DOCSLIB.ORG

Exact Solutions of Einstein's Field Equations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Exact Solutions of Einstein's Field Equations Exact Solutions of Einstein’s Field Equations Second Edition HANS STEPHANI Friedrich-Schiller-Universit¨at, Jena DIETRICH KRAMER Friedrich-Schiller-Universit¨at, Jena MALCOLM MACCALLUM Queen Mary, University of London CORNELIUS HOENSELAERS Loughborough University EDUARD HERLT Friedrich-Schiller-Universit¨at, Jena published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, NewYork, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, E. Herlt 2003 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2003 Printed in the United Kingdom at the University Press, Cambridge Typeface Computer Modern 11/13pt System LATEX2ε [tb] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Exact solutions of Einstein’s field equations. – 2nd ed. / H. Stephani ... [et al.]. p. cm. – (Cambridge monographs on mathematical physics) Includes bibliographical references and index. ISBN 0-521-46136-7 1. General relativity (Physics) 2. Gravitational waves. 3. Space and time. 4. Einstein field equations – Numerical solutions. I. Stephani, Hans. II. Series. QC173.6 .E96 2003 530.11–dc21 2002071495 ISBN 0 521 46136 7 hardback Contents Preface xix List of tables xxiii Notation xxvii 1 Introduction 1 1.1 What are exact solutions, and why study them? 1 1.2 The development of the subject 3 1.3 The contents and arrangement of this book 4 1.4 Using this book as a catalogue 7 Part I: General methods 9 2 Differential geometry without a metric 9 2.1 Introduction 9 2.2 Differentiable manifolds 10 2.3 Tangent vectors 12 2.4 One-forms 13 2.5 Tensors 15 2.6 Exterior products and p-forms 17 2.7 The exterior derivative 18 2.8 The Lie derivative 21 2.9 The covariant derivative 23 2.10 The curvature tensor 25 2.11 Fibre bundles 27 vii viii Contents 3 Some topics in Riemannian geometry 30 3.1 Introduction 30 3.2 The metric tensor and tetrads 30 3.3 Calculation of curvature from the metric 34 3.4 Bivectors 35 3.5 Decomposition of the curvature tensor 37 3.6 Spinors 40 3.7 Conformal transformations 43 3.8 Discontinuities and junction conditions 45 4 The Petrov classification 48 4.1 The eigenvalue problem 48 4.2 The Petrov types 49 4.3 Principal null directions and determination of the Petrov types 53 5 Classification of the Ricci tensor and the energy-momentum tensor 57 5.1 The algebraic types of the Ricci tensor 57 5.2 The energy-momentum tensor 60 5.3 The energy conditions 63 5.4 The Rainich conditions 64 5.5 Perfect fluids 65 6 Vector fields 68 6.1 Vector fields and their invariant classification 68 6.1.1 Timelike unit vector fields 70 6.1.2 Geodesic null vector fields 70 6.2 Vector fields and the curvature tensor 72 6.2.1 Timelike unit vector fields 72 6.2.2 Null vector fields 74 7 The Newman–Penrose and related formalisms 75 7.1 The spin coefficients and their transformation laws 75 7.2 The Ricci equations 78 7.3 The Bianchi identities 81 7.4 The GHP calculus 84 7.5 Geodesic null congruences 86 7.6 The Goldberg–Sachs theorem and its generalizations 87 Contents ix 8 Continuous groups of transformations; isometry and homothety groups 91 8.1 Lie groups and Lie algebras 91 8.2 Enumeration of distinct group structures 95 8.3 Transformation groups 97 8.4 Groups of motions 98 8.5 Spaces of constant curvature 101 8.6 Orbits of isometry groups 104 8.6.1 Simply-transitive groups 105 8.6.2 Multiply-transitive groups 106 8.7 Homothety groups 110 9Invariants and the characterization of geometries 112 9.1 Scalar invariants and covariants 113 9.2 The Cartan equivalence method for space-times 116 9.3 Calculating the Cartan scalars 120 9.3.1 Determination of the Petrov and Segre types 120 9.3.2 The remaining steps 124 9.4 Extensions and