[10] A. KrÃ³lak, A. Czyrka, J. Gaber and W. Rudnicki, in Proceedings of the Cornelius Lanczos International. Centenary Conference, eds. J. D. Brown et...

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arXiv:gr-qc/0203063v1 19 Mar 2002

WIESLAW RUDNICKI∗ Institute of Physics, University of Rzesz´ ow, Rejtana 16A, 35-959 Rzesz´ ow, Poland † ´ ROBERT J. BUDZYNSKI

Department of Physics, Warsaw University, Ho˙za 69, 00-681 Warsaw, Poland

WITOLD KONDRACKI‡ ´ Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-950 Warsaw, Poland

Abstract A new definition of a strong curvature singularity is proposed. This definition is motivated by the definitions given by Tipler and Kr´ olak, but is significantly different and more general. All causal geodesics terminating at these new singularities, which we call generalized strong curvature singularities, are classified into three possible types; the classification is based on certain relations between the causal structure and the curvature strength of the singularities. A cosmic censorship theorem is formulated and proved which shows that only one class of generalized strong curvature singularities, corresponding to a single type of geodesics according to our classification, can be naked. Implications of this result for the cosmic censorship hypothesis are indicated. Keywords: Spacetime singularities; Cosmic censorship; Causal structure; Geodesics PACS: 04.20 Dw; 04.20 Gz

∗ E-mail:

[email protected]

† E-mail:

[email protected]

‡ E-mail:

[email protected]

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1

Introduction

The cosmic censorship hypothesis (CCH) of Penrose [1, 2] says that in generic situations, all spacetime singularities arising from regular initial data are always hidden behind event horizons and hence invisible to outside observers (no naked singularities). This hypothesis plays a fundamental role in the theory of black holes and has been recognized as one of the most important open problems in classical general relativity. There exist many exact solutions of Einstein’s equations which admit naked singularities. However, Penrose [2] has stressed that the exact solutions with special symmetries have a rather limited value for veriﬁcation of the CCH and what is required here is an understanding of the generic case. One possible approach to this problem is to propose a class of generic singularities and then attempt to formulate and prove a censorship theorem which would constrain or prohibit the occurrence of naked singularities of the proposed class. Tipler et al. [3] and Kr´olak [4] have argued that all singularities arising in generic situations should be of the strong curvature type. These singularities have the property that all objects approaching them are crushed to zero volume. The idea of a strong curvature singularity was introduced by Ellis and Schmidt [5] and deﬁned in precise geometrical terms by Tipler [6] (a slightly diﬀerent deﬁnition was given by Kr´olak [4], see below). Kr´olak [4, 7] formulated and proved some censorship theorems that ruled out a class of naked singularities of the strong curvature type. Unfortunately, these results rely heavily on a further assumption (the so-called simplicity condition) which need not hold for generic spacetimes. It should be stressed that there is no hope for ﬁnding a proof of the CCH using only the assumption that all singularities occurring in a given spacetime are of the strong curvature type. This follows just from the fact that naked singularities of this type do occur in certain exact solutions of Einstein’s equations. For instance, they occur in the Tolman-Bondi solution representing spherically symmetric inhomogeneous collapse of dust (see, e.g., Ref. [8]; see also Ref. [9] and references therein). Such naked singularities also occur in more general models of dust collapse — namely, in the Szekeres spacetimes which do not have any Killing vectors [10, 11]. Unnikrishnan [12] has argued that the existence of naked strong curvature singularities in the Tolman-Bondi solution can be ruled out by imposing certain reasonable constraints on the initial distribution of the energy density of dust. This argument, however, depends crucially on the spherical symmetry of the solution, and so cannot be applied to naked singularities occurring in more general cases — e.g. in the Szekeres spacetimes. It is worth recalling here that Bonnor [13] remarked, in another context, that “the Szekeres solution has a good deal of symmetry, even though it has in general no Killing vectors.” It is thus possible that the existence of naked singularities of the strong curvature type will always be accompanied by spacetime symmetries and/or instabilities of some sort, and so one can still hope to prove a formulation of the CCH involving — besides the assumption that all singularities are of strong curvature — a suitable criterion of genericity or stability. To help identify such a criterion, it may be useful to establish and study various relations between strong curvature singularities and the causal structure in their neighborhood — such relations get to the heart of cosmic censorship. For this purpose, we need a new deﬁnition of a strong curvature singularity — one that not only describes the curvature strength of the singularity, but additionally enables one to relate the strength with

