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

More documents

Recommendations

Info

12 Introduction 3D (no symmetry at all). In parallel, a lot of studies have been devoted to formulating the 3+1 equations in a form suitable for numerical implementation. The authors who participated to this effort are too numerous to be cited here but it is certainly worth to mention Takashi Nakamura and his school, who among other things initiated the formulation which would become the popular BSSN scheme [193, 192, 233]. Needless to say, a strong motivation for the expansion of numerical relativity has been the development of gravitational wave detectors, either groundbased (LIGO, VIRGO, GEO600, TAMA) or in space (LISA project). Today, most numerical codes for solving Einstein equations are based on the 3+1 formalism. Other approaches are the 2+2 formalism or characteristic formulation, as reviewed by Winicour [269], the conformal field equations by Friedrich [134] as reviewed by Frauendiener [129], or the generalized harmonic decomposition used by Pretorius [206, 207, 208] for his recent successful computations of binary black hole merger. These lectures are devoted to the 3+1 formalism and theoretical foundations for numerical relativity. They are not covering numerical techniques, which mostly belong to two families: finite difference methods and spectral methods. For a pedagogical introduction to these techniques, we recommend the lectures by Choptuik [84] (finite differences) and the review article by Grandclément and Novak [150] (spectral methods). We shall start by two purely geometrical 2 chapters devoted to the study of a single hypersurface embedded in spacetime (Chap. 2) and to the foliation (or slicing) of spacetime by a family of spacelike hypersurfaces (Chap. 3). The presentation is divided in two chapters to distinguish clearly between concepts which are meaningful for a single hypersurface and those who rely on a foliation. In some presentations, these notions are blurred; for instance the extrinsic curvature is defined as the time derivative of the induced metric, giving the impression that it requires a foliation, whereas it is perfectly well defined for a single hypersurface. The decomposition of the Einstein equation relative to the foliation is given in Chap. 4, giving rise to the Cauchy problem with constraints, which constitutes the core of the 3+1 formalism. The ADM Hamiltonian formulation of general relativity is also introduced in this chapter. Chapter 5 is devoted to the decomposition of the matter and electromagnetic field equations, focusing on the astrophysically relevant cases of a perfect fluid and a perfect conductor (MHD). An important technical chapter occurs then: Chap. 6 introduces some conformal transformation of the 3-metric on each hypersurface and the corresponding rewriting of the 3+1 Einstein equations. As a byproduct, we also discuss the Isenberg-Wilson-Mathews (or conformally flat) approximation to general relativity. Chapter 7 details the various global quantities associated with asymptotic flatness (ADM mass and ADM linear momentum, angular momentum) or with some symmetries (Komar mass and Komar angular momentum). In Chap. 8, we study the initial data problem, presenting with some examples two classical methods: the conformal transverse-traceless method and the conformal thin sandwich one. Both methods rely on the conformal decomposition that has been introduced in Chap. 6. The choice of spacetime coordinates within the 3+1 framework is discussed in Chap. 9, starting from the choice of foliation before discussing the choice of the three coordinates in each leaf of the foliation. The major coordinate families used in modern numerical relativity are reviewed. Finally Chap. 10 presents various schemes for the time integration of the 3+1 Einstein equations, putting some emphasis on the most successful scheme to 2 by geometrical it is meant independent of the Einstein equation
date, the BSSN one. Two appendices are devoted to basic tools of the 3+1 formalism: the Lie derivative (Appendix A) and the conformal Killing operator and the related vector Laplacian (Appendix B). 13
