(2) ii.

(3) I declare that this thesis is, where not explicitly stated otherwise, the result of joint research, carried out by myself along with Prof. C. J. S. Clarke (xx1.3{8) and S. M. Scott (Appendix D; the original abstract boundary construction, a setting we use, is due to her and P. Szekeres). The collaboration with Clarke was in the form of (roughly) weekly discussions in the 1995-96 academic yesr in Southampton, where my ideas and proofs were checked and suggestions made. Collaboration with Scott was of a broadly similar form and took place from 1993 to 1995 at The Australian National University; see the Acknowledgements (p. v) for further details of relatively minor contributions. Signed: .............................................. C. J. Fama. iii.

(4) iv.

(5) Acknowledgments First of all I thank and praise God in all three Persons for His support and encouragement throughout this Ph.D., which has been (literally!) too long by half! However, not wishing to o end my non-Christian readers|after all I believe you have a God-given right to make your own faith choice|I shall move on to more traditional acknowledgements. My family, particularly my immediate family but also my more far- ung aunts, uncles and cousins in Europe, have been a real tower of strength throughout the period, especially over the last two years, a year of which I spent in Southampton. In particular, my mother Dr. Mary E. Duncan Fama, an applied mahematician herself, has been tirelessly supportive of a lot more than just my mathematical career. All my friends, personal and professional (or both!), in both Canberra and Southampton, have been both|well, friendly |bless you all!| and astoundingly helpful in personal and professional matters. Just a few names, in alphabetical order as I don't want to implicitly prioritise: Rosie Ackerley, Sergey Bakin, John Bryant, Yvonne Burns, Ruth Clay, Hannah Davies, Mike Dever, Melissa Dobbie, John and Francis Du and their family, Carl Fitchett, Marcus Kriele, Brenton Lemesurier, Giselle Lim, Dave Matthews, Peter and Trish McKay and their family, Peter and Bev Moyle, Rona Nadile, Masoud Nikoukar, Gabriel Rivara, Trevor and Judith Owen, Denise Silva, Charlie and Bev Stockley and their family, Pauline Stonebridge, Peter and Judy Thompson, Colin and Lorna Trend, and Nolan Virgo. And very warm thanks to all those friends whose names aren't represented here, either through oversight or space constraints. I owe an enormous debt|by no means grudgingly given, mind!| of gratitude to my supervisory panel here at the A.N.U., currently Profs. Derek W. Robinson and Neil S. Trudinger, and ex ocio Dr. Brenton J. Lemesurier (who is now in Charleston, U.S.A.). Although he is not ocially on this panel, it is appropriate to add Prof. Christopher J. S. Clarke, whose superb supervision brought my Ph.D. out of the doldrums and

(6) red long-missing enthusiasm in me. Many thanks also to the respective Dean of the Graduate School and Dean of Students at the A.N.U., Profs. Ray Spear and Selwyn Cornish, who, along with my good friend Dr. Giselle Lim and my current supervisory panel, were instrumental in sending me overseas. Finally, I have been

(7) lled with both appreciation and admiration for the generous and useful recent contributions of Prof. Robert A. Bartnik, Dr. John E. Hutchinson, v.

(8) Dr. Jingyu Shi,and particularly those of Dr. M. Kriele. I thank Susan M. Scott, my original supervisor, for inspiring an interest in relativity in me, and for introducing me to her work, with Peter Szekeres, on the abstract boundary. Finally, I dedicate this thesis to my grandfather Gordon Teal, who passed away mere weeks before its completion, and to my grandfather James Duncan, who left behind him several years ago a truly wonderful pair of generations, on whom I shower both thanks and fondness (see also above)!. vi.

(9) Contents Acknowledgements. v. Introduction. 1. I The abstract boundary, regular abstract boundaries and their characterisation in the smooth case 5 1 First de

(10) nitions 1.1 1.2 1.3 1.4. De

(11) ning the abstract boundary . . . . . . . . . . A broad classi

(12) cation of abstract boundary points Regular abstract boundaries . . . . . . . . . . . . Various regularity classes . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 7. 7 8 10 13. 2 Some properties of regular Lipschitz envelopments in the smooth case 15 2.1 Lipschitz boundaries . . . . . . . . . . . . . . . . . . . . . . . 2.2 The bundle metric|classical (C 1) case . . . . . . . . . . . . . 2.2.1 Using Lipschitz boundaries to imbed the Cauchy comc.................... pletion LM in LM 2.3 Application to regular abstract boundaries . . . . . . . . . . .. . 15 . 18 . 19 . 20. II Shock-transverse generalised parallel transport, and characterisation of the Lipschitz regular abstract boundary in the shock case: \toy" model for gravitational wave space-times 25 3 Generalised parallel transport along shock-transverse curves 27 3.1 Existence and uniqueness of generalised parallel transports along certain curves . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 From R n to manifolds . . . . . . . . . . . . . . . . . . . 3.2 The description of gravitational waves by metrics of shock type 3.3 Generalised parallel transport of frames along shock-transverse curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii. 29 31 31 34.