applications of the Cartan method 125 9.5 Limits of families of space-times 126 10 Generation techniques 129 10.1 Introduction 129 10.2 Lie symmetries of Einstein’s equations 129 10.2.1 Point transformations and their generators 129 10.2.2 How to find the Lie point symmetries of a given differential equation 131 10.2.3 How to use Lie point symmetries: similarity reduction 132 10.3 Symmetries more general than Lie symmetries 134 10.3.1 Contact and Lie–B¨acklund symmetries 134 10.3.2 Generalized and potential symmetries 134 10.4 Prolongation 137 10.4.1 Integral manifolds of differential forms 137 10.4.2 Isovectors, similarity solutions and conservation laws 140 10.4.3 Prolongation structures 141 10.5 Solutions of the linearized equations 145 10.6 B¨acklund transformations 146 10.7 Riemann–Hilbert problems 148 10.8 Harmonic maps 148 10.9 Variational B¨acklund transformations 151 10.10 Hirota’s method 152 x Contents 10.11 Generation methods including perfect fluids 152 10.11.1 Methods using the existence of Killing vectors 152 10.11.2 Conformal transformations 155 Part II: Solutions with groups of motions 157 11 Classification of solutions with isometries or homotheties 157 11.1 The possible space-times with isometries 157 11.2 Isotropy and the curvature tensor 159 11.3 The possible space-times with proper homothetic motions 162 11.4 Summary of solutions with homotheties 167 12 Homogeneous space-times 171 12.1 The possible metrics 171 12.2 Homogeneous vacuum and null Einstein-Maxwell space-times 174 12.3 Homogeneous non-null electromagnetic fields 175 12.4 Homogeneous perfect fluid solutions 177 12.5 Other homogeneous solutions 180 12.6 Summary 181 13 Hypersurface-homogeneous space-times 183 13.1 The possible metrics 183 13.1.1 Metrics with a G6 on V3 183 13.1.2 Metrics with a G4 on V3 183 13.1.3 Metrics with a G3 on V3 187 13.2 Formulations of the field equations 188 13.3 Vacuum, Λ-term and Einstein–Maxwell solutions 194 13.3.1 Solutions with multiply-transitive groups 194 13.3.2 Vacuum spaces with a G3 on V3 196 13.3.3 Einstein spaces with a G3 on V3 199 13.3.4 Einstein–Maxwell solutions with a G3 on V3 201 13.4 Perfect fluid solutions homogeneous on T3 204 13.5 Summary of all metrics with Gr on V3 207 14 Spatially-homogeneous perfect fluid cosmologies 210 14.1 Introduction 210 14.2 Robertson–Walker cosmologies 211 14.3 Cosmologies with a G4 on S3 214 14.4 Cosmologies with a G3 on S3 218 Contents xi 15 Groups G3 on non-null orbits V2. Spherical and plane symmetry 226 15.1 Metric, Killing vectors, and Ricci tensor 226 15.2 Some implications of the existence of an isotropy group I1 228 15.3 Spherical and plane symmetry 229 15.4 Vacuum, Einstein–Maxwell and pure radiation fields 230 15.4.1 Timelike orbits 230 15.4.2 Spacelike orbits 231 15.4.3 Generalized Birkhoff theorem 232 15.4.4 Spherically- and plane-symmetric fields 233 15.5 Dust solutions 235 15.6 Perfect fluid solutions with plane, spherical or pseudospherical symmetry 237 15.6.1 Some basic properties 237 15.6.2 Static solutions 238 15.6.3 Solutions without shear and expansion 238 15.6.4 Expanding solutions without shear 239 15.6.5 Solutions with nonvanishing shear 240 15.7 Plane-symmetric perfect fluid solutions 243 15.7.1 Static solutions 243 15.7.2 Non-static solutions 244 16 Spherically-symmetric perfect fluid solutions 247 16.1 Static solutions 247 16.1.1 Field equations and first integrals 247 16.1.2 Solutions 250 16.2 Non-static solutions 251 16.2.1 The basic equations 251 16.2.2 Expanding solutions without shear 253 16.2.3 Solutions with non-vanishing shear 260 17 Groups G2 and G1 on non-null orbits 264 17.1 Groups G2 on non-null orbits 264 17.1.1 Subdivisions of the groups G2 264 17.1.2 Groups G2I on non-null orbits 265 17.1.3 G2II on non-null orbits 267 17.2 Boost-rotation-symmetric space-times 268 17.3 Group G1 on non-null orbits 271 18 Stationary gravitational fields 275 18.1 The projection formalism 275 xii Contents 18.2 The Ricci tensor on Σ3 277 18.3 Conformal transformation of Σ3 and the field equations 278 18.4 Vacuum and Einstein–Maxwell equations for stationary fields 279 18.5 Geodesic eigenrays 281 18.6 Static fields 283 18.6.1 Definitions 283 18.6.2 Vacuum solutions 284 18.6.3 Electrostatic and magnetostatic Einstein–Maxwell fields 284 18.6.4 Perfect fluid solutions 286 18.7 The conformastationary solutions 287 18.7.1 Conformastationary vacuum solutions 287 18.7.2 Conformastationary Einstein–Maxwell fields 288 18.8 Multipole moments 289 19Stationary axisymmetric fields: basic concepts and field equations 292 19.1 The Killing vectors 292 19.2 Orthogonal surfaces 293 19.3 The metric and the projection formalism 296 19.4 The field equations for stationary axisymmetric Einstein– Maxwell fields 298 19.5 Various forms of the field equations for stationary axisym- metric vacuum fields 299 19.6 Field equations for rotating fluids 302 20 Stationary axisymmetric vacuum solutions 304 20.1 Introduction 304 20.2 Static axisymmetric vacuum solutions (Weyl’s class) 304 20.3 The class of solutions U = U(ω) (Papapetrou’s class) 309 20.4 The class of solutions S = S(A) 310 20.5 The Kerr solution and the Tomimatsu–Sato class 311 20.6 Other solutions 313 20.7 Solutions with factor structure 316 21 Non-empty stationary axisymmetric solutions 319 21.1 Einstein–Maxwell fields 319 21.1.1 Electrostatic and magnetostatic solutions
Load more

Recommended publications
  • A Mathematical Derivation of the General Relativistic Schwarzschild
    A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components .................
    [Show full text]
  • Einstein's Mistakes
    Einstein’s Mistakes Einstein was the greatest genius of the Twentieth Century, but his discoveries were blighted with mistakes. The Human Failing of Genius. 1 PART 1 An evaluation of the man Here, Einstein grows up, his thinking evolves, and many quotations from him are listed. Albert Einstein (1879-1955) Einstein at 14 Einstein at 26 Einstein at 42 3 Albert Einstein (1879-1955) Einstein at age 61 (1940) 4 Albert Einstein (1879-1955) Born in Ulm, Swabian region of Southern Germany. From a Jewish merchant family. Had a sister Maja. Family rejected Jewish customs. Did not inherit any mathematical talent. Inherited stubbornness, Inherited a roguish sense of humor, An inclination to mysticism, And a habit of grüblen or protracted, agonizing “brooding” over whatever was on its mind. Leading to the thought experiment. 5 Portrait in 1947 – age 68, and his habit of agonizing brooding over whatever was on its mind. He was in Princeton, NJ, USA. 6 Einstein the mystic •“Everyone who is seriously involved in pursuit of science becomes convinced that a spirit is manifest in the laws of the universe, one that is vastly superior to that of man..” •“When I assess a theory, I ask myself, if I was God, would I have arranged the universe that way?” •His roguish sense of humor was always there. •When asked what will be his reactions to observational evidence against the bending of light predicted by his general theory of relativity, he said: •”Then I would feel sorry for the Good Lord. The theory is correct anyway.” 7 Einstein: Mathematics •More quotations from Einstein: •“How it is possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” •Questions asked by many people and Einstein: •“Is God a mathematician?” •His conclusion: •“ The Lord is cunning, but not malicious.” 8 Einstein the Stubborn Mystic “What interests me is whether God had any choice in the creation of the world” Some broadcasters expunged the comment from the soundtrack because they thought it was blasphemous.
