3+1 Formalism and Bases of Numerical Relativity

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

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Unformatted text preview: 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 eric.gourgoulhon@obspm.fr 6 March 2007 2 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 4 CONTENTS 3.3.6 Evolution of the orthogonal projector . . . . . . . . . . . . . . . . . . . . ....
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