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 .