    [Show full text]
  • Riemannian Geometry and the General Theory of Relativity
    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1967 Riemannian geometry and the general theory of relativity Marie McBride Vanisko The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Vanisko, Marie McBride, "Riemannian geometry and the general theory of relativity" (1967). Graduate Student Theses, Dissertations, & Professional Papers. 8075. https://scholarworks.umt.edu/etd/8075 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. p m TH3 OmERAl THEORY OF RELATIVITY By Marie McBride Vanisko B.A., Carroll College, 1965 Presented in partial fulfillment of the requirements for the degree of Master of Arts UNIVERSITY OF MOKT/JTA 1967 Approved by: Chairman, Board of Examiners D e a ^ Graduante school V AUG 8 1967, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V W Number: EP38876 All rights reserved INFORMATION TO ALL USERS The quality of this reproduotion is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missir^ pages, these will be noted. Also, if matedal had to be removed, a note will indicate the deletion. UMT Oi*MMtion neiitNna UMi EP38876 Published ProQuest LLQ (2013).
    [Show full text]
  • Riemannian Curvature and the Petrov Classification G
    Riemannian Curvature and the Petrov Classification G. S. Hall Department of Mathematics, The Edward Wright Building, Dunbar Street, University of Aber- deen, Aberdeen AB9 2TY, Scotland Z. Naturforsch. 33a, 559-562 (1978); received November 11, 1977 A connection is shown between the Riemannian Curvatures of the wave surfaces of a null vector I at a point p in space-time and the Petrov type of the space-time at p. Some other results on Riemannian Curvature are discussed. 1. Introduction at the point p under consideration. (A full discussion of the Petrov classification and the geometry of Let M be a Lorentzian space-time manifold. If null congruences and wave surfaces can be found p G 31, let TV(M) denote the tangent space to M in the works of Ehlers and Kundt [5], Sachs [6, 7], at p and let I e Tp (M) be a null vector. Because of Pirani [8, 9] and Petrov [10]. The importance of the the identification of radiation in General Relativity Petrov classification in General Relativity in the with null geodesic congruences of curves in M, study of exact solutions of Einstein's equations is one is led to study the geometrical properties of the dealt with thoroughly in Ref. [5] and in a more wave surfaces of I. These are the two dimensional recent paper by Kinnersley [11].) subspaces of TP(M), each member of which is spacelike and orthogonal to I. There is in fact a two parameter family of such wave surfaces for a 2. Auxiliary Results given null vector leTv(M) which, from the physical viewpoint, might be thought of as the The notation will be the usual one.
    [Show full text]
  • Riemannian Geometry Learning for Disease Progression Modelling Maxime Louis, Raphäel Couronné, Igor Koval, Benjamin Charlier, Stanley Durrleman
    Riemannian geometry learning for disease progression modelling Maxime Louis, Raphäel Couronné, Igor Koval, Benjamin Charlier, Stanley Durrleman To cite this version: Maxime Louis, Raphäel Couronné, Igor Koval, Benjamin Charlier, Stanley Durrleman. Riemannian geometry learning for disease progression modelling. 2019. hal-02079820v2 HAL Id: hal-02079820 https://hal.archives-ouvertes.fr/hal-02079820v2 Preprint submitted on 17 Apr 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Riemannian geometry learning for disease progression modelling Maxime Louis1;2, Rapha¨elCouronn´e1;2, Igor Koval1;2, Benjamin Charlier1;3, and Stanley Durrleman1;2 1 Sorbonne Universit´es,UPMC Univ Paris 06, Inserm, CNRS, Institut du cerveau et de la moelle (ICM) 2 Inria Paris, Aramis project-team, 75013, Paris, France 3 Institut Montpelli`erainAlexander Grothendieck, CNRS, Univ. Montpellier Abstract. The analysis of longitudinal trajectories is a longstanding problem in medical imaging which is often tackled in the context of Riemannian geometry: the set of observations is assumed to lie on an a priori known Riemannian manifold. When dealing with high-dimensional or complex data, it is in general not possible to design a Riemannian geometry of relevance. In this paper, we perform Riemannian manifold learning in association with the statistical task of longitudinal trajectory analysis.