(13) 4 Frame bundle considerations: the shock case. 4.1 Generalised ane parameter length of shock-transverse curves 4.1.1 From R n to manifolds . . . . . . . . . . . . . . . . . . 4.2 A metric on the frame bundle . . . . . . . . . . . . . . . . . . 4.2.1 Step I: semi-metricity . . . . . . . . . . . . . . . . . . . 4.2.2 Step II: growth of norms . . . . . . . . . . . . . . . . . 4.2.3 Step III: bounding g.a.p. length with respect to Euclidean length . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Metrics on (M; d) associated with a given frame

(14) eld . 4.3 An application: well-de

(15) nedness of bundle completeness . . . . 4.4 Shock-regular envelopments with Lipschitz boundaries . . . . .. . . . . . . . . .. 35. 35 36 37 39 40 42 44 45 46. III Non-uniqueness of generalised parallel transports along curves lying on shocks, and characterising the \wave-regular" boundary: much more general gravitational wave space-times 49 5 Extending generalised parallel transport to non-shocktransverse curves 51 5.1 The essential range of a locally integrable function . . . . . . . . 5.2 A wider de

(16) nition of generalised parallel transport . . . . . . . . 5.3 Connectedness and compactness of the set of generalised parallel transports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Generalised parallel transport of frames . . . . . . . . . . . . . .. 51 53 55 56. 6 Extending generalised parallel transport to more general metrics 61 6.1 Metrics of Geroch{Traschen type . . . . . . . . . . . . . . . . . 6.1.1 The generalised parallel transport di erential inclusion revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 From R n to almost di erentiable manifolds . . . . . . . . . . . . 6.3 Generalising results on LM , and regular abstract boundaries . .. IV Geodesics of metrics of shock type. 61 63 67 68. 71. 7 Generalised geodesics as solutions of a di erential inclusion 73. 7.1 Correcting the \feature" . . . . . . . . . . . . . . . . . . . . . . 74 7.2 The generalised exponential map . . . . . . . . . . . . . . . . . 76. 8 Non-smooth variational problems. 79. 8.1 \Broken" variational geodesics and generalised Bolza problems . 80 8.2 Relating C 1 variational and generalised geodesics . . . . . . . . 81 8.2.1 Deriving the Euler-Lagrange equations for a Lipschitz Lagrangian with linear growth in one of the derivatives . 81 viii.

(17) 8.2.2 Relating variational and generalised geodesics . . . . . . 85 8.3 An \equivalent" problem, and the positive de

(18) nite case . . . . . 87 8.4 Variational geodesics of inde

(19) nite metrics of shock type . . . . . 89. 9 An application: asymptotic structure of shock manifolds. 91. Some concluding remarks. 97. 9.1 Essentiality and purity . . . . . . . . . . . . . . . . . . . . . . . 95. Appendices. 101. A Summary of Schmidt's b-boundary construction B Some results pertaining to di erential inclusions. 101 105. B.1 \Solutions", or absolutely continuous functions . . . . . . . . . . 105 B.2 Properties of set-valued maps and di erential inclusions . . . . . 106 B.2.1 De

(20) nitions . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.3 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 107 B.3.1 Upper semi-continuous set-valued maps with closed, convex values . . . . . . . . . . . . . . . . . . . . . . . . 107 B.3.2 Regularisation of di erential equations with discontinuous right hand side . . . . . . . . . . . . . . . . . . . . . 108 B.3.3 Integral representation . . . . . . . . . . . . . . . . . . . 108 B.3.4 C 1 solutions of Lipschitz set-valued maps with closed values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.4 Qualitative properties of solutions to di erential inclusions with upper semi-continuous, convex-compact-valued set-valued maps 109. C Non-smooth variational analysis C.1 C.2 C.3 C.4. A review of some measure theory Results on di erential inclusions . Generalised Bolza problems . . . Some non-smooth analysis . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 111. . 111 . 112 . 113 . 114. D Invariance properties of boundary sets with no regularity assumptions 117 D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . D.2 Compact boundary sets . . . . . . . . . . . . . . . . D.3 Isolated boundary sets . . . . . . . . . . . . . . . . . D.3.1 De

(21) nition and properties . . . . . . . . . . . . D.3.2 Invariance . . . . . . . . . . . . . . . . . . . . D.4 Connected boundary sets . . . . . . . . . . . . . . . . D.4.1 The connected neighbourhood property . . . . D.4.2 The T neighbourhood property and invariance ix. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . 118 . 119 . 123 . 123 . 126 . 127 . 128 . 131.

