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I Am Aware of My Ignorance , ) ~~(7 (7 ) / Ev otōa OT1 ouoev oloa -► i EWKpaTrjf I am aware of my ignorance" Sokrates " ITavtuv • XpI11..laTWV petpwv avOpWTrof eaT1V IIpWTayopaf "Man is a measure of all things" Protagoras 0 SINGULARITIES IN THEORIES OF GRAVITATION by THEODOSIOS CHRISTODOULAKIS A thesis submitted for the degree of Doctor of Philosophy (in Mathematical Physics) in the University of London Mathematics Department Imperial College of Science and Technology London S.W.7. DEDICATED TO MY PARENTS • ACKNOWLEDGEMENTS I wish to express my deep gratitude to my supervisor, Dr. Patrick Dolan, whose constant guidance and encouragement throughout the course of this work was invaluable. I also thank Mrs. T. Wright for typing this thesis. • ABSTRACT This work is divided into four Chapters. Each chapter is more or less self-contained. At the end a conclusion is given, which may as well serve as an introduction for the reader whose interest lies in the physical interpretation of the results contained in the work . A chapter consists of two parts (marked as A and B) each of which is divided into various sections (marked as a,b,c...). Sections (a) of each part of each chapter serve as a specialized introduction to the subject that follows. The references are quoted inside parentheses and in addition an alphabetical list of the references quoted, as well as relevant references, is given at the end of each chapter. Part A of Chapter I is concerned with the question of limits of space-times, and in part B the global properties of the space of all Lorentz metrics are considered. In a particular topology it then proven that the space of the solutions of Einstein's Field Equations contains no isolated points. Chapter II is concerned with causality: In part A, a differ- ential structure is induced in a full Causal space, and in part B a boundary for such a space is constructed, using only the causal relation. Chapter III deals with the b-boundary techniques: Part A contains the standard definitions while in B an attempt is made to reformulate b-boundary techniques in a way which makes use of structures of the space-time manifold only. In Chapter IV the problem of combining General Relativity with Quantum theory is considered. Part A is a general discussion on the problem, while part B contains two applications of the ADM formalism G. to the ROBINSON-BERTOTTI Universe and to the GODEL Universe. TABLE OF CONTENTS Page I. LIMITS AND TOPOLOGIES OF SPACE-TIMES. 1 A. Limits of space-times. 1 B. Topologies on space-times. 15 II. SPACE-TIME STRUCTURE. 35 A. Causal and differential structure 35 SUPPLEMENT 57 B. The Causal Boundary of Space-Time. 60 III. SPACE-TIME b-BOUNDARIES. 76 A. Definition and properties. 76 B. Further properties and examples. 95 IV. QUANTUM GRAVITY. 113 A. General discussion and examples. 113 B. Quantum Cosmology. 132 V. CONCLUSIONS AND SPECULATIONS. 160 • CHAPTER I LIMITS AND TOPOLOGIES OF SPACETIMES A. Limits of Spacetimes (a) A lot of statements are in common use, concerning the limit of some family of solutions of Einstein's field equations as some free parameter approaches a certain value. In all such statements there is a serious ambiguity, because they usually refer to a particular coordinate system; by changing it one obtains an entirely different limit space. On the other hand, the concept of a limit applied to a spacetime is a useful one. It seems thus worthwhile to try to formulate some unambiguous definition of this notion. By a space-time we understand a (connected, Hausdorff) 4-dimensional 09 manifold with a (C ) metric gab of signature (+,-,-,-). Consider a one-parameter family of such space-times (the results can very easily be generalized to many parameter families and perhaps to families depending on arbitrary functions). That is to say, for each value of a parameter 1 (>0) we have a space-time M1 and a metric gab (1) on M1. We want to find the limits as 1--?'0. It would be simpler to consider the gab(1) as a one parameter family of metrics on a fixed manifold M; this amounts in specifying when two points and p ►~M1 1 (1)dl') p1pM1 1 are to be considered as representing the "same point" of M. But, providing this information is not appropriate, since we are interested in finding all limits and studying their properties, because it always involves singling out a particular limit. An example will clarify this point: Consider the family of metrics 2 ) ds2 = (I - 2 )dt2 -(I- -I dr2 - r2(de2 + sin2 edd2) (1 ) 13r 13r depending on the single parameter 1(=m I"3). In the form (I) clearly 2 there is no limit as 1 0. If, however, we apply the coordinate transformation r = ir, t = 1-It, p = 1-Ie then (I) becomes ds2 = (1 2 - ?)dt2 -(1 2 - 2?) -Idr - r2(dp2 + 1-2sin2(1p)dt2) r r The limit as 1 0 now exists and gives the metric: ds = - -2„-dt2 + r- dr2 r2(dp2 + p2ds2) r 2 This is a nonflat solution of Einstein's equations discovered originally by Kasner and obtained by Robinson as a limit of the Schwarzschild solution. On the other hand, if we apply to the metric (I) the co- ordinate transformation x = r+1-4, p=1-4e, we obtain flat space in the limit 1-->0. Thus one cannot simply speak of "the limit of the Schwarzschild solution as 1 >O" for the limit one obtains depends on the choice of coordinates. The essential difference between the various limits above consists in different identifications of the M1. We seek to find a way to express the idea that (M1 , gab(1)) depend smoothly on 1 (This is essential in order to define limits) without at the same time prejudicing the particular limit we are to obtain. Let us assume that the different M1 may be put together to make a smooth (Hausdorff) 5-dimensional manifold M. Each M1 is to be a 4-dimensional submanifold of M. The parameter 1 now appears as a scalar field on M and gab(1) on M1 is uniquely extended to a tensor AB field gAB on M, which is assumed to be smooth. The signature of g is (0,+,-,-,-,); in fact, the singular direction is precisely the gradient of 1, i.e. we have gA6vB1=0. M contains all information ab about the original collection (M1, g (1)) but does not define a preferred correspondence between different Ml's (such a correspondence could be defined by giving a vector field on M, nowhere vanishing and nowhere tangent to the M1: and plICMI t are in correspondence p1EM1 if a trajectory of this vector field joins p1 and pi.. But no such vector field is in the structure of M). Now the problem of finding limits of the family (M1, gab( 1 )) reduces to that of placing a suitable boundary on M. We define as a limit space of M a 5-manifold M' with boundary GM', equipped with a tensor field g'AB, a scalar field 1', and a smooth, one-to-one map Y of M onto the interior of M' such that the following three conditions are satisfied: 1. Y is an isometry, i.e. Y takes gAB into g'AB and 1 into 1'. 2. GM' is the region given by 1' = 0. Furthermore it is required that GM' be connected, Hausdorff and non-empty. 3. g'AB has signature (0,+,-,-,-,) on 8M'. The first condition ensures that M' really represents M with a boundary attached; the second ensures that the boundary represents a limit as 1-'30; the third condition ensures that the metric on the boundary is nonsingular. At this point we must mention a complication with regard to Hausdorffness. Although the M1's and M are Hausdorff we cannot impose this to the limit spaces M' if we are to be able to deal with pathological situations;. and indeed pathological situations are the main reason for introducing boundaries and taking limits. The above definition, although corresponding to our intuitive idea of a limit of a collection of spacetimes, is not very useful for actually writing down limits. For this reason, we will try to show how certain structures on M may be used to characterize the limit spaces. By a family of frames in M we mean an orthonormal tetrad w(1) of vectors tangent to M1 and attached to a single point p1&M1, for each 1> 0, such that the w(1) vary smoothly along the smooth curve in M defined by the points pl. If M' is any limit space of M, we may ask whether or noithe given family of frames assumes a limit, i.e. approaches a frame w(0) at some point pof9M' as 1--3 O. In general, of course, the answer will be no. The reverse, however, is always true: given a limit space M', we can find some family of frames which does have a limit in M'. — 4 — Let M' be a limit space of M, and let w(1) be a family of frames assuming a limit as 1-- 0. Let us represent points in MI in a neighbourhood of p1 in terms of the system of normal coordinates based on w(1). In terms of these coordinates, the components of the metric tensor in M1 approach a limit as 1 --. 0, and the limiting components are precisely the components of gab(0) in 9M` in a neighbourhood of po. Thus, the family of frames • w(1) uniquely determines M`, at least in a sufficiently small neighbourhood of po.
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