    [Show full text]
  • Linearized Einstein Field Equations
    General Relativity Fall 2019 Lecture 15: Linearized Einstein field equations Yacine Ali-Ha¨ımoud October 17th 2019 SUMMARY FROM PREVIOUS LECTURE We are considering nearly flat spacetimes with nearly globally Minkowski coordinates: gµν = ηµν + hµν , with jhµν j 1. Such coordinates are not unique. First, we can make Lorentz transformations and keep a µ ν globally-Minkowski coordinate system, with hµ0ν0 = Λ µ0 Λ ν0 hµν , so that hµν can be seen as a Lorentz tensor µ µ µ ν field on flat spacetime. Second, if we make small changes of coordinates, x ! x − ξ , with j@µξ j 1, the metric perturbation remains small and changes as hµν ! hµν + 2ξ(µ,ν). By analogy with electromagnetism, we can see these small coordinate changes as gauge transformations, leaving the Riemann tensor unchanged at linear order. Since we will linearize the relevant equations, we may work in Fourier space: each Fourier mode satisfies an independent equation. We denote by ~k the wavenumber and by k^ its direction and k its norm. We have decomposed the 10 independent components of the metric perturbation according to their transformation properties under spatial rotations: there are 4 independent \scalar" components, which can be taken, for instance, ^i ^i^j to be h00; k h0i; hii, and k k hij { or any 4 linearly independent combinations thereof. There are 2 independent ilm^ ilm^ ^j transverse \vector" components, each with 2 independent components: klh0m and klhmjk { these are proportional to the curl of h0i and to the curl of the divergence of hij, and are divergenceless (transverse to the ~ TT Fourier wavenumber k).
    [Show full text]
  • An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity
    An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity Leonor Godinho and Jos´eNat´ario Lisbon, 2004 Contents Chapter 1. Differentiable Manifolds 3 1. Topological Manifolds 3 2. Differentiable Manifolds 9 3. Differentiable Maps 13 4. Tangent Space 15 5. Immersions and Embeddings 22 6. Vector Fields 26 7. Lie Groups 33 8. Orientability 45 9. Manifolds with Boundary 48 10. Notes on Chapter 1 51 Chapter 2. Differential Forms 57 1. Tensors 57 2. Tensor Fields 64 3. Differential Forms 66 4. Integration on Manifolds 72 5. Stokes Theorem 75 6. Orientation and Volume Forms 78 7. Notes on Chapter 2 80 Chapter 3. Riemannian Manifolds 87 1. Riemannian Manifolds 87 2. Affine Connections 94 3. Levi-Civita Connection 98 4. Minimizing Properties of Geodesics 104 5. Hopf-Rinow Theorem 111 6. Notes on Chapter 3 114 Chapter 4. Curvature 115 1. Curvature 115 2. Cartan’s Structure Equations 122 3. Gauss-Bonnet Theorem 131 4. Manifolds of Constant Curvature 137 5. Isometric Immersions 144 6. Notes on Chapter 4 150 1 2 CONTENTS Chapter 5. Geometric Mechanics 151 1. Mechanical Systems 151 2. Holonomic Constraints 160 3. Rigid Body 164 4. Non-Holonomic Constraints 177 5. Lagrangian Mechanics 186 6. Hamiltonian Mechanics 194 7. Completely Integrable Systems 203 8. Notes on Chapter 5 209 Chapter 6. Relativity 211 1. Galileo Spacetime 211 2. Special Relativity 213 3. The Cartan Connection 223 4. General Relativity 224 5. The Schwarzschild Solution 229 6. Cosmology 240 7. Causality 245 8. Singularity Theorem 253 9. Notes on Chapter 6 263 Bibliography 265 Index 267 CHAPTER 1 Differentiable Manifolds This chapter introduces the basic notions of differential geometry.
