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REVIEW Singularity Theorems and Their Consequences General Relativity and Gravitation, Vol. 29,30, No.No. 5,5, 1997 1998 REVIEW Singularity Theorems and Their Consequences Jos Âe M. M. Senov illa1 Received September 24, 1997. Rev. version February 26, 1998 A detailed study of the singularity theorems is presented. I discuss the plausibility and reasonability of their hypotheses, the applicability and implications of the theorems, as well as the theorems themselves. The consequences usually extracted from them, some of them without the necessary rigour, are widely and carefully analysed with many clarifying examples and alternative views. 1. INTRODUCTION It is now more than 30 years since the appearance of the Penrose singularity theorem [162], and more than 40 years since the fundamental Raychaud- huri equation and results were published [170]. Much work based on these important ideas was later developed and has become one of the main parts of classical general relativity: singularity theorems and related problems. It has even crossed the standard limits of general relativity and become a ® eld of interest both for other branches of physics or mathematics and for the general public. This is due to its possible consequences concerning gravitational collapse, astrophysics and cosmology, which are very popular nowadays. The singularity theorems can be termed as one of the greatest achievements within relativity, but they contain some mathematical and 1 Departament de FÂõ sica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain. E-mail: seno@hermes.Œn.ub.es 7 0 1 0001-7701/ 97/ 0500-0701$12.50/ 0 1997 P lenum P ublishing Corporation 7 0 2 S e n ov illa physical subtleties which cause them to be, on some occasions, wrongly in- terpreted or not well-understood. The main conclusions of the singularity theorems are usually interpreted as providing evidence of the (classical) singular beginning of the Universe and of the singular ® nal fate of some stars. To what extent this is truly so will be analysed here. In fact, the singularity theorems do not exactly lead to such conclu- sions, but given that singularity theory in general relativity is a di cult matter, there has always been a wish to simplify the `exact wording’ of the theorems Ð while keeping the essentials by pinpointing the important clues Ð such that their assumptions and conclusions could be easily un- derstood and applied to some simple situations. Unfortunat ely, all these simplifying attempts have not succeeded, even though sometimes they were devised very cleverly. I have the feeling that any other simplifying attempt yet to come is also doomed to failure, given that many explicit examples show that exact rigour Ð both in the assumptions and the conclusions Ð is absolut ely necessary. Many such examples will be presented in this paper. The lesson is that one has to rely on the true actual theorems, and this is sometimes unpleasant as they need a rather good knowledge of the standard global and causality techniques in general relativity. One of the purposes of this paper is to collect all that is needed for the singu- larity theorems in a manner as easy and simple as possible. Everything will be treated within classical relativity; semi-classical theories, quan- tum phenomena or quant um gravity will not be taken into account, as they essentially change the physics behind the singularity theorems and thereby drastically modify their conclusions. Alternative classical gravi- tation theories will not be considered either, although many results are easily recovered there (see the complete review, Ref. 84). The main questions to be treated here are: what do the singularity theorems say, and what do they not say? And, what are their actual consequences? To that end, on the one hand the paper is completely self- contained, and on the other hand it has many explicit examples illustrating the subtleties and problems behind most concepts and results. Concerning the ® rst point, only standard basic general relativity is needed to be able to read the whole paper. The global causality theory, the concepts of maximal curves and its existence, the focusing of geodesics, and all that is basic for the singularity theorems, are fully developed step-by-step in Section 2. Of course, there are excellent references for these mathematical developments, such as [107,165]; besides, there exist very good and complete reviews, for instance [220]. Nevertheless, and even though Section 2 is essentially standard as there are no new results, I have tried to keep the level as simple as possible compatible with absolute rigour, and many of the proofs have Singularity Theorems and Their Consequences 7 0 3 been modi® ed Ð and sometimes given in further detail Ð with respect to the standard references. People interested in these global techniques who nonetheless are