3+1 formalism and bases of numerical relativity - LUTh ...
3+1 formalism and bases of numerical relativity - LUTh ... 3+1 formalism and bases of numerical relativity - LUTh ...
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
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 Expression of the Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Conformal form of the 3+1 Einstein system . . . . . . . . . . . . . . . . . . . . . 95 6.5.1 Dynamical part of Einstein equation . . . . . . . . . . . . . . . . . . . . . 95 6.5.2 Hamiltonian constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.5.3 Momentum constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Asymptotic flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.2.2 Asymptotic coordinate freedom . . . . . . . . . . . . . . . . . . . . . . . . 105 7.3 ADM mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3.4 Positive energy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.3.5 Constancy of the ADM mass . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.4 ADM momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4.2 ADM 4-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.5 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Komar mass and angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.6.1 Komar mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.6.2 3+1 expression of the Komar mass and link with the ADM mass . . . . . 119 7.6.3 Komar angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 121
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4 CONTENTS<br />
3.3.6 Evolution <strong>of</strong> the orthogonal projector . . . . . . . . . . . . . . . . . . . . 46<br />
3.4 Last part <strong>of</strong> the <strong>3+1</strong> decomposition <strong>of</strong> the Riemann tensor . . . . . . . . . . . . . 47<br />
3.4.1 Last non trivial projection <strong>of</strong> the spacetime Riemann tensor . . . . . . . . 47<br />
3.4.2 <strong>3+1</strong> expression <strong>of</strong> the spacetime scalar curvature . . . . . . . . . . . . . . 48<br />
4 <strong>3+1</strong> decomposition <strong>of</strong> Einstein equation 51<br />
4.1 Einstein equation in <strong>3+1</strong> form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br />
4.1.1 The Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br />
4.1.2 <strong>3+1</strong> decomposition <strong>of</strong> the stress-energy tensor . . . . . . . . . . . . . . . . 52<br />
4.1.3 Projection <strong>of</strong> the Einstein equation . . . . . . . . . . . . . . . . . . . . . . 53<br />
4.2 Coordinates adapted to the foliation . . . . . . . . . . . . . . . . . . . . . . . . . 54<br />
4.2.1 Definition <strong>of</strong> the adapted coordinates . . . . . . . . . . . . . . . . . . . . . 54<br />
4.2.2 Shift vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56<br />
4.2.3 <strong>3+1</strong> writing <strong>of</strong> the metric components . . . . . . . . . . . . . . . . . . . . 57<br />
4.2.4 Choice <strong>of</strong> coordinates via the lapse <strong>and</strong> the shift . . . . . . . . . . . . . . 59<br />
4.3 <strong>3+1</strong> Einstein equation as a PDE system . . . . . . . . . . . . . . . . . . . . . . . 59<br />
4.3.1 Lie derivatives along m as partial derivatives . . . . . . . . . . . . . . . . 59<br />
4.3.2 <strong>3+1</strong> Einstein system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60<br />
4.4 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61<br />
4.4.1 General <strong>relativity</strong> as a three-dimensional dynamical system . . . . . . . . 61<br />
4.4.2 Analysis within Gaussian normal coordinates . . . . . . . . . . . . . . . . 61<br />
4.4.3 Constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64<br />
4.4.4 Existence <strong>and</strong> uniqueness <strong>of</strong> solutions to the Cauchy problem . . . . . . . 64<br />
4.5 ADM Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65<br />
4.5.1 <strong>3+1</strong> form <strong>of</strong> the Hilbert action . . . . . . . . . . . . . . . . . . . . . . . . 66<br />
4.5.2 Hamiltonian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67<br />
5 <strong>3+1</strong> equations for matter <strong>and</strong> electromagnetic field 71<br />
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
5.2 Energy <strong>and</strong> momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
5.2.1 <strong>3+1</strong> decomposition <strong>of</strong> the 4-dimensional equation . . . . . . . . . . . . . . 72<br />
5.2.2 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
5.2.3 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73<br />
5.2.4 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74<br />
5.3 Perfect fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
5.3.1 kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
5.3.2 Baryon number conservation . . . . . . . . . . . . . . . . . . . . . . . . . 78<br />
5.3.3 Dynamical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79<br />
5.3.4 Energy conservation law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81<br />
5.3.5 Relativistic Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . 81<br />
5.3.6 Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82<br />
5.4 Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82<br />
5.5 <strong>3+1</strong> magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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