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properties of the causal structure in a neighborhood of the singularity. In Section II of this paper, we shall propose such a deﬁnition. Our deﬁnition is motivated by the deﬁnition of a strong curvature singularity given by Tipler and its modiﬁcations by Kr´olak, but is signiﬁcantly diﬀerent and more general. The diﬀerence is in that our deﬁnition involves a certain focusing condition on solutions of the Raychaudhuri equation not only along a single causal geodesic, as in Tipler’s and Kr´olak’s case, but rather along all causal geodesics in some small neighborhood about a given geodesic that reaches the singularity. All causal geodesics terminating at the singularities described by our deﬁnition, which we will refer to as generalized strong curvature singularities, will next be classiﬁed into three possible types; the classiﬁcation is based on certain relations between the causal structure and the curvature strength of the singularities. Further on, in Sections III and IV, we shall formulate and prove a cosmic censorship theorem which shows that only one class of generalized strong curvature singularities, corresponding to a single type of geodesics according to our classiﬁcation, can be naked. This paper was inspired by ideas given in Ref. [14]. Some of the results presented here are reﬁnements of those announced without proofs in Ref. [15]. The notation and fundamental deﬁnitions are as those in the monograph of Hawking and Ellis [16].

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Generalized Strong Curvature Singularities

Before we give our deﬁnition, we need to recall one standard result on the behavior of geodesic congruences. Let λ(t) be an aﬃnely parametrized null (resp. timelike) geodesic. A congruence of null (timelike) geodesics inﬁnitesimally neighboring λ(t) and originating from a point on λ(t) is characterized by two parameters: the expansion θ and the shear σ. The rate of change of the expansion θ along λ(t) is given by the Raychaudhuri equation: dθ 1 = −Rab K a K b − 2σ 2 − θ2 , dt n

(1)

where Rab is the Ricci tensor, K a is the tangent vector to λ(t), n = 2 if λ(t) is a null geodesic and n = 3 in the case when it is timelike (see pp. 78–88 of Ref. [16]). Let us also recall the deﬁnition of a strong curvature singularity introduced by Kr´olak [4].

Definition 1: Let λ be a future-endless, future-incomplete null (timelike) geodesic. λ is said to terminate in the future at a strong curvature singularity if, for each point p ∈ λ, the expansion θ of every future-directed congruence of null (timelike) geodesics emanating from p and containing λ becomes negative somewhere on λ.

This deﬁnition is equivalent to condition K given in Ref. [19]. Kr´olak’s deﬁnition generalizes Tipler’s deﬁnition of a strong curvature singularity because it implies weaker restrictions on the divergence of the curvature near the singularity [19]. We can now introduce our deﬁnition of a generalized strong curvature singularity.

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Definition 2: Let λ be a future-endless, future-incomplete null (timelike) geodesic. We say that λ terminates in the future at a generalized strong curvature singularity if, for each sequence of endless null (timelike) geodesics, {λn }, for which λ is a limit curve, at least one of the following conditions holds: e ∈ {λn } and a point qe ∈ λ e∩N (i) for each point p ∈ λ and for each neighborhood N of p there exist a geodesic λ such that the expansion θ of a future-directed congruence of null (timelike) geodesics emanating from qe

e becomes negative somewhere on λ; e and containing λ

(ii) I − (λ) = I − (λn ) for almost all geodesics belonging to {λn }. Analogously, λ is said to terminate in the past at a generalized strong curvature singularity if the time-reverse versions of the above conditions hold for λ.