    [Show full text]
  • Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md
    International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 157 ISSN 2229-5518 Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md. Abdul Halim Sajal Saha Md Shafiqul Islam Abstract-In the paper some aspects of Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields are focused. The purpose of this paper is to develop the theory of manifolds equipped with Riemannian metric. I have developed some theorems on torsion and Riemannian curvature tensors using affine connection. A Theorem 1.20 named “Fundamental Theorem of Pseudo-Riemannian Geometry” has been established on Riemannian geometry using tensors with metric. The main tools used in the theorem of pseudo Riemannian are tensors fields defined on a Riemannian manifold. Keywords: Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields. —————————— —————————— I. Introduction (c) { } is a family of open sets which covers , that is, 푖 = . Riemannian manifold is a pair ( , g) consisting of smooth 푈 푀 manifold and Riemannian metric g. A manifold may carry a (d) ⋃ is푈 푖푖 a homeomorphism푀 from onto an open subset of 푀 ′ further structure if it is endowed with a metric tensor, which is a 푖 . 푖 푖 휑 푈 푈 natural generation푀 of the inner product between two vectors in 푛 ℝ to an arbitrary manifold. Riemannian metrics, affine (e) Given and such that , the map = connections,푛 parallel transport, curvature tensors, torsion tensors, ( ( ) killingℝ vector fields and conformal killing vector fields play from푖 푗 ) to 푖 푗 is infinitely푖푗 −1 푈 푈 푈 ∩ 푈 ≠ ∅ 휓 important role to develop the theorem of Riemannian manifolds.
    [Show full text]
  • 3+1 Formalism and Bases of Numerical Relativity
    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 .
    [Show full text]
  • Part 3 Black Holes
    Part 3 Black Holes Harvey Reall Part 3 Black Holes March 13, 2015 ii H.S. Reall Contents Preface vii 1 Spherical stars 1 1.1 Cold stars . .1 1.2 Spherical symmetry . .2 1.3 Time-independence . .3 1.4 Static, spherically symmetric, spacetimes . .4 1.5 Tolman-Oppenheimer-Volkoff equations . .5 1.6 Outside the star: the Schwarzschild solution . .6 1.7 The interior solution . .7 1.8 Maximum mass of a cold star . .8 2 The Schwarzschild black hole 11 2.1 Birkhoff's theorem . 11 2.2 Gravitational redshift . 12 2.3 Geodesics of the Schwarzschild solution . 13 2.4 Eddington-Finkelstein coordinates . 14 2.5 Finkelstein diagram . 17 2.6 Gravitational collapse . 18 2.7 Black hole region . 19 2.8 Detecting black holes . 21 2.9 Orbits around a black hole . 22 2.10 White holes . 24 2.11 The Kruskal extension . 25 2.12 Einstein-Rosen bridge . 28 2.13 Extendibility . 29 2.14 Singularities . 29 3 The initial value problem 33 3.1 Predictability . 33 3.2 The initial value problem in GR . 35 iii CONTENTS 3.3 Asymptotically flat initial data . 38 3.4 Strong cosmic censorship . 38 4 The singularity theorem 41 4.1 Null hypersurfaces . 41 4.2 Geodesic deviation . 43 4.3 Geodesic congruences . 44 4.4 Null geodesic congruences . 45 4.5 Expansion, rotation and shear . 46 4.6 Expansion and shear of a null hypersurface . 47 4.7 Trapped surfaces . 48 4.8 Raychaudhuri's equation . 50 4.9 Energy conditions . 51 4.10 Conjugate points .
