General Relativity Trimester Institut Henri Poincar´e http://www.luth.obspm.fr/IHP06/ Lecture notes 3+1 Formalism and Numerical Relativity Eric´ Gourgoulhon Laboratoire de l’Univers et de ses Th´eories, UMR 8102 du C.N.R.S., Observatoire de Paris, F-92195 Meudon Cedex, France [email protected] 22 November 2006 2 Contents 1 Geometry of hypersurfaces 7 1.1 Introduction ....................................... 7 1.2 Framework and notations ............................... 7 1.2.1 Spacetime and tensor fields .......................... 7 1.2.2 Scalar products and metric duality ...................... 8 1.2.3 Curvature tensor ................................ 10 1.3 Hypersurface embedded in spacetime ......................... 11 1.3.1 Definition .................................... 11 1.3.2 Normal vector ................................. 13 1.3.3 Intrinsic curvature ............................... 14 1.3.4 Extrinsic curvature ............................... 14 1.3.5 Examples: surfaces embedded in the Euclidean space R3 .......... 16 1.4 Spacelike hypersurface ................................. 20 1.4.1 The orthogonal projector ........................... 20 1.4.2 Relation between K and ∇n ......................... 22 1.4.3 Links between the ∇ and D connections ................... 23 1.5 Gauss-Codazzi relations ................................ 24 1.5.1 Gauss relation ................................. 25 1.5.2 Codazzi relation ................................ 27 2 Geometry of foliations 29 2.1 Introduction ....................................... 29 2.2 Globally hyperbolic spacetimes and foliations .................... 29 2.2.1 Globally hyperbolic spacetimes ........................ 29 2.2.2 Definition of a foliation ............................ 30 2.3 Foliation kinematics .................................. 30 2.3.1 Lapse function ................................. 30 2.3.2 Normal evolution vector ............................ 31 2.3.3 Eulerian observers ............................... 32 2.3.4 Gradients of n and m ............................. 34 2.3.5 Evolution of the 3-metric ........................... 34 2.3.6 Evolution of the orthogonal projector .................... 35 2.4 Last part of the 3+1 decomposition of the Riemann tensor ............. 36 2.4.1 Last non trivial projection of the spacetime Riemann tensor ........ 36 2.4.2 3+1 expression of the spacetime scalar curvature .............. 38 4 CONTENTS 3 3+1 decomposition of Einstein equation 39 3.1 Einstein equation in 3+1 form ............................. 39 3.1.1 The Einstein equation ............................. 39 3.1.2 3+1 decomposition of the stress-energy tensor ................ 40 3.1.3 Projection of the Einstein equation ...................... 41 3.2 Coordinates adapted to the foliation ......................... 42 3.2.1 Definition of the adapted coordinates ..................... 42 3.2.2 Shift vector ................................... 43 3.2.3 3+1 writing of the metric components .................... 44 3.2.4 Choice of coordinates via the lapse and the shift .............. 46 3.3 3+1 Einstein equation as a PDE system ....................... 46 3.3.1 Lie derivatives along m as partial derivatives ................ 46 3.3.2 3+1 Einstein system .............................. 47 3.4 The Cauchy problem .................................. 48 3.4.1 General relativity as a three-dimensional dynamical system ........ 48 3.4.2 Analysis within Gaussian normal coordinates ................ 49 3.4.3 Constraint equations .............................. 51 3.4.4 Existence and uniqueness of solutions for the Cauchy problem ....... 51 3.5 ADM Hamiltonian formulation ............................ 52 3.5.1 3+1 form of the Hilbert action ........................ 53 3.5.2 Hamiltonian approach ............................. 54 4 3+1 equations for matter and electromagnetic field 57 4.1 Introduction ....................................... 57 4.2 Energy and momentum conservation ......................... 57 4.2.1 3+1 decomposition of the 4-dimensional equation .............. 57 4.2.2 Energy conservation .............................. 58 4.2.3 Newtonian limit ................................ 59 4.2.4 Momentum conservation ............................ 60 4.3 Perfect fluid ....................................... 61 4.3.1 kinematics .................................... 61 4.3.2 Baryon number conservation ......................... 64 4.3.3 Dynamical quantities .............................. 65 4.3.4 Energy conservation law ............................ 66 4.3.5 Relativistic Euler equation .......................... 66 4.3.6 Further developments ............................. 67 4.4 Electromagnetic field .................................. 67 4.5 3+1 magnetohydrodynamics .............................. 