Remark: By the term limit curve we mean in the above a curve that satisﬁes the deﬁnition given on p. 185 of Ref. [16]. It is easy to notice that every causal geodesic that terminates at a strong curvature singularity as deﬁned by Kr´olak also terminates at a generalized strong curvature singularity as deﬁned above. To see this, consider, e.g., a null geodesic λ that terminates in the future at Kr´olak’s singularity. Next, take any sequence of endless null geodesics, {λn }, for which λ is a limit curve. Let p be an arbitrary point on λ, and let {pn } be an arbitrary sequence of points converging to p such that, for each n, pn ∈ λn . In accordance with Kr´olak’s deﬁnition, the expansion θ of every future-directed congruence of null geodesics outgoing from p and containing λ must become negative. It follows by continuity that there must exist certain points pk ∈ {pn } suﬃciently close to p such that, for each k, the expansion θ of a future-directed congruence of null geodesics outgoing from pk and containing λk ∈ {λn } shall become negative as well. This clearly means that {λn } shall obey condition (i) of Deﬁnition 2. Let us now turn to condition (ii) of Deﬁnition 2. This condition has no direct equivalent in either Tipler’s or Kr´olak’s deﬁnition; however, it appeals to the original idea behind the concept of a strong curvature singularity [5]. To see this, observe ﬁrst that the requirement I − (λ) = I − (λn ) found in condition (ii) means that both λ and λn must reach exactly the same point in the c-boundary of the spacetime (p. 218 of Ref. [16]). The geodesics λn represent trajectories of point particles in motion towards the singularity within some small neighborhood about λ. Following the original idea of a strong curvature singularity, all small physical objects should be crushed to zero volume as they approach the singularity. Condition (ii) corresponds to the particular case where they are crushed to a single point in the c-boundary. It is worth mentioning here that the c-boundary of cosmological models has been investigated by Tipler [18] in the context of limitations imposed on computation by general relativity. He has shown that a true universal Turing machine can be constructed only in a closed universe whose ﬁnal singularity is a single point in the c-boundary topology. Thus all causal geodesics reaching this singularity will satisfy condition (ii) of Deﬁnition 2 and hence terminate in our generalized strong curvature singularity; moreover, they will be of type A according to the classiﬁcation given below.

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The following deﬁnition provides a complete classiﬁcation of causal geodesics terminating at the singularities introduced in Deﬁnition 2.

Definition 3: Let λ be a future-endless null (timelike) geodesic terminating in the future at a generalized strong curvature singularity, and let {λn } be a sequence of endless null (timelike) geodesics for which λ is a limit curve. λ is said to be of type: • A, if condition (2) of Definition 2 holds for each {λn }; e ∈ {λn } • B, if, for each {λn } which does not satisfy condition (2) of Definition 2, there exist a geodesic λ e − I − (λ) such that the expansion θ of a future-directed congruence of null (timelike) and a point qe ∈ λ e becomes negative somewhere on λ; e geodesics emanating from qe and containing λ

• C, if λ is neither of type A nor B.

Remark: If the λ mentioned above is a null geodesic that admits a segment contained in the boundary, I˙− (λ), of its chronological past, one can always ﬁnd a sequence of endless null geodesics, {λn }, for which λ is a limit curve, such that none of the λn will be contained in the closure of I − (λ). Clearly, such a sequence cannot satisfy condition (ii) of Deﬁnition 2, and so λ cannot be of type A. In general, however, there may exist a future-endless null geodesic λ that never intersects the boundary of its chronological past. For example, if λ terminates in the future at a curvature singularity and the curvature diverges fast enough along λ, every future-endless segment of λ may admit a pair of conjugates points, which implies that no segment of λ can be contained in the achronal boundary I˙− (λ). In this case λ may be of type A.

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Censorship Theorem

For the sake of simplicity, we shall restrict our considerations to weakly asymptotically simple and empty (WASE) spacetimes (p. 225 of Ref. [16]). Such a spacetime (M, g) can be conformally imbedded in a larger spacetime f, e (M g) as a manifold with boundary M = M ∪ ∂M , where the boundary ∂M consists of two null surfaces J + and J − that represent future and past null inﬁnity, respectively. Moreover, there exists an open neighborhood f such that U ∩ M coincides with part of an asymptotically simple and empty spacetime (M ′ , g ′ ), U of ∂M in M which means that all possible singularities of (M, g) can only occur in the region M − U . Since the CCH is concerned with singularities that develop from an initially non-singular state, we shall deal here only with such WASE spacetimes (M, g) in which the region M − U does not extend arbitrarily far into the past and to a spatial inﬁnity.a To make this precise, we shall assume that (M, g) admits a partial Cauchy surface S for which the following two conditions hold: a Due

to its generality, the definition of a WASE spacetime given in Ref. [16] has the unwanted feature that the region M − U

might extend to a spatial infinity. For a deeper discussion of this problem, see Ref. [19].

