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8、al Department I Tiergartenstrasse 17 69121 Heidelberg/Germany ric Gourgoulhon 3+1 Formalism in General Relativity Bases of Numerical Relativity 123ric Gourgoulhon Lab. Univers et Thories (LUTH) UMR 8102 du CNRS Observatoire de Paris Universit Paris Diderot place Jules Janssen 5 92190 Meudon France I
9、SSN 0075-8450 e-ISSN 1616-6361 ISBN 978-3-642-24524-4 e-ISBN 978-3-642-24525-1 DOI 10.1007/978-3-642-24525-1 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2011942212Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved,
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13、ck (1955-2000)Preface This monograph originates from lectures given at the General Relativity Trimester at the Institut Henri Poincar in Paris 1; at the VII Mexican School on Gravi- tation and Mathematical Physics in Playa del Carmen (Mexico) 2; and at the 2008 International Summer School on Computa
14、tional Methods in Gravitation and Astrophysics held in Pohang (Korea) 3. It is devoted to the 3+1 formalism of general relativity, which constitutes among other things, the theoretical founda- tions for numerical relativity. Numerical techniques are not covered here. For a pedagogical introduction t
15、o them, we recommend instead the lectures by Choptuik 4 (nite differences) and the review article by Grandclment and Novak 5 (spectral methods), as well as the numerical relativity textbooks by Alcubierre 6, Bona, Palenzuela-Luque and Bona-Casas 7 and Baumgarte and Shapiro 8. The prerequisites are t
16、hose of a general relativity course, at the undergraduate or graduate level, like the textbooks by Hartle 9 or Carroll 10, or part I of Walds book 11, as well as track 1 of the book by Misner, Thorne and Wheeler 12. The fact that the present text consists of lecture notes implies two things: the cal
17、culations are rather detailed (the experienced reader might say too detailed), with an attempt to made them self-consistent and complete, trying to use as infrequently as possible the famous phrases as shown in paper XXX or see paper XXX for details; the bibliographical references do not constitute
18、an extensive survey of the lit- erature on the subject: articles have been cited in so far as they have a direct connection with the main text. The book starts with a chapter setting the mathematical background, which is differential geometry, at a basic level (Chap. 2). This is followed by two pure
19、ly geometrical chapters devoted to the study of a single hypersurface embedded in spacetime (Chap. 3) and to the foliation (or slicing) of spacetime by a family of spacelike hypersurfaces (Chap. 4). The presentation is divided in two chapters to distinguish between concepts which are meaningful for
20、a single hypersurface and those that rely on a foliation. The decomposition of the Einstein equation relative viito the foliation is given in Chap. 5, giving rise to the Cauchy problem with constraints, which constitutes the core of the 3+1 formalism. The ADM Hamil- tonian formulation of general rel
21、ativity is also introduced in this chapter. Chapter 6 is devoted to the decomposition of the matter and electromagnetic eld equations, focusing on the astrophysically relevant cases of a perfect uid and a perfect conductor (ideal MHD). An important technical chapter occurs then: Chap. 7 introduces s
22、ome conformal transformation of the 3-metric on each hypersurface and the corresponding rewriting of the 3+1 Einstein equations. As a by-product, we also discuss the Isenberg-Wilson-Mathews (or conformally at) approximation to general relativity. Chapter 8 details the various global quantities assoc
23、iated with asymptotic atness (ADM mass, ADM linear momentum and angular momentum) or with some symmetries (Komar mass and Komar angular momentum). In Chap. 9, we study the initial data problem, presenting with some examples two classical methods: the conformal transversetraceless method and the conf
24、ormal thin-sandwich one. Both methods rely on the conformal decomposition that has been introduced in Chap. 7. The choice of spacetime coordinates within the 3+1 framework is discussed in Chap. 10, starting from the choice of foliation before discussing the choice of the three coordinates in each le
25、af of the foliation. The major coordinate families used in modern numerical relativity are reviewed. Finally Chap. 11 presents various schemes for the time integration of the 3+1 Einstein equations, putting some emphasis on the most successful scheme to date, the BSSN one. Appendix A is devoted to b
26、asic tools of the 3+1 formalism: the conformal Killing operator and the related vector Laplacian, whereas Appendix B provides some computer algebra codes based on the Sage system. A web page is dedicated to the book, at the URL http:/relativite.obspm.fr/3p1 This page contains the errata, the clickab
27、le list of references, the computer algebra codes described in Appendix B and various supplementary material. Readers are invited to use this page to report any error that they may nd in the text. I am deeply indebted to Micha Bejger, Philippe Grandclment, Alexandre Le Tiec, Yuichiro Sekiguchi and N
28、icolas Vasset for the careful reading of a pre- liminary version of these notes. I am very grateful to my friends and colleagues Thomas Baumgarte, Micha Bejger, Luc Blanchet, Silvano Bonazzola, Brandon Carter, Isabel Cordero-Carrin, Thibault Damour, Nathalie Deruelle, Guillaume Faye, John Friedman,
