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3+1 Formalism and Bases of Numerical Relativity

3+1 Formalism and Bases of Numerical Relativity - 3 1...

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arXiv:gr-qc/0703035v1 6 Mar 2007 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´ e Paris 7 F-92195 Meudon Cedex, France [email protected] 6 March 2007
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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
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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 Einstein equation in 3+1 form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.1 The Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Shift vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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+1 Einstein equation as a PDE system . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.1 Lie derivatives along m as partial derivatives . . . . . . . . . . . . . . . . 59 4.3.2 3+1 Einstein system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hamiltonian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.3 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.4 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Perfect fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3.1 kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3.2 Baryon number conservation . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.3 Dynamical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3.4 Energy conservation law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.5 Relativistic Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.6 Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.5 3+1 magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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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 Background metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2.3 Conformal metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.4 Conformal connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3
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