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(i) S has an asymptotically simple past (p. 316 of Ref. [16]); (ii) every null geodesic µ generating J + admits a past-endless segment µ0 such that µ0 ⊂ I + (S, M ) and I − (µ0 , M ) ∩ I + (S) ⊂ D+ (S) ∩ U . Such a surface S will be called a regular partial Cauchy surface. (This deﬁnition is a slight modiﬁcation of that previously used in Ref. [20].) Let us now recall that a WASE spacetime (M, g) is said to be future asymptotically predictable from a partial Cauchy surface S if the future null inﬁnity J + is contained in the closure of the future Cauchy development D+ (S, M ) (p. 310 of Ref. [16]). Future asymptotic predictability is a mathematically precise statement of cosmic censorship for (M, g), since it ensures that there will be no singularities to the future of S which are naked, i.e. which are visible from J + . We are now in a position to state our cosmic censorship theorem. By this theorem, only singularities corresponding to null geodesics of type C can be naked.

Theorem 1: Let (M, g) be a WASE spacetime admitting a regular partial Cauchy surface S. Suppose furthermore that the following conditions hold: (i) the null convergence condition, i.e. Rab K a K b ≥ 0 for every null vector K a of (M, g); (ii) the generic condition, i.e. every endless null geodesic of (M, g) admits a point at which K[a Rb]cd[e Kf ] K c K d 6= 0, where K a is the tangent vector to the geodesic; (iii) (M, g) admits no naked points-at-infinity, i.e. for each point p ∈ J + , every future-endless, future-complete null geodesic of (M, g) contained in I¯− (p, M ) ∩ D(S) has a future endpoint on J + in M ; (iv) every incomplete null geodesic of (M, g) terminates at a generalized strong curvature singularity. If (M, g) is not future asymptotically predictable from S, then there must exist a null geodesic λ ⊂ intD(S) which terminates in the future at a generalized strong curvature singularity and is of type C.

Remarks: Conditions (i) and (ii) of Theorem 1 are reasonable requirements for any physically realistic model of a classical spacetime; they have been discussed extensively in the literature on the singularity theorems (see, e.g., Refs. [3, 16]). Condition (iii) is a slightly weakened version of that previously used by Newman and Joshi [21] in their censorship theorem. This condition ensures that any possible breakdown of future asymptotic predictability in (M, g) must be associated with the occurrence of a naked singularity and not of a naked “point-at-infinity”. Known examples of WASE spacetimes with naked singularities contain no naked points-at-inﬁnity in the sense of condition (iii). It should, however, be stressed that this condition restricts to some extent the generality of our result, since it is conceivable that some spacetimes might contain both naked singularities and naked points-at-inﬁnity. 6

Note also that no causality conditions, other than those implied by the existence of S, are imposed on (M, g) in Theorem 1. Thus this theorem may be applied to naked singularities associated with causality violation. The possible existence of such naked singularities results from Tipler’s singularity theorem [22].

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Proof of the Theorem

Now we shall prove our theorem; the following two lemmas serve this purpose.

Lemma 1: Under the assumptions of Theorem 1, suppose that (M, g) is not future asymptotically predictable from S. Then there must exist a past-incomplete null geodesic η ⊂ H + (S, M ) which has a future endpoint p ∈ J + . Moreover, the following conditions hold: (a) there exists a point r ∈ η ∩ M such that the closure of I − (r) ∩ S is compact; (b) for each point ci ∈ I − (p, M ) ∩ D+ (S), there exists a null geodesic λi ⊂ I˙+ (ci , M ) which extends from ci to some point on J + .

