofJ + (S 1 ) on which Ω initially increases when moving away from S 1 along the generators, and N R 1 the subset on which Ω initially decreases. As in [6, Theorem 6.1] there exists a null geodesic γ inJ + (S 1 ) connecting S 1 to I + 2 . The geodesic γ intersects ∂U 1 , and therefore cannot lie on N R 1 which is entirely included in U 1 by (3.3). Hence γ lies on N L 1 , which has negative divergence on S 1 , as claimed. But this contradicts Theorem 2.5. We conclude that To finish the proof, note that the collection of open sets The sets are pairwise disjoint by (3.5). Since M 1 is connected there is only one such set, contradicting the assumption of non-simple connectivity, and Theorem 3.1 is established.
As such, the least satisfactory condition in Theorem 3.1 appears to be (3.3), and the question arises whether it can be replaced by some other natural conditions. We list here some alternatives; we have not been able to verify that (3.3) can be gotten rid of altogether. is also globally hyperbolic, and there exists a foliation S τ as in Theorem 3.1 such that for all τ and for some ε small enough we have a
2. on e M there exists a function ⌦ with the property that the metric e g, defined as ⌦ 2 g on M , extends by continuity to the boundary of e M , with the extended metric maintaining its signature on the boundary, 3. ⌦ is (strictly) positive on M , di↵erentiable on e M , vanishes precisely on I := @ e M , with d⌦ nowhere vanishing on I .
In the case where I is a null hypersurface, Geroch and Horowitz [17] propose to add the conditions that 1) one can find a gauge such that the Hessian of ⌦ vanishes on I , and that 2) the integral curves of e r⌦ on I are complete in this gauge. While this is certainly useful for some purposes, these conditions do not seem to play any obvious role in our treatment of the problem at hand.
As usual, we set  Suppose hhI ii is not simply connected. Let U 1 and U 2 be as in the proof of Theorem 3.1. Let S 1 , S 0 1 , S and A 1 be the lifts of S, S 0 , S ⌧ 0 ,✏ and A, respectively, to U 1 . We may assume J + (A 1 ) meets I + 2 but, by Lemma 2.4, does not contain all of it. Then there exists a future complete null geodesic onJ(A 1 ) that extends from I + 2 to either S 1 or S 0 1 when followed to the past. If it extends to S 0 1 , it will have to cross D + ( c S 1 ) ⇢ U 1 at some point But then there will be a null generator, sayˆ of H + ( c S 1 ) from p that extends to S 1 . Now, generators of H + ( c S 1 ) are either past inextendible, or have an end point on the edge of c S 1 . We claim that the former case cannot occur. In order to see this, we first note that any point q lying on a generator of In the case where I is a null hypersurface, Geroch and Horowitz [17] propose to add the conditions that 1) one can find a gauge such that the Hessian of ⌦ vanishes on I , and that 2) the integral curves of e r⌦ on I are complete in this gauge. While this is certainly useful for some purposes, these conditions do not seem to play any obvious role in our treatment of the problem at hand.
As usual, we set The domain of outer communications hhI ii is defined as hhI ii := I + (I ) \ I (I + ) .
2 covers e M 1 . The sets are pairwise disjoint by (3.4). Since e M 1 is connected there is only one such set, and Theorem 3.1 is established. 2 As such, the least satisfactory condition in Theorem 3.1 appears to be (3.3), and the question arises whether it can be replaced by some other natural conditions. We list here some alternatives; we have not been able to verify that it can be gotten rid of altogether, as hinted-to in [11] (compare Lemma 1 there): is also globally hyperbolic, and it holds that

7)
where I S := I \ J + (S ) (cf.  is also globally hyperbolic, and it holds that

7)
where I S := I \ J + (S ) (cf. Figure 3.2); or 3. for each ⌧ and for all " small enough the connected component of the level set {⌦ = "} containing S ⌧," is a timelike hypersurface entirely contained in U and separating U . Suppose hhI ii is not simply connected. Let U 1 and U 2 be as in the proof of Theorem 3.1. Let S 1 , S 0 1 , S and A 1 be the lifts of S, S 0 , S ⌧ 0 ,✏ and A, respectively, to U 1 . We may assume J + (A 1 ) meets I + 2 but, by Lemma 2.4, does not contain all of it. Then there exists a future complete null geodesic onJ(A 1 ) that extends from I + 2 to either S 1 or S 0 1 when followed to the past. If it extends to S 0 1 , it will have to cross D + ( c S 1 ) ⇢ U 1 at some point p 2 H + ( c S 1 ) \ M (not on S 1 ). But then there will be a null generator, sayˆ of H + ( c S 1 ) from p that extends to S 1 . Now, generators of H + ( c S 1 ) are either past inextendible, or have an end point on the edge of c S 1 . We claim that the former case cannot occur. In order to see this, we first note that any point q lying on a generator of is also globally hyperbolic, and it holds that

