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

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102 Conformal decomposition Remark : In the original article [160], Isenberg has derived the system (6.129)-(6.131) from a variational principle based on the Hilbert action (4.95), by restricting γij to take the form (6.133) and requiring that the momentum conjugate to Ψ vanishes. That the IWM scheme constitutes some approximation to general relativity is clear because the solutions (N,Ψ,β i ) to the IWM system (6.129)-(6.131) do not in general satisfy the remaining equations of the full conformal 3+1 Einstein system, i.e. Eqs. (6.117) and (6.120). However, the IWM approximation • is exact for spherically symmetric spacetimes (the Cotton tensor vanishes for any spherically symmetric (Σt,γ)), as shown for the Schwarzschild spacetime in the example given in Sec. 6.2.3; • is very accurate for axisymmetric rotating neutron stars; [97] • is correct at the 1-PN order in the post-Newtonian expansion of general relativity. The IWM approximation has been widely used in relativistic astrophysics, to compute binary neutron star mergers [186, 121, 198] gravitational collapses of stellar cores [112, 113, 114, 215, 216], as well as quasi-equilibrium configurations of binary neutron stars or binary black holes (cf. Sec. 8.4).
Chapter 7 Asymptotic flatness and global quantities Contents 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2 Asymptotic flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.3 ADM mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.4 ADM momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.5 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.6 Komar mass and angular momentum . . . . . . . . . . . . . . . . . . 116 7.1 Introduction In this Chapter, we review the global quantities that one may associate to the spacetime (M,g) or to each slice Σt of the 3+1 foliation. This encompasses various notions of mass, linear momentum and angular momentum. In the absence of any symmetry, all these global quantities are defined only for asymptotically flat spacetimes. So we shall start by defining the notion of asymptotic flatness. 7.2 Asymptotic flatness The concept of asymptotic flatness applies to stellar type objects, modeled as if they were alone in an otherwise empty universe (the so-called isolated bodies). Of course, most cosmological spacetimes are not asymptotically flat.
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arXiv:gr-qc/0703035v1 6 Mar 2007 3+
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4 CONTENTS 3.3.6 Evolution of the o
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6 CONTENTS 8 The initial data probl
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8 CONTENTS
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10 CONTENTS
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12 Introduction 3D (no symmetry at
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14 Introduction
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16 Geometry of hypersurfaces in two
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18 Geometry of hypersurfaces 2.2.3
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20 Geometry of hypersurfaces Figure
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22 Geometry of hypersurfaces •
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24 Geometry of hypersurfaces The ei
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26 Geometry of hypersurfaces Figure
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28 Geometry of hypersurfaces The no
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30 Geometry of hypersurfaces Since
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32 Geometry of hypersurfaces 2.4.3
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34 Geometry of hypersurfaces 2.5 Ga
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36 Geometry of hypersurfaces Exampl
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38 Geometry of hypersurfaces
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40 Geometry of foliations Figure 3.
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42 Geometry of foliations 3.3.2 Nor
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44 Geometry of foliations means Eq.
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46 Geometry of foliations Remark :
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48 Geometry of foliations Note that
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50 Geometry of foliations
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Page 104 and 105: 104 Asymptotic flatness and global
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Page 146 and 147: 146 The initial data problem 8.4.1
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152 Choice of foliation and spatial
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176 Evolution schemes been used by
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178 Evolution schemes Comparing wit
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180 Evolution schemes Now the ∇-d
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182 Evolution schemes coordinates (
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184 Evolution schemes = 1 2 −∆
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186 Evolution schemes corresponds t
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188 Evolution schemes
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190 Lie derivative Figure A.1: Geom
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192 Lie derivative
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194 Conformal Killing operator and
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200 BIBLIOGRAPHY [13] A. Anderson a
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202 BIBLIOGRAPHY [43] T.W. Baumgart
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204 BIBLIOGRAPHY [75] M. Campanelli
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206 BIBLIOGRAPHY [105] G. Darmois :
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208 BIBLIOGRAPHY [135] H. Friedrich
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210 BIBLIOGRAPHY [163] J. Isenberg
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212 BIBLIOGRAPHY [195] S. Nissanke
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214 BIBLIOGRAPHY [226] M. Shibata :
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216 BIBLIOGRAPHY [255] K. Taniguchi
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Index 1+log slicing, 161 3+1 formal
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220 INDEX Ricci identity, 18 Ricci
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