Proof: Let us ﬁrst observe that from condition (ii) of the deﬁnition of S it follows that every generator of J + must intersect the closure of D+ (S, M ). In addition, as S has an asymptotically simple past by condition (i), J − must be contained in the closure of D− (S, M ). This implies that (M, g) is partially asymptotically predictable from S as deﬁned by Tipler [22]. Thus, as (M, g) is not future asymptotically predictable from S, by Proposition 2 of Ref. [22] there must exist a past-endless null geodesic generator η of H + (S) with future endpoint p ∈ J + . Following steps as in the proof of Theorem 1 of Ref. [22], we can now easily show, by making use of conditions (i) and (ii) of our theorem, that η must be incomplete in the past. We shall now show that condition (a) holds. Let us choose an arbitrary point r ∈ η∩M ; we shall demonstrate that the set R ≡ I − (r) ∩ S is compact. Since S has an asymptotically simple past, the proof of this fact will be quite similar to the proof of the Theorem of Ref. [23]. Assume, to the contrary, that R is not compact. Then, by Lemma 2 of Ref. [23], there must exist a sequence of null geodesics ηi with future endpoints ri converging to r, such that each ηi will be a generator of the achronal boundary I˙− (ri ) and will have a past endpoint on J − ; moreover, the geodesic η must be a limit curve of the sequence {ηi }. As η is past-incomplete, by condition (iv) of Theorem 1, it must terminate in the past at a generalized strong curvature singularity. Thus {ηi } must fulﬁll at least one of the time-reverse versions of conditions (i) and (ii) of Deﬁnition 2. Since each of the ηi has a past endpoint on J − , for each ηi one has I + (ηi ) 6= I + (η), as η ⊂ I + (S) while J − ⊂ I − (S, M ). It is thus clear that the time-reverse version of condition (ii) cannot be fulﬁlled, and so the time-reverse version of condition (i) must be. There must thus exist a geodesic ηe ∈ {ηi } and a point qe ∈ ηe such that the expansion θ of a past-directed null geodesic congruence emanating from qe and containing ηe must become negative somewhere on ηe. In this case, by Proposition 4.4.4 of Ref. [16] and condition (i) of our Theorem 1, there would exist a point conjugate to qe along ηe, since ηe is 7

past-complete as it has a past endpoint on J − . However, by Proposition 4.5.12 of Ref. [16], the existence of a pair of conjugate points on ηe would contradict the achronality of ηe. In view of this contradiction we must conclude that R is compact. We shall now show that condition (b) holds. Let ci be an arbitrary point in I − (p, M ) ∩ D+ (S), and let µ be a generator of J + that passes through p. We shall ﬁrst show that µ must leave I + (ci , M ). To see this, T suppose that µ were contained in I + (ci , M ). Then the past set X ≡ a∈µ0 [I − (a, M ) ∩ M ], where µ0 denotes the past-endless segment of µ mentioned in condition (ii) of the deﬁnition of S, would be non-empty, as it would contain ci . From the deﬁnition of X it follows that any null geodesic generator of the boundary of X cannot have its future endpoint on J + ; otherwise, such a point would have to coincide with a past endpoint of µ0 , which is impossible as µ0 is past-endless. But from the deﬁnition of the surface S it follows that X would have to be contained in an open neighborhood U of J + ∪ J − , such that U ∩ M coincides with part of an asymptotically simple and empty spacetime, which implies that all null geodesics generating the boundary of X would have to have their future endpoints on J + . This contradiction shows that µ must leave I + (ci , M ), so there must exist a point b ∈ µ ∩ I˙+ (ci , M ). Since b ∈ µ ∩ J − (p, M ), and p ∈ H + (S, M ), it follows that b must belong to the closure of D+ (S, M ). Therefore all the past-directed null geodesics outgoing from b, with the exception of µ, must enter the interior of D+ (S) immediately after leaving b; otherwise they would be generators of J + , which is impossible. Thus, as intD+ (S) is a causally simple set, there must exist a null geodesic λi ⊂ I˙+ (ci , M ) which extends from ci to b ∈ J + , as required in condition (b).

✷

Lemma 2: Let r be a point on H + (S), where S is a partial Cauchy surface. If the set I˙− (r)∩J + (S) is compact, then every null geodesic generator of H + (S) through r is geodesically complete in the past direction.

Proof: The proof of this lemma is identical to the proof of Lemma 8.5.5 of Ref. [16]. In the course of the proof we only need to consider the set I˙− (r) ∩ J + (S) instead of the Cauchy horizon H + (S).