not experts on them may ® nd this section of some help, for most of the developm ents are carefully proved while keeping always the maximum simplicity. Moreover, the basic traditional structure has been partially reorganized so that every single result appears at its natural place, and some of the fundamental steps have been split into simpler smaller ones, making the way to the singularity theorems more practicable. However, readers familiar with the global techniques may ® nd it more useful to skip Section 2 and go directly to Section 3. The rest of the sections, with the exception of Section 5 which is de- voted to the full proof of the theorems themselves, contain many original explicit and simple examples, together with a deep analysis of the concepts involved, showing how some intuitive or naive conclusions concerning sin- gularity theorems may be false. The emphasis is placed on the alternative views with regard to what can be considered the standard opinion, and specially on the doors left open by the theorems in order to construct re- alistic and reasonable singularity -free spacetimes. Section 3 is devoted to the study, de® nition, classi® cation and properties of singularities. Several explicit examples are shown, and the problem of choosing extensions of given incomplete spacetimes, and how to construct them, will be exam- ined. The concept of a singular extension is introduced and shown to be both necessary in some situations and a non-orthodox alternative possi- bility in some other cases. A de® nition of big-bang singularity, together with the unusual propert ies they may have, is also provided. In Section 4 the fundamental concepts of trapped sets and closed trapped surfaces are introduced. This is mainly standard but some explicit examples are given. The consequences of their existence relevant to the singularity theorems are explicit ly proved here. Again, Sections 4 and 5 may be skipped by connoisseurs. The last two sections are completely original and may be controver- sial. Section 6 is devoted to a thorough analysis of the assumptions and conclusions of the theorems. A skeleton pattern-theorem for the singu- larity theorems is shown, and the intuitive ideas behind their proofs are explicit ly described. A full detailed outspoken criticism of the singularity theorems is presented, considering each of its parts on its own, and ending with a summary of what I think their main unsolved weaknesses are. Most of this criticism is supported by many explicit and illustrative examples given in the last Section 7, with a total of nine diŒerent subsections. All the examples are simple and clear, and each of them sheds light on a particular part, or on a particular widespread Ð but incorrect Ð belief concerning 7 0 4 S e n ov illa the theorems; in other cases, the examples show which particular things are taken for granted, sometimes incorrect ly. A tentative de® nition of a classical cosmological model is put forward, and many non-singular space- times are explicit ly given. Sometimes, the existence of such examples has been considered surprising, but that is precisely the purpose: they show that many doors for constructing regular realistic spacetimes are still open. They also allow us to control the degrees of necessity of the theorem as- sumptions, proving manifest ly that they cannot be relaxed and that they may be more demanding than is usually thought . Before entering intothe main subject, let us brie¯ y review the notation to be used in this paper. The closure, interior and boundary of a set f are denoted by f, int f and ¶ f, respectively. The proper and normal inclusions are denoted by Ì and Í , as usual. Einstein’ s equations are written as 1 Rm u ± 2 gm u R = Tm u , where Tm u is the energy-momentum tensor, R m u is the Ricci tensor and R is the scalar curvature. The geometrized units 8pG = c = 1 are used throughout the paper. An equality by de® nition is denoted by º , while the end (or the absence) of a proof is signaled by . Greek indices run from 0 to 3, small latin indices from 1 to 3 and capital latin indices take the a values 2 and 3. The sign of the Riemann tensor Rb lm is de® ned by eq. (26). Boldface letters and arrowed symbols indicate one-forms and vectors, re- spectively, and the exterior diŒerential is denoted by d. Square or round brackets enclosing some indices denote the usual (anti-)symmetrization. The metric tensor ® eld is denoted by g, and the line-element in any local coordinat e chart xm is the quantity f g 2 m u ds = gm u dx dx . 2. BASIC RESULTS IN GENERAL RELATIVITY AND CAUSALITY The basic arena in General Relativity is the spacetim e .A C k space- time (V4 , g) is a paracompact, connected, oriented, HausdorŒ, 4-dimen- k sional diŒerentiable manifold V4 endowed with a C (k ³ 2 ± ) Lorentzian metric g (signature ± , + , + , + ) [12,65,107,187,2 39].
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