(22) D.4.3 Isolated boundary points and the C NP . . . . . . . . . . 134 D.5 Simple connectedness and vanishing of higher homotopy groups 135 D.5.1 Simply connected boundary sets . . . . . . . . . . . . . . 136 D.5.2 The simply connected neighbourhood property . . . . . . 138 D.5.3 Isolated boundary points and the k NP . . . . . . . . . 140 D.6 An application to isolated boundary submanifolds . . . . . . . . 141 D.7 Summary of this appendix . . . . . . . . . . . . . . . . . . . . . 146. Bibliography. 148. x.

(23) List of Figures 2.1 A typical Lipschitz boundary, on which a pair of proper (inward and outward) cones are shown. . . . . . . . . . . . . . . . . . . . 16 2.2 d^(~x; y~) < d(~x; y~). . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1 G.a.p. length of a curve in Rn is bounded below by a multiple of its (small enough) Euclidean length. . . . . . . . . . . . . . . 43 D.1 Proof of Lemma D.1. . . . . . . . . . . . . . . . . . . . . . . . 120 D.2 Proof of Theorem D.2. . . . . . . . . . . . . . . . . . . . . . . . 122 D.3 A schematic representation of the compact, connected surface of genus 4 as discussed in Example D.3. The shaded areas represent one side of the surface, and the black lines represent the boundary sets in question. . . . . . . . . . . . . . . . . . . 129 D.4 The g = 2 case of Example D.3. . . . . . . . . . . . . . . . . . 129 D.5 Illustrating the construction of U 0 in the proof of Lemma D.7. . 132 D.6 An illustration of Example D.7. On the left is depicted M = c, and on the right is shown a cross-section of 0(M), (M) M through a plane containing the z0 -axis. The boundary sets B and fp0g are equivalent. . . . . . . . . . . . . . . . . . . . . . . 137 D.7 An illustration of Example D.9, in which B satis

(24) es the SC NP but is not simply connected. . . . . . . . . . . . . . . . . . . . 139 D.8 An illustration for the proof of Theorem D.16. Here V is k nice, W0 and W1 are normal tubular neighbourhoods of B , and W1 V W0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144. xi.

(25) xii.

(26) Introduction This thesis is largely concerned with the changing representations of \boundary" or \ideal" points of a pseudo-Riemannian manifold|and our primary interest is in the space-times of general relativity. In particular, we are interested in the following question: What assumptions about the \nature" of \portions" of a certain \ideal boundary" construction (essentially the \abstract boundary" of Scott and Szekeres (1994)) allow us to de

(27) ne precisely the topological type of these \portions", i.e., to show that di erent representations of this ideal boundary, corresponding to di erent embeddings of the manifold into others, have corresponding \portions" that are homeomorphic? Certain topological properties of these \portions" are preserved, even allowing for quite unpleasant properties of the metric (Fama and Scott 1995). These results are given in Appendix D, since they are not used elsewhere and, as well as representing the main portion of work undertaken under the supervision of Scott, which deserves recognition, may serve as an interesting example of the relative ease with which certain simple results about the abstract boundary can be obtained. An answer to a more precisely formulated version of this question appears very dicult in general. However, we can give a rather complete answer in certain cases, where we dictate certain \generalised regularity" requirements for our embeddings, but make no demands on the precise functional form of our metrics apart from these. For example, we get a complete answer to our question for abstract boundary sets which do not \wiggle about" too much| i.e., they satisfy a certain Lipschitz condition|and through which the metric can be extended in a manner which is not required to be di erentiable (C 1), but is continuous and non-degenerate. We allow similar freedoms on the interior of the manifold, thereby bringing gravitational wave space-times within our sphere of discussion. In fact, in the course of developing these results in progressively greater generality, we get, almost \free", certain abilities to begin looking at geodesic structure on quite general pseudo-Riemannian manifolds. It is possible to delineate most of this work cleanly into two major parts. Firstly, there are results which use classical geometric constructs and can be given for the original abstract boundary construction, which requires di erentiability of both manifolds and metrics, and which we summarise below. The 1.