    [Show full text]
  • Geometric GSI’19 Science of Information Toulouse, 27Th - 29Th August 2019
    ALEAE GEOMETRIA Geometric GSI’19 Science of Information Toulouse, 27th - 29th August 2019 // Program // GSI’19 Geometric Science of Information On behalf of both the organizing and the scientific committees, it is // Welcome message our great pleasure to welcome all delegates, representatives and participants from around the world to the fourth International SEE from GSI’19 chairmen conference on “Geometric Science of Information” (GSI’19), hosted at ENAC in Toulouse, 27th to 29th August 2019. GSI’19 benefits from scientific sponsor and financial sponsors. The 3-day conference is also organized in the frame of the relations set up between SEE and scientific institutions or academic laboratories: ENAC, Institut Mathématique de Bordeaux, Ecole Polytechnique, Ecole des Mines ParisTech, INRIA, CentraleSupélec, Institut Mathématique de Bordeaux, Sony Computer Science Laboratories. We would like to express all our thanks to the local organizers (ENAC, IMT and CIMI Labex) for hosting this event at the interface between Geometry, Probability and Information Geometry. The GSI conference cycle has been initiated by the Brillouin Seminar Team as soon as 2009. The GSI’19 event has been motivated in the continuity of first initiatives launched in 2013 at Mines PatisTech, consolidated in 2015 at Ecole Polytechnique and opened to new communities in 2017 at Mines ParisTech. We mention that in 2011, we // Frank Nielsen, co-chair Ecole Polytechnique, Palaiseau, France organized an indo-french workshop on “Matrix Information Geometry” Sony Computer Science Laboratories, that yielded an edited book in 2013, and in 2017, collaborate to CIRM Tokyo, Japan seminar in Luminy TGSI’17 “Topoplogical & Geometrical Structures of Information”.
    [Show full text]
  • Equivalence Principle (WEP) of General Relativity Using a New Quantum Gravity Theory Proposed by the Authors Called Electro-Magnetic Quantum Gravity Or EMQG (Ref
    WHAT ARE THE HIDDEN QUANTUM PROCESSES IN EINSTEIN’S WEAK PRINCIPLE OF EQUIVALENCE? Tom Ostoma and Mike Trushyk 48 O’HARA PLACE, Brampton, Ontario, L6Y 3R8 [email protected] Monday April 12, 2000 ACKNOWLEDGMENTS We wish to thank R. Mongrain (P.Eng) for our lengthy conversations on the nature of space, time, light, matter, and CA theory. ABSTRACT We provide a quantum derivation of Einstein’s Weak Equivalence Principle (WEP) of general relativity using a new quantum gravity theory proposed by the authors called Electro-Magnetic Quantum Gravity or EMQG (ref. 1). EMQG is manifestly compatible with Cellular Automata (CA) theory (ref. 2 and 4), and is also based on a new theory of inertia (ref. 5) proposed by R. Haisch, A. Rueda, and H. Puthoff (which we modified and called Quantum Inertia, QI). QI states that classical Newtonian Inertia is a property of matter due to the strictly local electrical force interactions contributed by each of the (electrically charged) elementary particles of the mass with the surrounding (electrically charged) virtual particles (virtual masseons) of the quantum vacuum. The sum of all the tiny electrical forces (photon exchanges with the vacuum particles) originating in each charged elementary particle of the accelerated mass is the source of the total inertial force of a mass which opposes accelerated motion in Newton’s law ‘F = MA’. The well known paradoxes that arise from considerations of accelerated motion (Mach’s principle) are resolved, and Newton’s laws of motion are now understood at the deeper quantum level. We found that gravity also involves the same ‘inertial’ electromagnetic force component that exists in inertial mass.
    [Show full text]