Szekeres quasispherical collapse, in Proceedings of the Cornelius Lanczos International Centenary. Conference, eds. J. D. Brown et al. (SIAM, 1994), p...

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arXiv:gr-qc/0606007v1 1 Jun 2006

Wieslaw Rudnicki Institute of Physics, University of Rzesz´ow, Rejtana 16A 35-959 Rzesz´ow, Poland [email protected] Robert J. Budzy´ nski Department of Physics, Warsaw University, Ho˙za 69 00-681 Warsaw, Poland [email protected] Witold Kondracki ´ Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8 00-950 Warsaw, Poland Abstract This paper is a further development of the approach to weak cosmic censorship proposed by the authors in Ref. [5]. We state and prove a modified version of that work’s main result under significantly relaxed assumptions on the asymptotic structure of space–time. The result, which imposes strong constraints on the occurrence of naked singularities of the strong curvature type, is in particular applicable to physically realistic cosmological models. Keywords: Space–time singularities; cosmic censorship; black holes; causal structure. PACS Nos.: 04.20.Dw, 04.20.Gz

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Introduction

The cosmic censorship hypothesis (CCH) proposed by Penrose [1, 2] plays a fundamental role in the theory of black holes and is recognized to be one of the most important open problems in classical general relativity. This hypothesis states that, in generic situations, all space–time singularities formed in gravitational collapse, which develops from a regular initial state, should be invisible to sufficiently distant observers — by being hidden behind black-hole event horizons. To be precise, this statement is usually referred to as the weak CCH, while its strong version requires that singularities resulting from generic gravitational collapse must be invisible to all possible observers [2]. The difficulty of the cosmic censorship problem stems from the existence of certain exact solutions of Einstein’s equations which do admit naked (i.e. not hidden) singularities. For instance, such sigularities occur in the Tolman–Bondi solution representing spherically symmetric inhomogeneuos collapse of dust (see, e.g., Ref. [3]). Naked singularities also occur in more general models of dust collapse — namely, in the Szekeres space–times which do not have any Killing vectors [4]. All these models, however, are highly idealized (e.g. they deal only with the collapse of non-rotating matter), so it is possible that their naked singularities will disappear under sufficiently generic perturbations. There is, therefore, the expectation that a suitably refined statement of the CCH may be found, under which all the possible examples of naked singularities will prove to correspond to rather exceptional situations. Certainly, such

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a refinement would involve a precise criterion of the genericity of space–times and singularities to be considered. To help identify such a criterion, it may be useful to establish and study various relations between the curvature strength of singularities and the causal structure in their neighborhood, as such relations underlie the mechanism of cosmic censorship. In Ref. [5] we have defined a new class of singularities, i.e. the so-called generalized strong curvature singularities, that seem to be especially interesting in this context. Our definition is a generalization of earlier definitions of strong curvature singularities proposed in Refs. [6] and [7]; for further motivation for our approach, see the introduction to Ref. [5]. As was found in the setting considered in that paper, the assumption that all singularities are of the generalized strong curvature type cannot be expected to fully rule out naked singularities — however, it does significantly constrain the possibility of their occurrence. Namely, Ref. [5] proposed and motivated a classification of such singularities into three types, based on certain relations between the causal structure of space–time and the focusing properties of solutions to the Raychaudhuri equation along all causal geodesics in the vicinity of a geodesic that reaches the singularity. It was then found, under certain additional assumptions imposed on the asymptotic structure of space–time, that only one of those types of singularities might evade cosmic censorship. The main result of the present paper is that a result analogous to Ref. [5] can be proven to hold (Theorem 1) under significantly relaxed asymptotic assumptions. Namely, Definition 1 provides a natural generalization of the concept of “external region” of black holes to a highly general class of space–times. This is then employed in the statement of Definition 2, which gives a precise formulation of the concept of a naked singularity arising from regular initial data. These assumptions are then found to suffice to prove, as in Ref. [5], that for a wide class of generalized strong curvature singularities (types A and B of the classification proposed therein), uncensored singularities are ruled out. Our notation and fundamental definitions will be the same as those in the monograph of Hawking and Ellis [8].

