The Conformal Structure of Space-Time: Geometry, Analysis, Numerics PDF
Preview The Conformal Structure of Space-Time: Geometry, Analysis, Numerics
Lecture Notes in Physics EditorialBoard R.Beig,Wien,Austria B.-G.Englert,Ismaning,Germany U.Frisch,Nice,France P.Ha¨nggi,Augsburg,Germany K.Hepp,Zu¨rich,Switzerland W.Hillebrandt,Garching,Germany D.Imboden,Zu¨rich,Switzerland R.L.Jaffe,Cambridge,MA,USA R.Lipowsky,Golm,Germany H.v.Lo¨hneysen,Karlsruhe,Germany I.Ojima,Kyoto,Japan D.Sornette,Nice,France,andLosAngeles,CA,USA S.Theisen,Golm,Germany W.Weise,Trento,Italy,andGarching,Germany J.Wess,Mu¨nchen,Germany J.Zittartz,Ko¨ln,Germany 3 Berlin Heidelberg NewYork Barcelona HongKong London Milan Paris Tokyo EditorialPolicy TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelopmentsin physicsresearchandteaching--quickly,informallybutwithahighquality.Manuscripts to be considered for publication are topical volumes consisting of a limited number of contributions,carefullyeditedandcloselyrelatedtoeachother.Eachcontributionshould containatleastpartlyoriginalandpreviouslyunpublishedmaterial,bewritteninaclear, pedagogical style and aimed at a broader readership, especially graduate students and nonspecialistresearcherswishingtofamiliarizethemselveswiththetopicconcerned.For thisreason,traditionalproceedingscannotbeconsideredforthisseriesthoughvolumes toappearinthisseriesareoftenbasedonmaterialpresentedatconferences,workshops and schools (in exceptional cases the original papers and/or those not included in the printedbookmaybeaddedonanaccompanyingCDROM,togetherwiththeabstracts of posters and other material suitable for publication, e.g. large tables, colour pictures, programcodes,etc.). 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OnlineVersion/LNPHomepage LNPhomepage(listofavailabletitles,aimsandscope,editorialcontactsetc.): http://www.springer.de/phys/books/lnpp/ LNPonline(abstracts,full-texts,subscriptionsetc.): http://link.springer.de/series/lnpp/ J. Frauendiener H. Friedrich (Eds.) The Conformal Structure of Space-Time Geometry, Analysis, Numerics 1 3 Editors Jo¨rgFrauendiener Universita¨tTu¨bingen TheoretischeAstrophysik AufderMorgenstelle10 72076Tu¨bingen,Germany HelmutFriedrich Albert-Einstein-Institut MPIfu¨rGravitationsphysik AmMu¨hlenberg1 14476Golm,Germany CoverPicture:(Thefigureonthecoverisarenderingofthefiniterepresentationofspace- likeinfinityasacylinder.FormoredetailsseethearticlesbyFriedrichandbyKroon.The figurewasobtainedbyray-tracing.) Cataloging-in-PublicationDataappliedfor AcatalogrecordforthisbookisavailablefromtheLibraryofCongress. BibliographicinformationpublishedbyDieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataisavailableintheInternetathttp://dnb.ddb.de CRSubjectClassification(1998):D.4.5,E.3,C.2.0,H.2.0,K.6.5,K.4.4 ISSN0075-8450 ISBN3-540-44280-4Springer-VerlagBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustra- tions, recitation, broadcasting, reproduction on microfilm or in any other way, and storageindatabanks.Duplicationofthispublicationorpartsthereofispermittedonly undertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrent version,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violations areliableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagBerlinHeidelbergNewYork amemberofBertelsmannSpringerScience+BusinessMediaGmbH http://www.springer.de (cid:1)c Springer-VerlagBerlinHeidelberg2002 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readybytheauthors/editor Camera-dataconversionbySteingraeberSatztechnikGmbHHeidelberg Coverdesign:design&production,Heidelberg Printedonacid-freepaper SPIN:10893706 54/3141/du-543210 Preface Controlling the behaviour of the solutions to Einstein’s field equations on large scales is still the most important technical task in classical general relativity. The long standing problem of ‘cosmic censorship’ will not be resolved without a sufficiently general and deep understanding of the solutions. The unexpected emergenceoftheBartnik-McKinnonsolutionsandChoptuik’sdisclosureofcrit- ical collapse phenomena show us that Einstein’s theory still has surprises in store. The last two discoveries also demonstrate clearly the important role of nu- merical techniques in the analysis of specific solutions and in the study of the manifold of solutions. The interplay of analytical with numerical methods is bound to become the most important strategy in exploring the content of the theory. Moreover, since various projected gravitational wave antennae will soon be- come operational, templates with gravitational waves forms will urgently be needed for analysing the recorded data. This makes the interaction between an- alyticalandnumericaltechniquesamostexcitingandimportantproject,because itmayhelpopennewvistasofouruniverse.Thefactthatthecalculationofthe form of gravitational waves generated by astrophysical processes turned out so much more difficult than expected hints at a lack of insight into the analytical basis of the theory or at an insufficient exploitation of the present analytical knowledge. Any general analysis of asymptotically flat space-times in the large needs to takeintoaccountthecausalandtheunderlyingnullconeorconformalstructure of the field. These structures should thus also be critical