✷

Proof of Theorem 1: Assume that (M, g) is not future asymptotically predictable from S. Then, by Lemma 1, there exists a past-incomplete null geodesic η ⊂ H + (S) which has a future endpoint p ∈ J ; moreover, there exists a point r ∈ η ∩ M such that R ≡ I − (r) ∩ S is compact. As η is past-incomplete, by Lemma 2 the set Q ≡ I˙− (r) ∩ J + (S) cannot be compact. Let us put a timelike vector ﬁeld on M (such a ﬁeld will always exist since M admits a Lorentz metric g). If every integral curve of this ﬁeld that intersects the set R also intersects the set Q, we would have a continuous one-to-one mapping of R onto Q, and hence Q would have to be compact. Therefore there must exist a future-endless timelike curve α ⊂ I − (r). Let P be the family of all sets of the form I − (α), where α is a future-endless timelike curve contained in I − (r); and let Pb be a maximal chain determined in P by the relation of inclusion. Denote now by P0 the set T {P : P ∈ Pb }. This set is non-empty. To see this, note ﬁrst that no member of Pb can be contained in I − (S);

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otherwise, there would exist a future-endless timelike curve α ⊂ I − (S), which is impossible as I − (S) = D− (S). Note now that, for each P ∈ Pb , the set P ∩ S must be compact, as it is a closed subset of the compact set R. This clearly implies, by the deﬁnition of P0 , that the set A ≡ P0 ∩ S is non-empty and compact. In addition, from the deﬁnition of P0 it follows that there must exist a future-endless timelike curve α0 such that P0 = I − (α0 ). Note also that P0 must be a minimal element of Pb , i.e. we will have P0 = I − (β) for every future-endless timelike curve β ⊂ P0 . Let us now ﬁx some future-endless timelike curve α0 such that P0 = I − (α0 ). Let {ci } be a sequence of points on α0 ∩ D+ (S) such that, for each i, ci+1 ∈ I + (ci ); assume also that {ci } has no accumulation point on α0 (such a sequence can always be found as α0 has no future endpoint). By condition (b) of Lemma 1, there must exist, for each i, a null geodesic λi running from ci to some point on J + . Moreover, each λi shall be a generator of the achronal boundary I˙+ (ci , M ) and shall be future-complete as it reaches J + . Let us now extend each of the λi maximally into the past. Since each λi passes through ci ∈ α0 ∩ D+ (S), each of the extended λi must intersect S at some point ai ∈ A. As the set A is compact, the sequence {ai } must have an accumulation point a ∈ A. Therefore, by Lemma 6.2.1 of Ref. [16], through a there is a non-spacelike curve λ which is future-inextendible in M and which is a limit curve of the sequence {λi }. Since all the λi are null geodesics, λ must be a null geodesic as well. Note also that λ must be contained in P0 ; otherwise λ would have to intersect the curve α0 at some point that would be an accumulation point of the sequence {ci }, which is impossible as {ci } has no accumulation point on α0 . Moreover, as P0 is a minimal element of Pb , we must have P0 = I − (λ). Since λ intersects the surface S, it cannot be a generator of H + (S). Thus, as λ ⊂ P0 ⊂ D(S), we will have λ ⊂ intD(S). Since λ ⊂ P0 ⊂ I − (r), and r ∈ M , λ cannot have a future endpoint on J + . In addition, as r ∈ J − (p, M ), λ must be contained in the closure of I − (p, M ). Therefore, as p ∈ J + , by condition (iii) of Theorem 1, λ must be incomplete in the future. By condition (iv) of Theorem 1, λ must terminate in the future at a generalized strong curvature singularity. Thus the sequence {λi } must satisfy at least one of conditions (i) and (ii) of Deﬁnition 2. Since λ ⊂ P0 , and each of the λi must leave P0 as it has a future endpoint on J + , we will have I − (λ) 6= I − (λi ) for each i. This means that condition (ii) of Deﬁnition 2 cannot hold for the {λi }, hence λ cannot be of type A. Suppose that λ e ∈ {λi } and a point qe ∈ λ e − I − (λ) were of type B. Then, according to Deﬁnition 3, there would exist a geodesic λ

e such that the expansion θ of a future-directed congruence of null geodesics outgoing from qe and containing λ

e As λ e is future-complete, by condition (i) of our theorem and Proposition would be negative somewhere on λ. e Consequently, by 4.4.4 of Ref. [16], there would thus exist some point x ∈ J + (e q ) conjugate to qe along λ.