29、Philippe Grandclment, Jos Maria Ibez, Jos Luis Jaramillo, Jean-Pierre Lasota, Jrme Novak, Jean-Philippe Nicolas, Motoyuki Saijo, Masaru Shibata, Keisuke Taniguchi, Koji Uryu, Nicolas Vasset and Loc Villain, for the numerous and fruitful discussions that we had about general rel- ativity and the 3+1
30、formalism. I also warmly thank Robert Beig and Christian Caron for their invitation to publish this text in the Lecture Notes in Physics series. Meudon, September 2011 ric Gourgoulhon viii PrefaceReferences 1. http:/www.luth.obspm.fr/IHP06/ 2. http:/www.smf.mx/*dgfm-smf/EscuelaVII/ 3. http:/apctp.or
31、g/conferences/ 4. Choptuik, M.W.: Numerical analysis for numerical relativists, lecture at the VII Mexican school on gravitation and mathematical physics, Playa del Carmen (Mexico), 26 November-1 December 2006 2; available at http:/laplace.physics.ubc.ca/People/matt/Teaching/ 06Mexico/ 5. Grandclmen
32、t, P. and Novak, J.: Spectral methods for numerical relativity, Living Rev. Relat. 12, 1 (2009); http:/www.livingreviews.org/lrr-2009-1 6. Alcubierre, M.: Introduction to 3+1 Numerical Relativity. Oxford University Press, Oxford (2008) 7. Bona, C., Palenzuela-Luque, C. and Bona-Casas, C.: Elements o
33、f numerical relativity and relativistic hydrodynamics: from einsteins equations to astrophysical simulations (2nd edition). Springer, Berlin (2009) 8. Baumgarte, T. W. and Shapiro, S. L.: Numerical relativity. Solving Einsteins equations on the computer, Cambridge University Press, Cambridge (2010)
34、9. Hartle, J.B.: Gravity: An introduction to Einsteins general relativity, Addison Wes- ley(Pearson Education), San Fransisco (2003); http:/ 0,6533,512494-,00.html 10. Carroll, S.M.: Spacetime and geometry: an introduction to general relativity, Addison Wesley (Pearson Education), San Fransisco (200
35、4); http:/ spacetimeandgeometry/ 11. Wald, R.M.: General relativity, University of Chicago Press, Chicago (1984) 12. Misner, C.W., Thorne, K.S. and Wheeler, J.A.: Gravitation, Freeman, New York (1973) Preface ixContents 1 Introduction. 1 References . . . . . . . . . . . . . . . . . . . . . . . . . .
36、 . . . . . . . . . . . . . . . 2 2 Basic Differential Geometry . 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Notion of Manifold. . . . . . . . . . . . . . . . .
37、 . . . . . . . . 6 2.2.2 Vectors on a Manifold . . . . . . . . . . . . . . . . . . . . . . 8 2.2.3 Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.5 Fields on a Manifold. . . . . . . . .
38、. . . . . . . . . . . . . . . 13 2.3 Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Signature and Orthonormal Bases. . . . . . . . . . . . . . . 14 2.3.3 Metric Duality . . . . . . . .
39、. . . . . . . . . . . . . . . . . . . . 15 2.3.4 LeviCivita Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.1 Affine Connection on a Manifold . . . . . . . . . . . . . . . 17 2.4.2 LeviCivita Conn
40、ection. . . . . . . . . . . . . . . . . . . . . . 20 2.4.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.4 Weyl Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41、25 2.5.1 Lie Derivative of a Vector Field. . . . . . . . . . . . . . . . 25 2.5.2 Generalization to Any Tensor Field . . . . . . . . . . . . . 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Geometry of Hypersurfaces . 2 9 3.1 Introduction . . . .
42、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Framework and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Hypersurface Embedded in Spacetime. . . . . . . . . . . . . . . . . . 30 xi3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . .
43、 . . . . 30 3.3.2 Normal Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.3 Intrinsic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.4 Extrinsic Curvature. . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.5 Examples: Surfaces Embedded in the Euclidean S
44、paceR 3 . 3 6 3.3.6 An Example in Minkowski Spacetime: The Hyperbolic SpaceH 3 . 4 0 3.4 Spacelike Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.1 The Orthogonal Projector . . . . . . . . . . . . . . . . . . . . 44 3.4.2 Relation Between K andrn . 4 6 3.4.3 Links Between
45、 ther and D Connections . . . . . . . . . 47 3.5 GaussCodazzi Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5.1 Gauss Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5.2 Codazzi Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 52 References . .
46、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Geometry of Foliations. 5 5 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Globally Hyperbolic Spacetimes and Foliations . . . . . . . . . . . 55 4.2.1 Globally Hyperb
47、olic Spacetimes. . . . . . . . . . . . . . . . 55 4.2.2 Definition of a Foliation . . . . . . . . . . . . . . . . . . . . . 56 4.3 Foliation Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.1 Lapse Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.2
48、Normal Evolution Vector . . . . . . . . . . . . . . . . . . . . 57 4.3.3 Eulerian Observers . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.4 Gradients of n and m . 6 3 4.3.5 Evolution of the 3-Metric . . . . . . . . . . . . . . . . . . . . 64 4.3.6 Evolution of the Orthogonal Projector. . . . . . . . . . . . 66 4.4 Last Part of the 3+1 Decomposition of the Riemann Tensor . . . 67 4.4.1 Last Non Trivial Projection of the Spacetime Rieman