in the semi-global or global numerical calculation of space-times. Not surprisingly, the two numerical techniquespresentlyusedtoperformsuchcalculations,thecharacteristicmethod and the method based on the conformal field equations, employ basic features of the conformal structure in the definition of their procedures. Theanalyticalandthenumericalmethodsbothofferpossibilitiesnotaccessi- ble to the other one, and each of them asks questions and poses problems likely to initiate interesting research with the other method. For researchers in the fields to get an insight into the potential and the problems of the other field, we thereforeorganizedaworkshoponanalytical,geometrical,andnumericalstudies which make explicit use of conformal or related structures. This book contains extended versions of the contributions to this workshop. VI Preface Following the suggestion of the publisher we tried to avoid the traditional style of proceedings and aimed at a book which will help a newcomer find ac- cess to the field, which offers new results to the experienced researcher, and which provides a comprehensive source of references. In particular, one of us (H.F.) wrote an extensive introductory chapter in which he tries to introduce the newcomer to the field and to provide a general perspective by pointing our the relations to the other studies represented here. Because this perspective may be clouded by the author’s ignorance and per- sonal taste, however, we do urge the reader to understand these references as anencouragementtocarefullystudythosearticlesthemselves.Theintricatenet of relations between the different parts of the work discussed in this book will then become evident. A complete picture of the present situation can only be obtained by trying to understand the full scope of those articles and the specific views of their author’s. Were different opinions occur the reader is invited to search for solutions of the corresponding open problems. While we have tried to maintain to some extent the division of this area of research into the three subfields indicated in the title of this book, it is clear that the assignment of a single article to any of these subfields is not sharply defined. This fact should be seen as a virtue since it was the expressed purpose of the workshop to have researchers in different areas interact with each other andseehowtheycanprofitbyviewingtheirsubjectfromdifferentperspectives. There remains the pleasant task to thank the speakers and the participants forhelpingcreateaninspiringatmosphereattheworkshopandthecontributors of these proceedings for helping create a picture of the present situation of the field which illustrates its richness and its potential. We also thank the Deutsche Forschungsgemeinschaft and the ‘Unibund’ of the University of Tu¨bingen for their financial support. It is a pleasure to thank Prof.HannsRuderforhissupportandHeikeandBettinaFrickefortheirhelpin administrative matters during and after the workshop. Finally, we acknowledge help from T. Mu¨ller, A. King and M. King during the preparation of this book. Tu¨bingen and Golm, J¨org Frauendiener June 2002 Helmut Friedrich List of Contributors Lars Andersson Universidad de Valencia Department of Mathematics 46100 Burjassot (Valencia) University of Miami Spain Coral Gables, FL 33124 [email protected] USA [email protected] J¨org Frauendiener Institut fu¨r Theoretische Astrophysik Robert Bartnik Universita¨t Tu¨bingen School of Mathematics and Statistics 72076 Tu¨bingen University of Canberra, Germany ACT 2601, Australia [email protected] [email protected] Helmut Friedrich MPI fu¨r Gravitationsphysik Adrian Butscher Albert-Einstein-Institut MPI fu¨r Gravitationsphysik 14476 Golm Albert-Einstein-Institut Germany 14476 Golm [email protected] Germany [email protected] Simonetta Frittelli Physics Department Duquesne University Piotr Chru´sciel Pittsburgh, PA 15282 D´epartement de Math´ematiques USA Facult´e des Sciences [email protected] F 37200 Tours France Greg Galloway [email protected] University of Miami Coral Gables Sergio Dain FL 33124 MPI fu¨r Gravitationsphysik USA Albert-Einstein-Institut [email protected] 14476 Golm David Garfinkle Germany Department of Physics [email protected] Oakland University Rochester, Michigan 48309 Jos´e A. Font USA Dep. de Astronom´ıa y Astrof´ısica [email protected] VIII List of Contributors Sascha Husa Andrew Norton MPI fu¨r Gravitationsphysik School of Mathematics and Statistics Albert-Einstein-Institut University of Canberra, 14476 Golm ACT 2601, Australia Germany [email protected] Omar E. Ortiz Universidad Nacional de C´ordoba Facultad de Matem´atica, Niky Kamran Astronom´ıa y F´ısica, Mathematics Department (5000) Co´rdoba McGill University Argentina Montreal [email protected] Canada Roger Penrose Heinz-Otto Kreiss Mathematical Institute Department of Mathematics University of Oxford University of California 24–29 St. Giles’ Los Angeles, CA 90095 Oxford OX1 3LB USA England [email protected] Bernd G. Schmidt Juan Valiente Kroon MPI fu¨r Gravitationsphysik MPI fu¨r Gravitationsphysik Albert-Einstein-Institut Albert-Einstein-Institut 14476 Golm 14476 Golm Germany Germany [email protected] [email protected] Walter Simon Institut fu¨r theoretische Physik Luis Lehner der Universit¨at Wien Department of Physics and Astronomy Boltzmanngasse 5 The University of British Columbia A-1090 Wien Vancouver, BC V6T 1Z1 Austria Canada [email protected] [email protected] K. P. Tod Ezra T. Newman Mathematical Institute Department of Physics and Astronomy University of Oxford University of Pittsburgh 24–29 St. Giles’ Pittsburgh PA 15260 Oxford OX1 3LB USA England [email protected] [email protected] Contents 1 Conformal Einstein Evolution Helmut Friedrich................................................... 