e ∩ J + (x). But Proposition 4.5.12 of Ref. [16], there would also exist a timelike curve from qe to some point y ∈ λ

this is impossible because from the above construction it follows that the points qe and y would have to lie on

e contained in the boundary I˙+ (α0 ∩ λ). e This contradiction shows that the geodesic λ the achronal segment of λ cannot be of type B. Thus, as λ is neither of type A nor B, it must be of type C, which completes the proof. ✷

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5

Concluding Remarks

In Theorem 1 we have demonstrated that the only possible naked singularities of strong curvature are those corresponding to null geodesics of type C, according to the classiﬁcation proposed in this paper. As a consequence, one only needs to consider the stability of type C singularities as relevant to the cosmic censorship problem, which is a signiﬁcant restriction. One argument showing that these singularities have a tendency to be unstable under generic perturbations was given in Ref. [15]. We may conclude that a further study of the properties of type C singularities might bring us closer to formulating the genericity and stability criterium which is needed, as mentioned in the Introduction, to arrive at a satisfactory statement of the CCH. As we mentioned above, naked strong curvature singularities are present in certain exact solutions of the Einstein equations — solutions that represent gravitational collapse. In view of our result, it would therefore be of interest to verify whether in those solutions, the naked singularity is also associated in every case with the existence of type C geodesics. It should be stressed that this does not follow directly from Theorem 1, since it is not clear whether the solutions in question necessarily satisfy all the assumptions of that theorem.

Acknowledgements We wish to thank Andrzej Kr´olak for many valuable discussions. This work was supported in part by the Polish Committee for Scientiﬁc Research (KBN) under Grant No. 2 P03B 130 16.

References [1] R. Penrose, Riv. Nuovo Cimento 1, 252 (1969). [2] R. Penrose, in Theoretical Principles in Astrophysics and Relativity, eds. N. R. Lebovitz et al. (Univ. of Chicago Press, Chicago, 1978), pp. 217–243. [3] F. J. Tipler, C. J. S. Clarke and G. F. R. Ellis, in General Relativity and Gravitation, ed. A. Held (Plenum, New York, 1980), Vol. 2, pp. 97–206. [4] A. Kr´olak, Class. Quantum Grav. 3, 267 (1986). [5] G. F. R. Ellis and B. G. Schmidt, Gen. Rel. Grav. 8, 915 (1977). [6] F. J. Tipler, Phys. Lett. A64, 8 (1977). [7] A. Kr´olak, J. Math. Phys. 28, 138 (1987); J. Math. Phys. 33, 701 (1992). [8] S. S. Deshingkar, P. S. Joshi and I. H. Dwivedi, Phys. Rev. D59, 044018 (1999). [9] P. S. Joshi, Global Aspects in Gravitation and Cosmology (Clarendon Press, Oxford, 1993).

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[10] A. Kr´olak, A. Czyrka, J. Gaber and W. Rudnicki, in Proceedings of the Cornelius Lanczos International Centenary Conference, eds. J. D. Brown et al. (SIAM, Philadelphia, 1994), pp. 518–520. [11] P. S. Joshi and A. Kr´olak, Class. Quantum Grav. 13, 3069 (1996). [12] C. S. Unnikrishnan, Gen. Rel. Grav. 26, 655 (1994); Phys. Rev. D53, R580 (1996). [13] W. B. Bonnor, Commun. Math. Phys. 51, 191 (1976). [14] W. Rudnicki, Acta Cosm. 18, 77 (1992). [15] W. Kondracki, A. Kr´olak and W. Rudnicki, Math. Proc. Camb. Phil. Soc. 114, 379 (1993). [16] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge Univ. Press, Cambridge, 1973). [17] C. J. S. Clarke and A. Kr´olak, J. Geom. Phys. 2, 127 (1985). [18] F. J. Tipler, Int. J. Theor. Phys. 25, 617 (1986). [19] C.-M. Claudel, LANL archive preprint gr-qc/0005031. [20] W. Rudnicki, Phys. Lett. A208, 53 (1995). [21] R. P. A. C. Newman and P. S. Joshi, Ann. Phys. (N.Y.) 182, 112 (1988). [22] F. J. Tipler, Phys. Rev. Lett. 37, 879 (1976). [23] A. Kr´olak and W. Rudnicki, Gen. Rel. Grav. 25, 423 (1993).

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