(28) second|and signi

(29) cantly longer|part involves extensions of those constructs and results to more general metrics. This is, actually, the natural way the work developed, and is also a sensible way to approach later results without the possibility of getting \bogged down" in a mish-mash of geometry and analysis. The less analytical part is done in Part I. Why might we be particularly interested in the \stability of form" of these results on regular abstract boundaries? As noted at the end of chapter 2, it would cast grave doubts over the \viability" of our \regular abstract boundary" constructions if it turned out that \non-smooth perturbations" of a smooth metric could result in a space-time where the topological type of a portion of \boundary hypersurface", through which extension is possible, could, again, depend on the precise way in which the extension is performed, unlike the smooth case. Thus extending our results to non-smooth space-times is a crucial matter for the \reasonableness" of regular abstract boundaries of \classical" (non-quantum) space-times. It would appear to also be a crucial matter for the study of abstract boundaries of \quantum space-times", were this to be of interest in a theory of quantum gravity. Actually, due to length the second block of work has been, in turn, split into several parts, in which the constructions and results are progressively generalised. First, in Part II, we treat \shock manifolds", pseudo-Riemannian manifolds where the metric is, roughly speaking, merely required to be Lipschitz, rather than C 2; or even C 1 . We use analytical concepts developed in the last thirty yeas in Chapter 3 to develop \generalised parallel transport" along \shock-transverse" curves. Admittedly, the generality and wide applicability of this material would probably not be necessary were this our sole aim. However, our techniques will admit almost instant generalisations to, e.g., non-linear equations (that governing generalised parallel transport is linear), parallel transport where uniqueness of solution is not assured, and certain non-Lipschitz metrics, besides allowing a lucid, rigorous treatment of the issues involved. In Chapter 4, this generalised parallel transport is used to mimic the outcome of the classical construction of the \bundle metric" on the orthogonal frame bundle LM of (M; g), where now (M; g) is a shock manifold, i.e., g is non-degenerate and Lipschitz. This is then used in the same manner as for the smooth case, to give an answer to the question above. We might ask ourselves how reasonable it is to treat a space-time containing gravitational waves as described by a continuous metric, with discontinuities arising merely in its derivatives. There seems to be a general acceptance that this is so, backed up by strong evidence in the form of papers like Lichnerowicz (1993). The continuous, but not di erentiable, form for our metrics has long been assumed to be an appropriate one for describing space-times containing gravitational shock waves (Penrose 1972a). It should not surprise one to be told that a non-smooth (but continuous, unlike the situation with metrics of many gravitational impulse waves, c.f. x3.2) coordinate transform may often 2.

(30) be found which, according to the usual tensor transformation law. @x @x g ; g 0

(31) 0 = @x 0 @x

(32) 0 . applied at least where the coordinate transform is di erentiable, can exhibit a metric that is discontinuous, merely on some submanifold, in continuous form (i.e., g 0

(33) 0 continuous) (Aichelburg and Balasin 1996, apply x4). We can certainly multiply a function f : R n ! R that is discontinuous across the zero set of a Lipschitz function g, but C 1 elsewhere, by g itself to get a continuous function, and subject to certain requirements on g (e.g., that its proximal subgradient where it vanishes never contain zero|a non-smooth version of a familiar \non-vanishing Jacobian" property (Clarke 1989)), we lose no information in the process. [This proposal of Penrose's, \putting the metric in Rosen form", is also used for many distributional metrics, giving a continuous form for the metric, but destroying the topological structure of the underlying manifold|not just any di erentiable structure. We do not consider such cases here. The future of rigorous study in this area may lie in using Colombeau's \generalised functions" (see, e.g., Clarke, Vickers and Wilson (1996), Aichelburg and Balasin (1996), Balasin (1996), Balasin (1997) or Wilson (1997), the second of these containing interesting material about the \Rosen form" mentioned), which the author, alas, has not had time to go into here.] Part III begins with a way of generalising parallel transport to cope with non-uniqueness|while not enlarging the class of metrics which we consider. This generalisation, to parallel transport along so-called non-shock-transverse curves, is where our \di erential inclusions" method really comes into its own, as it permits a rather elegant treatment (Chapter 5). Only after this alternative exposition do we seek results, in Chapter 6, on manifolds whose Levi-Civita connection components are allowed to be merely square-integrable, though with metrics which are C 1 almost everywhere. In particular, this class can be seen to include space-times with, in addition, suitably \mild" singularities. Perhaps more signi

(34) cant than allowing more general metrics, we also here allow a breakdown of di erentiable structure of our manifolds across \shock surfaces". Our main question is again settled. Finally, in Part IV we consider geodesics of these shock manifolds. Further study is warranted, but in Chapter 7 we solve initial value problems and construct an exponential map at all points (to be more precise, a setvalued map). In the tantalising but time-limited Chapter 8, we then apply recent advances in the calculus of variations/optimal control theory to try to solve the boundary value problem of

(35) nding a geodesic between two points. Unfortunately, a solution eludes us in the interesting, inde

(36) nite case, and, as the author discovered after having solved the de

(37) nite case, the latter has already been done for even more general metrics, using slightly di erent means (Miranda 1996). (This is not too surprising as the method and tools have been around for several years.) 3.

(38) An application of most of the material here to the rigorous de

(39) nition of asymptotic structure appears in Chapter 9. Appendices A{C are largely summaries of material in the literature, while Appendix D is, as mentioned, a body of earlier, complementary work (Fama and Scott 1995).. 4.

(40)