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External Region and Naked Singularities

In order to define a naked singularity, we first need to give a precise definition of the external region of black holes occurring in a given space–time. The usual approaches tend to rely on rather detailed assumptions concerning the asymptotic structure of space–time. This includes Theorem 1 of Ref. [5] which applies to the so-called weakly asymptotically simple and empty (WASE) space–times (see p. 225 of Ref. [8]). Unfortunately such assumptions, in addition to often being technically awkard, may very well be unduly restrictive and physically unrealistic, at least in a cosmological setting (see Ref. [9] for a discussion of some problems with the standard definitions of black-hole region). Considering the current consensus that the actual universe contains the mysterious dark energy and matter, one would prefer an approach that reduces assumptions about space–time asymptotic to the extent possible. Our approach will be based on the following definition which is inspired by Penrose’s concept of “external region” (p. 236 of Ref. [2]). Definition 1 Let (M, g) be a maximally extended space–time. We denote by E(M ) the set of all points x ∈ M such that there exists at least one future-complete, achronal null geodesic with past endpoint at x. The interior of E(M ) will be called the external region of black holes in (M, g). By the requirement that (M, g) be maximally extended we assure the absence of artificial black-hole regions, i.e. the regions M − intE(M ), which could easily be created even in Minkowski space by simply removing some 3-dimensional, closed, spacelike subsets. The region E(M ) is defined as the locus of points in space–time from which an “infinitely long” light ray may be emitted; in other words: that part of space–time which is visible to some “arbitrarily distant” observers. These light rays are required to be achronal, in order to rule out the rather pathological situation where all future-directed null geodesics are complete without leaving some spatially bounded region (an example of such a geodesic is that orbiting at radius 3m in the Schwarzschild solution). Moreover, by the same requirement we exclude the possible presence of causality violations outside the black-hole region. Clearly, this leaves outside the scope of our considerations the class of naked 2

singularities that are known to be associated with causality violations [10]. However, many familiar examples of naked singularities, such as those that occur in the mentioned above Tolman–Bondi and Szekeres solutions, are not associated with causal pathologies. Let us compare our definition of the external region with that corresponding to the classical black hole in the maximally extended Schwarzschild solution. As one can easily see by direct inspection of the Penrose diagram of this solution (see, e.g., p. 154 of Ref. [8]), for any point p with the radial coordinate r > 2m, the achronal boundary, J˙+ (p, M ), of the causal future of p intersects the conformal null infinity J + . Fix a point q ∈ J˙+ (p, M ) ∩ J + , and let µ be a null geodesic generator of J˙+ (p, M ) outgoing into the past from q. As q ∈ J + , µ must be future-complete. Moreover µ, when maximally extended into the past, must intersect p, because the Schwarzschild space–time is globally hyperbolic, and hence causally simple. We thus see that p is a past endpoint of the future-complete, achronal, null geodesic segment µ ∩ J˙+ (p). This means, according to our definition, that p ∈ E(M ). As p is an arbitrary point with r strictly greater than 