The Formation Of Black Holes In General Relativity

EMM_Black_Holes_titelei 24.11.2008 10:35 Uhr Seite 1 EMM_Black_Holes_titelei 24.11.2008 10:35 Uhr Seite 2 EMS Monographs in Mathematics Edited by Ivar Ekeland (Pacific Institute, Vancouver, Canada) Gerard van der Geer (University of Amsterdam, The Netherlands) Helmut Hofer (Courant Institute, New York, USA) Thomas Kappeler (University of Zürich, Switzerland) EMS Monographs in Mathematics is a book series aimed at mathematicians and scientists. It publishes research monographs and graduate level textbooks from all fields of mathematics. The individual volumes are intended to give a reasonably comprehensive and selfcontained account of their particular subject. They present mathematical results that are new or have not been accessible previously in the literature. Previously published in this series: Richard Arratia, A.D. Barbour and Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach Demetrios Christodoulou, The Formation of Shocks in 3-Dimensional Fluids Sergei Buyalo and Viktor Schroeder, Elements of Asymptotic Geometry EMM_Black_Holes_titelei 24.11.2008 10:35 Uhr Seite 3 Demetrios Christodoulou The Formation of Black Holes in General Relativity EMM_Black_Holes_titelei 24.11.2008 10:35 Uhr Seite 4 Author: Prof. Demetrios Christodoulou Department of Mathematics ETH-Zentrum 8092 Zürich Switzerland 2000 Mathematical Subject Classification (primary; secondary): 83C57; 35L70, 35Q75, 58J45, 83C75 ISBN 978-3-03719-068-5 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Printed in Germany 987654321 To my wife Nikoleta Contents Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The Optical Structure Equations 1.1 The basic geometric setup . . 1.2 The optical structure equations 1.3 The Bianchi equations . . . . 1.4 Canonical coordinate systems . . . . . 29 33 54 59 2 The Characteristic Initial Data 2.1 The characteristic initial data . . . . . . . . . . . . . . . . . . . . . . . 2.2 Construction of the solution in an initial domain . . . . . . . . . . . . . 63 88 3 4 L∞ 3.1 3.2 3.3 3.4 3.5 3.6 . . . . . . . . . . . . . . . . . . . . . . . . Estimates for the Connection Coefficients Introduction . . . . . . . . . . . . . . . . L ∞ estimates for χ  . . . . . . . . . . . . L ∞ estimates for χ  . . . . . . . . . . . . L ∞ estimates for η, η . . . . . . . . . . . L ∞ estimates for ω, ω . . . . . . . . . . The smallness requirement on δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .