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The Nonsyrnrnetric Gravitational Theory This manuscript has been reproduced frorn the microfilm rnaster. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewrïterface, while others may be frorn any type of cornputer pnnter. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedttirough, substandard margins, and irnproper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a camplete manuscript and there are rnissing pages, these will be noted. Also. if unauthorized copyright matenal had to be removed, a note will indicate the deletion. Ovenize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing hmleft to right in equal sections with small overfaps. Photographs included in the original manuscript have ken reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographie prints are available for any photographs or illustrations appearing in mis copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Leaming 300 North Zeeb Road, Ann Abor, MI 48106-1346 USA The Nonsyrnrnetric Gravitational Theory: by Jacques Légaré A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy, Graduate Department of Physics, University of Toronto. Copyright @ 1998 by Jacques Légaré National Library Bibliothèque nationale I*I of Canada du Canada Acquisitions and Acquisitions et Bibliogaphic Services services bibliographiques 395 Wellington Street 395. rue Wellington OttawaON KtAON4 Ottawa ON K1A ON4 Canada Canada Your file Votre réference Our file Notre rehrence The author has granted a non- L'auteur a accordé une licence non exclusive Licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fkom it Ni la thèse ni des extraits substantiels may be p~tedor othewise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Wyman Sector Wave ColIapse in The Nonsymmetric Gravitational Theory: An Application of Smd-Scale Numerical Relativity Doctor of Philosophy, 1998 Jacques Légaré Graduate Department of Physics, University of Toronto Abstract A brief account of the current state of the Nonsymmetric Gravitational Theory is given, with emphasis placed on the formal and interpretational obstacles that might interfere with the acceptance of the theory as a mode1 of gravitational physics. The 3 + 1 decomposition of the theory is reviewed, dong with the reduction to the spherically- symmetric Wyman sector. Following a reparameterization, a connection is made between the field equations of the spherically-symmetric, Wyman sector, and a Klein-Gordon field coupled to Einstein gravity. Possible slicing conditions and gauge choices are discussed. A numerical investigation of both the initial-value problem and the evolution problem for the spherically-symmetric Wyman sector is undertaken. Conditions on the well-posedness of the initial- value problem are discussed. Gravitational wave pulses are evolved for various field strengths, and critical behaviour is observed in the coilapse of certain pulses. The stability of a static, sphericdy-symmetric solution to the field equations is studied. It is shown that the solution is unstable, and consequently cannot be the eadstate of a generic collapse. It is demonstrated that the ody two possible solutions that have a regular origin or an event horizon i are the Minkowski space solution and the Schwarzschild soiution, respectively. It is then argued that any sphericalIy-symmetric, Wyman sector gravitationai wave coiiapse must end in a standard Einsteinian configuration or a nakedly singular alternative. Several people have infiuenced me during my studies. The full list of aU who helped me complete this dissertation would most likely f2l a tome of comparable size. Evidently, 1cannot hope to succeed in ascribing credit where credit is surely due. 1 must instead content myself to thank only those who inAuenced me or helped me most recently or most greatly. This work could not have been completed without funding provided by the University of Toronto, the Government of Ontario, and the National Sciences and Engineering Research Council of Canada. 1 would especially Iike to thank the Walter C. Sumner Foundation for the hancial support they provided me during my doctoral studies. I would also like to thank the various goveniing bodies and cornmittees that supplied the occasionally needed diversion from this work: Associate Chair Professor Rashmi Desai and the de- partmental Graduate Liaison Committee, Associate Dean Professor Don Cormack and the Executive Committee of Division III, and Deans Professor Jon Cohen and Professor MichaeI R. Mmsand the Councii of the School of Graduate Studies. As well, 1would iike to thank Professor Ralph Scane and my colleagues of the Graduate Academic Appeals Board for showing me the not-so-nice side of academic administration, and for always being willing to listen to the opinions of a student, despite what may have been at stake. On the question of the substance or content of this dissertation, 1 benefited greatly from my discussions or correspondences with Drs. Neil J. Cornish, Pierre Savaria, and Igor Sokolov, and Pr* fessors M. W. Choptuik, G. F. R. Ellis, N. 6 Murchadha, and E. Seidel. A special acknowledgement goes to Karen Camarda, without whose help 1 could never have gotten started on this project. But perhaps the greatest influences were my supervisor Professor John W. Moffat, and my two colieagues Dr. Michael A. Clayton and Terry Demopoulos. Professor Moffat taught me the value of an idea and the importance of pursuing that idea despite any and all opposition. The incomparable enthusiasm he brings to his work is an inspiration to aU who work with him. Meanwhile, 1codd always count on the subtle but significant analyses of Terry, and the often all-too brutal but always all-too correct analyses of Michael. The latter two also taught me the danger of being the only member of a group to know how to use the tools at hand ("Hey Jacques, can you plot .. .") - On the question of the form or format of this work, 1 must extend my deepest gratitude to Terry Dernopoulos and Cardee Légaré for their nurnerous helpful suggestions, without whose help my crippled sense of aesthetics wodd have produced a much more inferior work, and also to Patrick Nadeau, when the QjX macros 1 wrote for this project didn't work as designed, or at all ("Bey Pat, ever try writing citation macros that collapse neighbouriig references automatically?" ). Finaliy corne those people who didn't influence this work directly, but rather helped me dong in sorne indirect fashion, either through their understanding enthusiasm when things were going weli, or through their exnotional support when things weren't going so well. Among these 1 count my close &ends Nancy Légrné, Paola Barbarossa, Patrick Nadeau, Showwei Chu, and Adrian Butscher, Pstrs. G. Pettipas, P. Chin, E. Garcia, J. James, and D. Talbot, my in-laws, my sister Josée and her husband, Professor John F. Corrigan, ail the Fnends, and especidy my wife Caralee Légaré, and my parents, Thérèse and ~aul-ÉmileLégaré. For the LORD God is a sun and shield: the LORD will give grace and glory: no good t hing will He withhold fiom them that walk uprightly. O LORD of hosts, blessed is the man that trusteth in Thee. Psalm 84:11, 12 Conventions and For convenience, we will introduce some conventions here, to be assumed throughout the re- mainder of this work. The acronyms GR and NGT refer to General Relativity and the Nonsymmetric Gravitational Theory, while the acronym UFT refers to the Unified Field Theory. We use the term "Wyman sector" to denote the spatial-spatial antisymmetric sector of NGT, whereas we use the term "Papapetrou sector" to refer to the temporal-spatial antisymmetric sector. Throughout, we use units where G = c = 1, and adopt a Minkowski metric with signature -2. The choice of the Riemann tensor is made explicit in (A.3a), and corresponds to the Landau-Lifshitz "timelike convention" [l, inside front cover] . The Einstein summation convention is assumed throughout. Indices from the lowercase greek alphabet (pl v, .. ) are used for holonomic bases in a four-dimensional space. Indices from the uppercase latin alphabet (A, B, .. .) denote anholonomic bases in any dimension. Indices from the middle of the Iowercase latin alphabet (i, j, . .) are used for spatial quantities, and are summed ody over spatial values. {8,}are used as the basis vectors of a holonomic frame, whereas {eA} are used for aaholonomic ftames. The corresponding dual bases are {&pl and {OA}. Boldface fonts denote tensors, with corresponding fonts of regular weight used for their components. Thus a tensor T might have components T~~,so that T = TABeA8 OB. Finally, parentheses
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