2m, the set of all such points is open, and so is contained in the interior of E(M ). Since all future-directed null geodesics lying behind the event horizon at r = 2m will reach the singularity at r = 0, none of them can be future-complete. Therefore no point with r < 2m, i.e. no point inside the Schwarzschild black hole, can belong to E(M ). We thus see that our definition of the external region coincides exactly with the most standard model of black hole. Although we will not do it here, we are able to show that the region intE(M ) coincides exactly with the outer region of the Kerr black hole as well. Moreover, we are able to prove a singularity theorem for our black-hole region, i.e. we can show that if a space–time (M, g) admits a non-empty region M − intE(M ), then there must also exist at least one incomplete causal geodesic, provided that (M, g) does not contain closed timelike curves and satisfies the generic and timelike convergence conditions. As is well known, according to the celebrated Hawking–Penrose singularity theorem [11, 8], the same holds true in the case of any space–time containing a black hole with trapped surfaces. Observe also that the definition of the region intE(M ) involves no assumptions on the asymptotic structure of space–time — other than the minimal assumption that future-complete and achronal null geodesics do exist. Thus intE(M ) would appear to provide a natural and general definition of the external region of black holes in open cosmological models free of causal pathologies. We now proceed to formulate a definition of a naked singularity that arises from some regular initial data given on a partial Cauchy surface S. Clearly, the occurrence of a naked singularity to the future of S must always lead to the formation of a future Cauchy horizon H + (S). As usual, a space–time singularity to the future of S is taken to be implied by the existence of a future-incomplete (null, in this case) geodesic in the causal future of S. By nakedness of the singularity we mean that such an incomplete geodesic is “visible from infinity” — i.e., remains within the external region intE(M ). Since the singularity is required to have evolved from some initial data on S, we shall assume that the intersection of the causal past of the future-incomplete null geodesic with the causal future of S will lie within the future Cauchy development, D+ (S), of the surface S. Definition 2 We say that a naked singularity arises in the future from regular initial data on a partial Cauchy surface S when the future Cauchy horizon H + (S) is not empty, and for some point r ∈ H + (S) there exists a future-endless, future-incomplete null geodesic λ ⊂ J + (S) ∩ intE(M ) ∩ J − (r) such that (i) J − (λ) ∩ J + (S) ⊂ D+ (S); (ii) J − (r) ∩ S is compact. The condition of compactness of J − (r) ∩ S assures the absence of singularities on S and rules out trivial Cauchy horizons due to a poor choice of the surface S — namely, such a choice where some tangents to S tend to null vectors within a region that is of interest here, a simple example being a spacelike hyperboloid in Minkowski space. It should be stressed here that Definition 2 is not particulary restrictive and agrees with other definitions of naked singularities considered in the context of the weak CCH. For example, all the requirements of Definition 2 are satisfied in the WASE class of nakedly singular space–times, for which we have proven our earlier censorship theorem [5]. To see this, let us first recall that if a WASE space– time (M, g) admits a naked singularity to the future of a partial Cauchy surface S ⊂ M , then it cannot 3