7)
where I S := I \ J + (S ) (cf. In the case where I is a null hypersurface, Geroch and Horowitz [17] propose to add the conditions that 1) one can find a gauge such that the Hessian of ⌦ vanishes on I , and that 2) the integral curves of e r⌦ on I are complete in this gauge. While this is certainly useful for some purposes, these conditions do not seem to play any obvious role in our treatment of the problem at hand.
As usual, we set The domain of outer communications hhI ii is defined as hhI ii := I + (I ) \ I (I + ) .
2 covers e M 1 . The sets are pairwise disjoint by (3.4). Since e M 1 is connected there is only one such set, and Theorem 3.1 is established. 2 As such, the least satisfactory condition in Theorem 3.1 appears to be (3.3), and the question arises whether it can be replaced by some other natural conditions. We list here some alternatives; we have not been able to verify that it can be gotten rid of altogether, as hinted-to in [11] (compare Lemma 1 there): is also globally hyperbolic, and it holds that where I S := I \ J + (S ) (cf.   that component of {Ω = ε} which contains S τ,ε . It is, by hypothesis, a timelike hypersurface. It contains a compact achronal spacelike surface S τ,ε and therefore constitutes a globally hyperbolic space-time of its own, with Cauchy surface S τ,ε , by [4,12]. Since {Ω = ε} 0 separates U by hypothesis, every generator oḟ J + (S τ,ε ) which starts withΩ < 0 at S τ,ε and which leaves U has to cross {Ω = ε} 0 again. But global hyperbolicity of {Ω = ε} 0 implies that through every point of {Ω = ε} 0 there exists a timelike curve meeting S τ,ε , which is not possible since the generators ofJ + (S τ,ε ) do not intersect I + (S τ,ε ).
3. Let τ 0 , as well as the remaining notation, be as in the proof of Theorem 3.1.
Let S τ 0 , = S τ 0 ∩ {Ω ≤ }, and set S = S τ 0 , ∪ I − Suppose I is not simply connected. Let U 1 and U 2 be as in the proof of Theorem 3.1. Let S 1 , S 1 , S 1 , S 1 and A 1 be the lifts of S, S , S τ 0 , , S and A, respectively, to U 1 . We may assume J + (A 1 ) meets I + 2 but, by Lemma 2.4,

8
does not contain all of it. Then there exists a future complete null geodesic γ onJ(A 1 ) that extends from I + 2 to either S 1 or S 1 when followed to the past. Suppose, first, that γ extends to S 1 . Then it will have to cross D + ( S 1 ) ⊂ U 1 at some point, say p ∈ H + ( S 1 )∩M (not on S 1 ). Let us denote byγ a generator of H + ( S 1 ) passing through p. Now, generators of H + ( S 1 ) are either past inextendible, or have an end point on the edge of S 1 . We wish to show thatγ extends to S 1 , and therefore the former case cannot occur. In order to see this, we first note that any point q lying on a generator of H + ( S 1 ) is in J + (S 1 , D − 1 ), where Indeed, since q ∈ H + ( S 1 ) there exists a causal curve connecting q with S 1 = S 1 ∪ I − S . If the curve connects q with S 1 there is nothing to prove; otherwise it connects to I − S 1 , but then we can concatenate it with a segment of generator of I − 1 which slides down from the intersection point to I − S 1 . We conclude that the part of the generator of H + ( S 1 ) which lies to the past of p is included in The set K is compact by global hyperbolicity of D − 1 . Sinceγ is contained in the compact set K , it has to end on the edge of S 1 to avoid a violation of strong causality, as claimed. Now, since the edge of S 1 is S 1 ,γ clearly enters the timelike future of A 1 . But this implies that γ enters the timelike future of A 1 , which is not possible for a curve lying onJ + (A 1 ), a contradiction. Thus γ extends to S 1 , and necessarily to the inside. Hence γ lies on N L 1 , and the proof continues as in the proof of Theorem 3.1.