67 5 Conformal decomposition 69 5.1 Introduction ....................................... 69 5.2 Conformal decomposition of the 3-metric ...................... 70 5.2.1 Unit-determinant conformal “metric” .................... 70 5.2.2 Background metric ............................... 71 5.2.3 Conformal metric ................................ 72 5.2.4 Conformal connection ............................. 73 5.3 Expression of the Ricci tensor ............................. 75 CONTENTS 5 5.3.1 General formula relating the two Ricci tensors ............... 75 5.3.2 Expression in terms of the conformal factor ................. 76 5.3.3 Formula for the scalar curvature ....................... 76 5.4 Conformal decomposition of the extrinsic curvature ................. 77 5.4.1 Traceless decomposition ............................ 77 5.4.2 Conformal decomposition of the traceless part ................ 77 5.5 Conformal form of the 3+1 Einstein system ..................... 80 5.5.1 Dynamical part of Einstein equation ..................... 80 5.5.2 Hamiltonian constraint ............................ 82 5.5.3 Momentum constraint ............................. 83 5.5.4 Summary: conformal 3+1 Einstein system .................. 83 5.6 Isenberg-Wilson-Mathews approximation to General Relativity .......... 84 6 Asymptotic flatness and global quantities 87 6.1 Introduction ....................................... 87 6.2 Asymptotic flatness ................................... 87 6.2.1 Definition .................................... 87 6.2.2 Asymptotic coordinate freedom ........................ 88 6.3 ADM mass ....................................... 89 6.3.1 Definition from the Hamiltonian formulation of GR ............. 89 6.3.2 Expression in terms of the conformal decomposition ............ 92 6.3.3 Newtonian limit ................................ 94 6.3.4 Positive energy theorem ............................ 94 6.3.5 Constancy of the ADM mass ......................... 95 6.4 ADM momentum .................................... 95 6.4.1 Definition .................................... 95 6.4.2 ADM 4-momentum ............................... 96 6.5 Angular momentum .................................. 96 6.5.1 The supertranslation ambiguity ........................ 96 6.5.2 The “cure” ................................... 97 6.5.3 ADM mass in the quasi-isotropic gauge ................... 98 6.6 Komar mass and angular momentum ......................... 99 6.6.1 Komar mass .................................. 99 6.6.2 3+1 expression of the Komar mass and link with the ADM mass ..... 102 6.6.3 Komar angular momentum .......................... 104 7 The initial data problem 107 7.1 Introduction ....................................... 107 7.2 Conformal transverse-traceless method ........................ 108 7.2.1 The starting point ............................... 108 7.2.2 Longitudinal/transverse decomposition of Aˆij ................ 109 7.2.3 Conformal transverse-traceless form of the constraints ........... 110 7.2.4 Decoupling on hypersurfaces of constant mean curvature .......... 111 7.2.5 Conformally flat and momentarily static initial data ............ 112 7.2.6 Bowen-York initial data ............................ 116 7.3 Conformal thin sandwich method ........................... 116 7.4 Unifying all conformal methods: the weighted conformal decomposition ..... 116 6 CONTENTS 8 Time evolution 117 9 Treatment of black holes 119 A Lie derivative 121 A.1 Lie derivative of a vector field ............................. 121 A.1.1 Introduction .................................. 121 A.1.2 Definition .................................... 121 A.2 Generalization to any tensor field ........................... 123 B Conformal Killing operator and conformal vector Laplacian 125 B.1 Conformal Killing operator .............................. 125 B.1.1 Definition .................................... 125 B.1.2 Behavior under conformal transformations .................. 126 B.1.3 Conformal Killing vectors ........................... 126 B.2 Conformal vector Laplacian .............................. 127 B.2.1 Definition .................................... 127 B.2.2 Elliptic character ................................ 127 B.2.3 Kernel ...................................... 128 B.2.4 Solutions to the conformal vector Poisson equation ............. 129 Chapter 1 Geometry of hypersurfaces Date: 2006-11-22 12:06:40 +0100 (mer, 22 nov 2006) Contents 1.1 Introduction ................................. 7 1.2 Framework and notations ......................... 7 1.3 Hypersurface embedded in spacetime .................. 11 1.4 Spacelike hypersurface ........................... 20 1.5 Gauss-Codazzi relations .......................... 24 1.1 Introduction The notion of hypersurface