1 1.1 Introduction .............................................. 1 1.2 The Conformal Field Equations ............................. 4 1.2.1 Conformal Geometry ................................ 6 1.2.2 Derivation of the Conformal Field Equations ........... 9 1.3 The Penrose Proposal ...................................... 21 1.4 Asymptotic Behaviour of Vacuum Fields with Vanishing Cosmological Constant ....................... 29 1.4.1 The Hyperboloidal Initial Value Problem............... 30 1.4.2 On the Existence of Asymptotically Simple Vacuum Solutions ............ 31 1.4.3 The Regular Finite Cauchy Problem................... 33 1.4.4 Time-Like Infinity................................... 42 References ..................................................... 46 2 Some Global Results for Asymptotically Simple Space-Times Gregory J. Galloway................................................ 51 2.1 Introduction .............................................. 51 2.2 The Null Splitting Theorem................................. 53 2.3 Proof of Theorem 2.1 ...................................... 55 2.4 Concluding Remarks ....................................... 58 References ..................................................... 59 3 Black Holes Piotr T. Chru´sciel ................................................. 61 3.1 Introduction .............................................. 61 3.2 Experimental Evidence..................................... 62 3.3 Causality for Symmetric Hyperbolic Systems ................. 65 3.3.1 Dumb Holes........................................ 69 3.3.2 Optical Holes....................................... 70 3.3.3 Trapped Surfaces ................................... 70 3.4 Standard Black Holes ...................................... 71 3.4.1 Scri Regularity Conditions, and the Area Theorem ...... 75 3.5 Horizons ................................................. 77 3.6 Apparent Horizons......................................... 79 X Contents 3.7 Classification of Stationary Solutions (“No Hair Theorems”) .... 80 3.8 Black Holes Without Scri................................... 83 3.8.1 Naive Black Holes................................... 84 3.8.2 Quasi-local Black Holes.............................. 86 3.8.3 Finding Horizons ................................... 90 References ..................................................... 96 4 Conformal Geometry, Differential Equations and Associated Transformations Simonetta Frittelli, Niky Kamran, Ezra T. Newman .................... 103 4.1 Introduction .............................................. 103 4.2 Example of Contact-Envelope Transformation................. 105 4.3 Generalizations............................................ 109 4.3.1 Three-Dimensional Conformal Lorentzian Geometries.... 109 4.3.2 Four-Dimensional Conformal Lorentzian Geometries..... 110 References ..................................................... 111 5 Twistor Geometry of Conformal Infinity Roger Penrose ..................................................... 113 5.1 Non-linear Gravitons....................................... 113 5.2 The Reasonableness of I+.................................. 115 5.3 The Construction of Projective Twistor Space PT from I+ ..... 115 5.4 The Construction of the Full Twistor Space T from I+ ........ 117 5.5 The Local Structure of Twistor Space PT .................... 118 5.6 Present Status of the Role of T in Encoding Ricci-Flatness.................................. 119 References ..................................................... 120 6 Isotropic Cosmological Singularities K. Paul Tod....................................................... 123 6.1 Introduction .............................................. 123 6.2 Formalism and Extensions .................................. 125 6.3 Review of Polytropic Perfect Fluid Case ...................... 128 6.4 Further Matter Models..................................... 131 6.4.1 Massive Einstein-Vlasov ............................. 132 6.4.2 Scalar Fields ....................................... 132 6.4.3 Einstein-Yang-Mills-Vlasov........................... 132 6.4.4 Einstein-Boltzmann ................................. 132 6.5 Conclusion and Future Possibilities .......................... 133 References ..................................................... 133 7 Polyhomogeneous Expansions Close to Null and Spatial Infinity Juan Antonio Valiente Kroon........................................ 135 7.1 Introduction .............................................. 135 7.2 Minkowski Space-Time Close to Null and Spatial Infinity ....... 136 7.3 Linearised Gravity in the F-Gauge........................... 138
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