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3+1 Formalism and Bases of Numerical Relativity

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3+1 Formalism and Bases of Numerical Relativity 3+1 Formalism and Bases of Numerical Relativity Lecture notes Eric´ Gourgoulhon Laboratoire Univers et Th´eories, UMR 8102 du C.N.R.S., Observatoire de Paris, Universit´eParis 7 arXiv:gr-qc/0703035v1 6 Mar 2007 F-92195 Meudon Cedex, France [email protected] 6 March 2007 2 Contents 1 Introduction 11 2 Geometry of hypersurfaces 15 2.1 Introduction.................................... 15 2.2 Frameworkandnotations . .... 15 2.2.1 Spacetimeandtensorfields . 15 2.2.2 Scalar products and metric duality . ...... 16 2.2.3 Curvaturetensor ............................... 18 2.3 Hypersurfaceembeddedinspacetime . ........ 19 2.3.1 Definition .................................... 19 2.3.2 Normalvector ................................. 21 2.3.3 Intrinsiccurvature . 22 2.3.4 Extrinsiccurvature. 23 2.3.5 Examples: surfaces embedded in the Euclidean space R3 .......... 24 2.4 Spacelikehypersurface . ...... 28 2.4.1 Theorthogonalprojector . 29 2.4.2 Relation between K and n ......................... 31 ∇ 2.4.3 Links between the and D connections. .. .. .. .. .. 32 ∇ 2.5 Gauss-Codazzirelations . ...... 34 2.5.1 Gaussrelation ................................. 34 2.5.2 Codazzirelation ............................... 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 Foliationkinematics .. .. .. .. .. .. .. .. ..... 41 3.3.1 Lapsefunction ................................. 41 3.3.2 Normal evolution vector . 42 3.3.3 Eulerianobservers ............................. 42 3.3.4 Gradients of n and m ............................. 44 3.3.5 Evolution of the 3-metric . 45 4 CONTENTS 3.3.6 Evolution of the orthogonal projector . ....... 46 3.4 Last part of the 3+1 decomposition of the Riemann tensor . ............ 47 3.4.1 Last non trivial projection of the spacetime Riemann tensor. 47 3.4.2 3+1 expression of the spacetime scalar curvature . .......... 48 4 3+1 decomposition of Einstein equation 51 4.1 Einsteinequationin3+1form. ...... 51 4.1.1 TheEinsteinequation . 51 4.1.2 3+1 decomposition of the stress-energy tensor . .......... 52 4.1.3 Projection of the Einstein equation . ....... 53 4.2 Coordinates adapted to the foliation . ......... 54 4.2.1 Definition of the adapted coordinates . ....... 54 4.2.2 Shiftvector ................................... 56 4.2.3 3+1 writing of the metric components . ..... 57 4.2.4 Choice of coordinates via the lapse and the shift . ......... 59 4.3 3+1EinsteinequationasaPDEsystem . ...... 59 4.3.1 Lie derivatives along m as partial derivatives . 59 4.3.2 3+1Einsteinsystem ............................. 60 4.4 TheCauchyproblem................................ 61 4.4.1 General relativity as a three-dimensional dynamical system . 61 4.4.2 Analysis within Gaussian normal coordinates . ......... 61 4.4.3 Constraintequations . 64 4.4.4 Existence and uniqueness of solutions to the Cauchy problem . 64 4.5 ADM Hamiltonian formulation . ..... 65 4.5.1 3+1 form of the Hilbert action . 66 4.5.2 Hamiltonianapproach . 67 5 3+1 equations for matter and electromagnetic field 71 5.1 Introduction.................................... 71 5.2 Energy and momentum conservation . ...... 72 5.2.1 3+1 decomposition of the 4-dimensional equation . .......... 72 5.2.2 Energyconservation . 72 5.2.3 Newtonianlimit ................................ 73 5.2.4 Momentum conservation . 74 5.3 Perfectfluid..................................... 75 5.3.1 kinematics.................................... 75 5.3.2 Baryon number conservation . 78 5.3.3 Dynamicalquantities. 79 5.3.4 Energyconservationlaw . 81 5.3.5 Relativistic Euler equation . ..... 81 5.3.6 Furtherdevelopments . 82 5.4 Electromagneticfield.. .. .. .. .. .. .. .. ..... 82 5.5 3+1magnetohydrodynamics. ..... 82 CONTENTS 5 6 Conformal decomposition 83 6.1 Introduction.................................... 83 6.2 Conformal decomposition of the 3-metric . ......... 85 6.2.1 Unit-determinant conformal “metric” . ....... 85 6.2.2 Backgroundmetric.............................. 85 6.2.3 Conformalmetric............................... 86 6.2.4 Conformalconnection . 88 6.3 ExpressionoftheRiccitensor . ....... 89 6.3.1 General formula relating the two Ricci tensors . ......... 90 6.3.2 Expression in terms of the conformal factor . ........ 90 6.3.3 Formula for the scalar curvature . ..... 91 6.4 Conformal decomposition of the extrinsic curvature . .............. 91 6.4.1 Traceless decomposition . .... 91 6.4.2 Conformal decomposition of the traceless part . .......... 92 6.5 Conformalformofthe3+1Einsteinsystem . ........ 95 6.5.1 Dynamical part of Einstein equation . ...... 95 6.5.2 Hamiltonian constraint . 98 6.5.3 Momentumconstraint . 