be future asymptotically predictable from S (see p. 310 of Ref. [8]). One can then show, under certain reasonable conditions of regularity of S,1 that there must exist a past-endless, past-incomplete null geodesic generator η of the Cauchy horizon H + (S), which has a future endpoint p ∈ J + , and which admits a point r ∈ η ∩ M such that the set J − (r) ∩ S is compact; see Lemma 1 of Ref. [5]. Moreover, from condition (b) of that lemma it is evident that the set I − (p, M ) ∩ D+ (S) must be contained in our external region intE(M ). We shall now show that there must also exist a future-endless, future-incomplete null geodesic λ ⊂ J + (S) ) ∩ J − (r), as required in Definition 2. To this end, let us first define the past set T∩ intE(M − X ≡ q∈η I (q). As r ∈ η, we must have X ⊂ J − (r). Thus, as J − (r) ∩ S is compact, X ∩ S is compact as well. This implies that X ∩ J + (S) is a non-empty subset of D+ (S). In addition, by the definition of X, X ∩ J + (S) must be a proper subset of D+ (S) ∩ I − (p, M ), because p is a future endpoint of η and the closure of I − (p, M ) ∩ S cannot be compact, as p ∈ J + . Accordingly, X ∩ J + (S) must be a proper subset of the external region intE(M ). Since X ∩ S is compact and S has an asymptotically simple past, by the time-reverse version of Lemma 6.9.3 of Ref. [8], the past null infinity J − must intersect J˙− (X ∩ S, M ), where the overdot indicates the boundary in M . We clearly have J − (X ∩ S) = X ∩ J − (S), so J − must intersect the achronal boundary X˙ as well. Let λ′ be a null geodesic generator of X˙ maximally extended into the future from some point in X˙ ∩ J − . From the definition of X it follows that λ′ must be future-endless, as the geodesic η is past-endless. Since λ′ has a past endpoint on J − , it must be pastcomplete; and so it cannot be complete in the future. Otherwise, as the generic and null convergence conditions are assumed to be satisfied, λ′ would have to contain, by Proposition 4.4.5 of Ref. [8], a pair ˙ of conjugate points, but this would contradict, by Proposition 4.5.12 of Ref. [8], the achronality of X. The asymptotically simple past of S implies that J − (S) = D− (S), hence λ′ must intersect S and enter ˙ we have Y ⊂ X ∩ J + (S). Thus J + (S). Let us now denote by Y the set I − (λ′ ) ∩ J + (S). As λ′ ⊂ X, + Y must be a proper subset of D (S) ∩ intE(M ), which follows from what has already been established for X. There must therefore exist a null geodesic generator, λ, of the boundary Y˙ contained entirely in D+ (S) ∩ intE(M ). By the definition of Y , λ must be future-endless, since the geodesic λ′ is futureendless. Moreover, as λ′ is future-incomplete, by condition (iii) of Theorem 1 of Ref. [5], λ must be future-incomplete as well. From the above proof it is also clear that the geodesic λ does satisfy all the other requirements of our Definition 2. In much the same way, one can show that Definition 2 is also satisfied in other classes of WASE space– times which are nakedly singular in the sense of weak cosmic censorship — e.g. in those investigated by Kr´olak [7]. In particular, it is satisfied in the familiar Tolman–Bondi solutions with naked singularities [3], because these solutions are WASE space–times in which future asymptotic predictability breaks down and they fulfill all the additional conditions used in the above proof. From the above proof it may also be observed that the breakdown of future asymptotic predictability is accompanied by the existence of the two types of incomplete null geodesics in J + (S) ∩ J − (J + , M ) — i.e., besides the future-incomplete geodesic λ there also exists the past-incomplete geodesic η ⊂ H + (S) which has a future endpoint on J + . This may suggest that our Definition 2 could equally well be formulated in a different way, such that some past-incomplete null geodesic η would be required to exist in J + (S) ∩ intE(M ) instead of the future-incomplete null geodesic λ. Recall, however, that future asymptotic predictability is only a necessary condition on WASE space–times assuring that such spaces are free of naked singularities. Namely, it may happen that some WASE space is future asymptotically predictable from a surface S but fails to be strongly future asymptotically predictabe from S as defined by Hawking and Ellis (p. 313 of Ref. [8]).2 Then there are no singularities to the future of S which are naked, i.e. which are visible from J + ; there must, however, exist some singularities to the future of S which lie on the event horizon J˙− (J + , M ), i.e. some generators of this horizon must be incomplete.3 It is not difficult to see that in this case there are no past-incomplete null geodesics in the region J + (S) ∩ J − (J + , M ), whereas there should still exist in this region some future-incomplete null geodesics that 1 These conditions require, in essence, that S should have an asymptotically simple past (p. 316 of Ref. [8]) and that (M, g) should be at least partially future asymptotically predictable from S as defined by Tipler [10]. Such requirements are expected to be satisfied in any model of gravitational collapse that develops from an initially non-singular state. 2 In this case one can expect that the smallest perturbation could lead to the breakdown of future asymptotic predictability, and so one can require that such cases should not occur if cosmic censorship holds in stable, physically realistic space–times. 3 The existence of such generators results from the standard argument with conjugate points.