Page 1: arXiv:gr-qc/0703035v1 6 Mar 2007 3+
Page 4 and 5: 4 CONTENTS 3.3.6 Evolution of the o
Page 6 and 7: 6 CONTENTS 8 The initial data probl
Page 8 and 9: 8 CONTENTS
Page 10 and 11: 10 CONTENTS
Page 14 and 15: 14 Introduction
Page 16 and 17: 16 Geometry of hypersurfaces in two
Page 18 and 19: 18 Geometry of hypersurfaces 2.2.3
Page 20 and 21: 20 Geometry of hypersurfaces Figure
Page 22 and 23: 22 Geometry of hypersurfaces •
Page 24 and 25: 24 Geometry of hypersurfaces The ei
Page 26 and 27: 26 Geometry of hypersurfaces Figure
Page 28 and 29: 28 Geometry of hypersurfaces The no
Page 30 and 31: 30 Geometry of hypersurfaces Since
Page 32 and 33: 32 Geometry of hypersurfaces 2.4.3
Page 34 and 35: 34 Geometry of hypersurfaces 2.5 Ga
Page 36 and 37: 36 Geometry of hypersurfaces Exampl
Page 38 and 39: 38 Geometry of hypersurfaces
Page 40 and 41: 40 Geometry of foliations Figure 3.
Page 42 and 43: 42 Geometry of foliations 3.3.2 Nor
Page 44 and 45: 44 Geometry of foliations means Eq.
Page 46 and 47: 46 Geometry of foliations Remark :
Page 48 and 49: 48 Geometry of foliations Note that
Page 50 and 51: 50 Geometry of foliations
Page 52 and 53: 52 3+1 decomposition of Einstein eq
Page 62 and 63:
62 3+1 decomposition of Einstein eq
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
72 3+1 equations for matter and ele
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
84 Conformal decomposition equivale
Page 86 and 87:
86 Conformal decomposition As an ex
Page 88 and 89:
88 Conformal decomposition 6.2.4 Co
Page 90 and 91:
90 Conformal decomposition 6.3.1 Ge
Page 92 and 93:
92 Conformal decomposition where K
Page 94 and 95:
94 Conformal decomposition to write
Page 96 and 97:
96 Conformal decomposition hence Lm
Page 98 and 99:
98 Conformal decomposition 6.5.2 Ha
Page 100 and 101:
100 Conformal decomposition discuss
Page 102 and 103:
102 Conformal decomposition Remark
Page 104 and 105:
104 Asymptotic flatness and global
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
126 The initial data problem Notice
Page 128 and 129:
128 The initial data problem where
Page 130 and 131:
130 The initial data problem 8.2.3
Page 132 and 133:
132 The initial data problem where
Page 134 and 135:
134 The initial data problem Figure
Page 136 and 137:
136 The initial data problem Figure
Page 138 and 139:
138 The initial data problem In par
Page 140 and 141:
140 The initial data problem Accord
Page 142 and 143:
142 The initial data problem Remark
Page 144 and 145:
144 The initial data problem Remark
Page 146 and 147:
146 The initial data problem 8.4.1
Page 148 and 149:
148 The initial data problem Since
Page 150 and 151:
150 The initial data problem • fo
Page 152 and 153:
152 Choice of foliation and spatial
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
176 Evolution schemes been used by
Page 178 and 179:
178 Evolution schemes Comparing wit
Page 180 and 181:
180 Evolution schemes Now the ∇-d
Page 182 and 183:
182 Evolution schemes coordinates (
Page 184 and 185:
184 Evolution schemes = 1 2 −∆
Page 186 and 187:
186 Evolution schemes corresponds t
Page 188 and 189:
188 Evolution schemes
Page 190 and 191:
190 Lie derivative Figure A.1: Geom
Page 192 and 193:
192 Lie derivative
Page 194 and 195:
194 Conformal Killing operator and
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
200 BIBLIOGRAPHY [13] A. Anderson a
Page 202 and 203:
202 BIBLIOGRAPHY [43] T.W. Baumgart
Page 204 and 205:
204 BIBLIOGRAPHY [75] M. Campanelli
Page 206 and 207:
206 BIBLIOGRAPHY [105] G. Darmois :
Page 208 and 209:
208 BIBLIOGRAPHY [135] H. Friedrich
Page 210 and 211:
210 BIBLIOGRAPHY [163] J. Isenberg
Page 212 and 213:
212 BIBLIOGRAPHY [195] S. Nissanke
Page 214 and 215:
214 BIBLIOGRAPHY [226] M. Shibata :
Page 216 and 217:
216 BIBLIOGRAPHY [255] K. Taniguchi
Page 218 and 219:
Index 1+log slicing, 161 3+1 formal
Page 220:
220 INDEX Ricci identity, 18 Ricci
show all

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

Create successful ePaper yourself

Delete template?

Save as template?