98 6.5.4 Summary: conformal 3+1 Einstein system . ...... 98 6.6 Isenberg-Wilson-Mathews approximation to General Relativity . 99 7 Asymptotic flatness and global quantities 103 7.1 Introduction.................................... 103 7.2 Asymptoticflatness............................... .... 103 7.2.1 Definition .................................... 104 7.2.2 Asymptotic coordinate freedom . ..... 105 7.3 ADMmass ....................................... 105 7.3.1 Definition from the Hamiltonian formulation of GR . ......... 105 7.3.2 Expression in terms of the conformal decomposition . .......... 108 7.3.3 Newtonianlimit ................................ 110 7.3.4 Positive energy theorem . 111 7.3.5 ConstancyoftheADMmass . 111 7.4 ADMmomentum.................................... 112 7.4.1 Definition .................................... 112 7.4.2 ADM4-momentum............................... 112 7.5 Angularmomentum ................................. 113 7.5.1 The supertranslation ambiguity . ...... 113 7.5.2 The“cure” ................................... 114 7.5.3 ADM mass in the quasi-isotropic gauge . ..... 115 7.6 Komarmassandangularmomentum. .... 116 7.6.1 Komarmass .................................. 116 7.6.2 3+1 expression of the Komar mass and link with the ADM mass ..... 119 7.6.3 Komarangularmomentum . 121 6 CONTENTS 8 The initial data problem 125 8.1 Introduction.................................... 125 8.1.1 Theinitialdataproblem. 125 8.1.2 Conformal decomposition of the constraints . ......... 126 8.2 Conformal transverse-traceless method . ........... 127 8.2.1 Longitudinal/transverse decomposition of Aˆij ................ 127 8.2.2 Conformal transverse-traceless form of the constraints ........... 129 8.2.3 Decoupling on hypersurfaces of constant mean curvature.......... 130 8.2.4 Lichnerowiczequation . 130 8.2.5 Conformally flat and momentarily static initial data . ........... 131 8.2.6 Bowen-York initial data . 136 8.3 Conformalthinsandwichmethod . ...... 139 8.3.1 The original conformal thin sandwich method . ........ 139 8.3.2 Extended conformal thin sandwich method . ...... 141 8.3.3 XCTS at work: static black hole example . ..... 142 8.3.4 Uniquenessofsolutions . 144 8.3.5 ComparingCTT,CTSandXCTS . 144 8.4 Initialdataforbinarysystems. ........ 145 8.4.1 Helicalsymmetry............................... 146 8.4.2 Helical symmetry and IWM approximation . ..... 147 8.4.3 Initial data for orbiting binary black holes . .......... 147 8.4.4 Initial data for orbiting binary neutron stars . .......... 149 8.4.5 Initial data for black hole - neutron star binaries . ........... 150 9 Choice of foliation and spatial coordinates 151 9.1 Introduction.................................... 151 9.2 Choiceoffoliation ............................... .... 152 9.2.1 Geodesicslicing............................... 152 9.2.2 Maximalslicing................................ 153 9.2.3 Harmonicslicing ............................... 159 9.2.4 1+logslicing .................................. 161 9.3 Evolution of spatial coordinates . ......... 162 9.3.1 Normalcoordinates ............................. 162 9.3.2 Minimaldistortion . .. .. .. .. .. .. .. .. 163 9.3.3 Approximate minimal distortion . ..... 167 9.3.4 Gammafreezing ................................ 168 9.3.5 Gammadrivers................................. 170 9.3.6 Otherdynamicalshiftgauges . .... 172 9.4 Full spatial coordinate-fixing choices . ........... 173 9.4.1 Spatial harmonic coordinates . ..... 173 9.4.2 Diracgauge................................... 174 CONTENTS 7 10 Evolution schemes 175 10.1Introduction................................... .... 175 10.2 Constrainedschemes . ..... 175 10.3 Freeevolutionschemes. ...... 176 10.3.1 Definitionandframework . 176 10.3.2 Propagation of the constraints . ...... 176 10.3.3 Constraint-violating modes . ...... 181 10.3.4 Symmetric hyperbolic formulations . ........ 181 10.4BSSNscheme ..................................... 181 10.4.1 Introduction ................................. 181 10.4.2 Expression of the Ricci tensor of the conformal metric........... 181 10.4.3 Reducing the Ricci tensor to a Laplace operator . ......... 184 10.4.4 Thefullscheme................................ 186 10.4.5 Applications ................................. 187 A Lie derivative 189 A.1 Lie derivative of a vector field . ....... 189 A.1.1 Introduction .................................. 189 A.1.2 Definition .................................... 189 A.2 Generalization to any tensor field . ........ 191 B Conformal Killing operator and conformal vector Laplacian 193 B.1 ConformalKillingoperator . ...... 193 B.1.1 Definition .................................... 193 B.1.2 Behavior under conformal transformations . ......... 194 B.1.3 ConformalKillingvectors
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