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terminate at the singularities occurring on the event horizon. In the same way as above, these geodesics can be shown to obey all the other requirements of our Definition 2. We can thus say, in accordance with this definition, that any WASE space–time, which fails to be strongly future asymptotically predictable from a surface S, does possess a naked singularity to the future of S, which agrees with the widely accepted view (see, e.g., p. 301 of Ref. [12]). Clearly, this would not be the case if Definition 2 were formulated to require the existence of some past-incomplete null geodesic in J + (S) ∩ intE(M ) instead of the future-incomplete one.

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Censorship Theorem

We now turn to the statement of the theorem that forms the main result of the present paper. To begin, let us recall that Ref. [5] introduced the concept of generalized strong curvature singularity and proposed a classification of the possible behaviors of geodesic congruences in the vicinity of such a singularity (the types A, B and C of Definition 3 in Ref. [5]). One of the results of Ref. [5] was Theorem 1, stating that — under certain assumptions, including some rather strong assumptions on the asymptotic structure of space-time — the only generalized strong curvature singularities that might evade cosmic censorship are those characterized by type C incomplete null geodesics. The theorem we state and prove below provides a significant generalization of that result, by doing without any detailed assumptions on the asymptotic structure of space–time. Theorem 1 Let (M, g) be a space–time admitting a partial Cauchy surface S and satisfying the null convergence condition, i.e. Rab K a K b ≥ 0 for every null vector K a of (M, g). Assume also that the following conditions hold: (i) If there exists a future-incomplete null geodesic λ ⊂ D+ (S), then every future-endless null geodesic α ⊂ J − (λ) ∩ J + (S) is future-incomplete as well (no “internal infinity” in the past of a singularity); (ii) Every future-incomplete null geodesic terminates in the future at a generalized strong curvature singularity and is of type A or B. Then there exist no naked singularities arising in the future from regular initial data on S. It should be noted that the assumption of absence of internal infinity, while it holds for all explicitly known models of gravitational collapse, cannot at present be guaranteed to hold in the generic case. Proof The proof proceeds by contradiction. That is, we assume the existence of a naked singularity, according to our definition, and demonstrate that this implies that the assumptions of our theorem cannot hold. First, we will show that, under the assumption of a naked singularity, there must exist a futureincomplete null geodesic terminating in a generalized strong curvature singularity but that is not of type A. According to Definition 2, there exist a point r ∈ H + (S) and a future-endless, future-incomplete null geodesic λ ⊂ J + (S) ∩ intE(M ) ∩ J − (r), such that J − (λ) ∩ J + (S) ⊂ D+ (S) and the set J − (r) ∩ S is compact. Fix such a point r, and let Λ(r) denote the corresponding family of all null geodesics λ with the above-mentioned properties. Let P(r) be the family of all past sets P ≡ I − (λ), where λ ∈ Λ(r); and let Pˆ be a maximal chain determined in P(r) by the relation of inclusion. Denote now by P0 the ˆ Clearly this set is a minimal element of the chain P. ˆ Note intersection of all sets P belonging to P. ˆ also that P0 is non-empty. This is because from the construction of P it follows that each member of Pˆ intersects the surface S and the closure of this intersection must be compact, as it is a closed subset of the compact set J − (r) ∩ S. Thus there exists a corresponding future-endless, future-incomplete null geodesic λ0 ∈ Λ(r) such that P0 = I − (λ0 ). Let {qi } be a sequence of points on λ0 with no accumulation point on λ0 (such a sequence can always be found, as λ0 has no future endpoint). Since λ0 ⊂ J + (S)∩intE(M ), for each of the points qi there must exist a future-complete, achronal null geodesic αi running from qi to the future. Fix a sequence {αi } of such geodesics and extend each of them maximally into the past. As we have J − (λ0 ) ∩ J + (S) ⊂ D+ (S), 5

