3+1 formalism and bases of numerical relativity - LUTh ...

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Contents 1 Introduction 11 2 Geometry of hypersurfaces 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Framework and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Spacetime and tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Scalar products and metric duality . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Hypersurface embedded in spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Intrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.4 Extrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.5 Examples: surfaces embedded in the Euclidean space R 3 . . . . . . . . . . 24 2.4 Spacelike hypersurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.1 The orthogonal projector . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.2 Relation between K and ∇n . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.3 Links between the ∇ and D connections . . . . . . . . . . . . . . . . . . . 32 2.5 Gauss-Codazzi relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5.1 Gauss relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5.2 Codazzi relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 Geometry of foliations 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Globally hyperbolic spacetimes and foliations . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Globally hyperbolic spacetimes . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Definition of a foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Foliation kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.1 Lapse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.2 Normal evolution vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.3 Eulerian observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.4 Gradients of n and m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.5 Evolution of the 3-metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Page 1: arXiv:gr-qc/0703035v1 6 Mar 2007 3+
Page 5 and 6: CONTENTS 5 6 Conformal decompositio
Page 7 and 8: CONTENTS 7 10 Evolution schemes 175
Page 9 and 10: Preface These notes are the written
Page 11 and 12: Chapter 1 Introduction The 3+1 form
Page 13 and 14: date, the BSSN one. Two appendices
Page 15 and 16: Chapter 2 Geometry of hypersurfaces
Page 17 and 18: 2.2 Framework and notations 17 is a
Page 19 and 20: 2.3 Hypersurface embedded in spacet
Page 23 and 24: 2.3.4 Extrinsic curvature 2.3 Hyper
Page 29 and 30: 2.4.1 The orthogonal projector 2.4
Page 31 and 32: 2.4.2 Relation between K and ∇n 2
Page 33 and 34: 2.4 Spacelike hypersurface 33 Figur
Page 35 and 36: or equivalently, since v µ = γ µ
Page 37 and 38: 2.5 Gauss-Codazzi relations 37 Exam
Page 39 and 40: Chapter 3 Geometry of foliations Co
Page 41 and 42: 3.3 Foliation kinematics 41 Figure
Page 43 and 44: 3.3 Foliation kinematics 43 Figure
Page 45 and 46: 3.3.5 Evolution of the 3-metric 3.3
Page 47 and 48: 3.4 Last part of the 3+1 decomposit
Page 49 and 50: 3.4 Last part of the 3+1 decomposit
Page 51 and 52: Chapter 4 3+1 decomposition of Eins
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4.1 Einstein equation in 3+1 form 5
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4.2 Coordinates adapted to the foli
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4.2 Coordinates adapted to the foli
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4.3 3+1 Einstein equation as a PDE
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4.4 The Cauchy problem 61 PDE syste
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4.4 The Cauchy problem 63 hightest
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4.5 ADM Hamiltonian formulation 65
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Chapter 5 3+1 equations for matter
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5.2 Energy and momentum conservatio
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5.3 Perfect fluid 75 where use has
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5.3 Perfect fluid 77 Remark : Where
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Combining Eqs. (5.31) and (5.43), w
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5.3 Perfect fluid 81 Finally, by ap
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Chapter 6 Conformal decomposition C
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6.2 Conformal decomposition of the
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6.3 Expression of the Ricci tensor
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Now 6.4 Conformal decomposition of
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6.5 Conformal form of the 3+1 Einst
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6.5 Conformal form of the 3+1 Einst
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6.6 Isenberg-Wilson-Mathews approxi
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6.6 Isenberg-Wilson-Mathews approxi
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Chapter 7 Asymptotic flatness and g
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7.2.2 Asymptotic coordinate freedom
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7.3 ADM mass 107 Example : Let us c
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7.3 ADM mass 109 and ˜γij = fij +
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7.3.4 Positive energy theorem 7.3 A
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7.5 Angular momentum 113 behaves as
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7.5 Angular momentum 115 Figure 7.1
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7.6 Komar mass and angular momentum
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Chapter 8 The initial data problem
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8.2 Conformal transverse-traceless
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with 8.2 Conformal transverse-trace
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8.3 Conformal thin sandwich method
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8.4 Initial data for binary systems
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where [cf. Eq. (8.90)] 8.4 Initial
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8.4 Initial data for binary systems
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Chapter 9 Choice of foliation and s
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9.2 Choice of foliation 153 Figure
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Now from the Gauss-Ostrogradsky the
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9.2 Choice of foliation 157 Figure
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9.2 Choice of foliation 159 type
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9.2 Choice of foliation 161 Remark
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9.3 Evolution of spatial coordinate
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9.4 Full spatial coordinate-fixing
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Chapter 10 Evolution schemes Conten
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i.e. 10.3 Free evolution schemes 17
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10.3 Free evolution schemes 179 Rem
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10.3.3 Constraint-violating modes 1
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Replacing ˜ Γ k ij by ∆k ij +
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10.4 BSSN scheme 185 Remark : Actua
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10.4 BSSN scheme 187 the constraint
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Appendix A Lie derivative Contents
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A.2 Generalization to any tensor fi
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Appendix B Conformal Killing operat
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B.2 Conformal vector Laplacian 195
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B.2 Conformal vector Laplacian 197
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Bibliography [1] http://www.luth.ob
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BIBLIOGRAPHY 201 [28] L. Baiotti, I
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BIBLIOGRAPHY 203 [59] C. Bona, L. L
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BIBLIOGRAPHY 205 [90] Y. Choquet-Br
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BIBLIOGRAPHY 207 [120] C.R. Evans :
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BIBLIOGRAPHY 209 [148] P. Grandclé
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BIBLIOGRAPHY 211 [179] F. Limousin,
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BIBLIOGRAPHY 213 [211] B.L. Reinhar
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BIBLIOGRAPHY 215 [241] M. Shibata,
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BIBLIOGRAPHY 217 [269] J. Winicour
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foliation, 40 free evolution scheme
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