each of the (past-extended) αi must intersect the partial Cauchy surface S. The sequence {ai } of intersection points ai ≡ αi ∩ S must have some accumulation point a0 ∈ S, since all the points ai belong to I − (λ0 ) ∩ S which is a compact set. It then follows, by Lemma 6.2.1 of Ref. [8], that there exists a limit curve, α0 , of the sequence {αi } with a past endpoint at a0 . Since all the αi are future-endless null geodesics, α0 must be a future-endless null geodesic as well. Note that α0 must be contained in J − (λ0 ) ∩ J + (S); otherwise α0 would have to intersect the geodesic λ0 at some point that would be an accumulation point of the sequence {qi }, which is impossible as {qi } has no accumulation points on λ0 . Thus, by virtue of condition (i) of our Theorem 1, α0 must be future-incomplete. We must also have ˆ I − (α0 ) ⊂ I − (λ0 ) and I − (λ0 ) coincides with I − (α0 ) = I − (λ0 ), since I − (α0 ) belongs to the chain P, ˆ P0 which is a minimal element of P. As α0 is future-incomplete, it must terminate in the future at a generalized strong curvature singularity. However, α0 cannot possibly be of type A: being a limit curve of the sequence {αi } whose all elements are future-complete geodesics that leave I − (α0 ), it fails to obey the defining property of type-A geodesics (see Definition 3 in Ref. [5]). As the next step, we shall now show that the same null geodesic α0 cannot be of type B, either. Suppose that α0 were of type B. Then, according to the definition of type B geodesics given in Ref. [5], there would exist a geodesic α ˜ belonging to the sequence {αi } and a point q˜ ∈ α ˜ − I − (α0 ), such that the expansion θ of some congruence of future-directed null geodesics outgoing from q˜ and containing α ˜ would become negative somewhere on α ˜ ∩ J + (˜ q ). As α ˜ is future-complete and the null convergence condition holds, by Proposition 4.4.4 of Ref. [8] there would then exist some point x ∈ α ˜ ∩ J + (˜ q ) conjugate to q˜ along α ˜ . Consequently, by Proposition 4.5.12 of Ref. [8], there would have to exist a timelike curve from q˜ to some point y ∈ α ˜ ∩ J + (x). But this is impossible, because from the construction of the sequence {αi } it follows that the whole geodesic segment α ˜ ∩ J + (˜ q ) must be contained in the achronal part of α ˜. Hence α0 fails to be of type B, as well as of type A. Therefore, having shown that the assumptions of our Theorem 1 fail to hold when its claim is negated, the theorem is proved.

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Concluding Remarks

While the result of the present work certainly falls short of resolving the problem of weak cosmic censorship, we have found significant constraints on the types of generalized strong curvature singularities which might remain naked. Moreover, we found it possible to extend the result of Ref. [5] to a setting where no detailed assumptions concerning the asymptotic structure of space–time are needed. A key element of our approach is the definition of the external region intE(M ). It is clear that an approach based on this definition is problematic in the presence of causality violations in the outer region of black-hole horizons. We hope to address this issue in the future. As argued by Penrose [2], if cosmic censorship is indeed a physical principle, any arguments supporting the validity of the weak version of cosmic censorship should also support the validity of the strong version — as physics is governed by local laws. It would be therefore interesting to see whether it is possible to prove that generalized strong curvature singularities of type A and B are subject to strong censorship as well. This remains an open problem for future work.

Acknowledgements We are greatly indebted to an anonymous referee for several helpful comments. Two of us (RJB and WK) wish to thank P. S. Joshi for valuable discussions and hospitality at the Tata Institute of Fundamental Research, where part of this work was carried out. This work was supported by the Polish Committee for Scientific Research (KBN) under Grant No. 2 P03B 073 24.

References [1] R. Penrose, Riv. Nuovo Cimento 1, 252 (1969). [2] R. Penrose, Singularities of space–time, in Theoretical Principles in Astrophysics and Relativity, eds. N. R. Lebovitz et al. (Univ. of Chicago Press, 1978), pp. 217-243. 6

[3] P. S. Joshi, Global Aspects in Gravitation and Cosmology (Oxford Univ. Press, 1997). [4] A. Kr´olak, A. Czyrka, J. Gaber and W. Rudnicki, On the nature and strength of singularities in the Szekeres quasispherical collapse, in Proceedings of the Cornelius Lanczos International Centenary Conference, eds. J. D. Brown et al. (SIAM, 1994), pp. 518-520. [5] W. Rudnicki, R. J. Budzy´ nski and W. Kondracki, Mod. Phys. Lett. A17, 387 (2002); gr-qc/0203063. [6] F. J. Tipler, Phys. Lett. A64, 8 (1977). [7] A. Kr´olak, Class. Quantum Grav. 3, 267 (1986). [8] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space–Time (Cambridge Univ. Press, 1973). [9] P. T. Chru´sciel, Black holes, in Lect. Notes Phys., eds. J. Frauendiener and H. Friedrich, 604, pp. 61-102 (2002); gr-qc/0201053. [10] F. J. Tipler, Phys. Rev. Lett. 37, 879 (1976). [11] S. W. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A314, 529 (1970). [12] R. M. Wald, General Relativity (Univ. of Chicago Press, 1984).

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