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Catalogue of Spacetimes Catalogue of Spacetimes e2 e1 x2 = 2 x = 2 q ∂x2 1 x2 = 1 ∂x1 x1 = 1 x1 = 0 x2 = 0 M Authors: Thomas Müller Visualisierungsinstitut der Universität Stuttgart (VISUS) Allmandring 19, 70569 Stuttgart, Germany [email protected] Frank Grave formerly, Universität Stuttgart, Institut für Theoretische Physik 1 (ITP1) Pfaffenwaldring 57 //IV, 70550 Stuttgart, Germany [email protected] URL: http://go.visus.uni-stuttgart.de/CoS Date: 21. Mai 2014 Co-authors Andreas Lemmer, formerly, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Alcubierre Warp Sebastian Boblest, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart deSitter, Friedmann-Robertson-Walker Felix Beslmeisl, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Petrov-Type D Heiko Munz, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Bessel and plane wave Andreas Wünsch, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Majumdar-Papapetrou, extreme Reissner-Nordstrøm dihole, energy momentum tensor Many thanks to all that have reported bug fixes or added metric descriptions. Contents 1 Introduction and Notation1 1.1 Notation...............................................1 1.2 General remarks...........................................1 1.3 Basic objects of a metric......................................2 1.4 Natural local tetrad and initial conditions for geodesics....................3 1.4.1 Orthonormality condition.................................3 1.4.2 Tetrad transformations...................................4 1.4.3 Ricci rotation-, connection-, and structure coefficients.................4 1.4.4 Riemann-, Ricci-, and Weyl-tensor with respect to a local tetrad...........4 1.4.5 Null or timelike directions.................................5 1.4.6 Local tetrad for diagonal metrics.............................5 1.4.7 Local tetrad for stationary axisymmetric spacetimes..................5 1.5 Newman-Penrose tetrad and spin-coefficients..........................6 1.6 Coordinate relations........................................7 1.6.1 Spherical and Cartesian coordinates...........................7 1.6.2 Cylindrical and Cartesian coordinates..........................7 1.7 Embedding diagram........................................8 1.8 Equations of motion and transport equations..........................9 1.8.1 Geodesic equation.....................................9 1.8.2 Fermi-Walker transport..................................9 1.8.3 Parallel transport......................................9 1.8.4 Euler-Lagrange formalism.................................9 1.8.5 Hamilton formalism.................................... 10 1.9 Special topics............................................ 10 1.9.1 Timelike circular geodesics................................ 10 1.10 Units................................................. 10 1.11 Energy momentum tensor..................................... 11 1.11.1 Energy conditions..................................... 11 1.11.2 Examples for energy momentum tensors........................ 11 1.12 Tools................................................. 13 1.12.1 Maple/GRTensorII..................................... 13 1.12.2 Mathematica........................................ 13 1.12.3 Maxima........................................... 15 1.12.4 Sympy............................................ 16 2 Spacetimes 19 2.1 Minkowski.............................................. 19 2.1.1 Cartesian coordinates................................... 19 2.1.2 Cylindrical coordinates.................................. 19 2.1.3 Spherical coordinates.................................... 20 2.1.4 Conform-compactified coordinates............................ 20 2.1.5 Rotating coordinates.................................... 21 2.1.6 Rindler coordinates..................................... 22 2.2 Schwarzschild spacetime...................................... 23 i ii CONTENTS 2.2.1 Schwarzschild coordinates................................ 23 2.2.2 Schwarzschild in pseudo-Cartesian coordinates.................... 25 2.2.3 Isotropic coordinates.................................... 25 2.2.4 Eddington-Finkelstein................................... 27 2.2.5 Kruskal-Szekeres...................................... 28 2.2.6 Tortoise coordinates.................................... 29 2.2.7 Painlevé-Gullstrand.................................... 30 2.2.8 Israel coordinates...................................... 32 2.3 Alcubierre Warp........................................... 33 2.4 Barriola-Vilenkin monopol..................................... 34 2.5 Bertotti-Kasner........................................... 36 2.6 Bessel gravitational wave..................................... 38 2.6.1 Cylindrical coordinates.................................. 38 2.6.2 Cartesian coordinates................................... 38 2.7 Cosmic string in Schwarzschild spacetime............................ 39 2.8 Einstein-Rosen wave with Weber-Wheeler-Bonnor pulse................... 41 2.9 Ernst spacetime........................................... 42 2.10 Extreme Reissner-Nordstrøm dihole............................... 44 2.11 Friedman-Robertson-Walker.................................... 47 2.11.1 Form 1............................................ 47 2.11.2 Form 2............................................ 48 2.11.3 Form 3............................................ 49 2.12 Gödel Universe........................................... 53 2.12.1 Cylindrical coordinates.................................. 53 2.12.2 Scaled cylindrical coordinates............................... 54 2.13 Halilsoy standing wave...................................... 56 2.14 Janis-Newman-Winicour...................................... 57 2.15 Kasner................................................ 59 2.16 Kastor-Traschen........................................... 60 2.17 Kerr.................................................. 61 2.17.1 Boyer-Lindquist coordinates............................... 61 2.18 Kottler spacetime.......................................... 64 2.19 Majumdar-Papapetrou spacetimes................................ 66 2.20 Melvin universe........................................... 70 2.21 Morris-Thorne............................................ 71 2.22 Oppenheimer-Snyder collapse................................... 73 2.22.1 Outer metric......................................... 73 2.22.2 Inner metric......................................... 74 2.23 Petrov-Type D – Levi-Civita spacetimes............................. 76 2.23.1 Case AI............................................ 76 2.23.2 Case AII........................................... 76 2.23.3 Case AIII........................................... 77 2.23.4 Case BI............................................ 77 2.23.5 Case BII........................................... 78 2.23.6 Case BIII........................................... 78 2.23.7 Case C............................................ 78 2.24 Plane gravitational wave...................................... 81 2.25 Reissner-Nordstrøm........................................ 82 2.26 de Sitter spacetime......................................... 84 2.26.1 Standard coordinates.................................... 84 2.26.2 Conformally Einstein coordinates............................ 84 2.26.3 Conformally flat coordinates............................... 85 2.26.4 Static coordinates...................................... 85 2.26.5 Lemaître-Robertson form................................. 87 2.26.6 Cartesian coordinates................................... 88 CONTENTS iii 2.27 Stationary axisymmetric spacetimes in Weyl Coordinates................... 89 2.28 Straight spinning string...................................... 90 2.29 Sultana-Dyer spacetime...................................... 92 2.30 TaubNUT............................................... 94 Bibliography 95 Chapter 1 Introduction and Notation The Catalogue of Spacetimes is a collection of four-dimensional Lorentzian spacetimes in the context of the General Theory of Relativity (GR). The aim of the catalogue is to give a quick reference for students who need some basic facts of the most well-known spacetimes in GR. For a detailed discussion of a metric, the reader is referred to the standard literature or the original articles. Important resources for exact solutions are the book by Stephani et al[SKM+03] and the book by Griffiths and Podolský[GP09]. Most of the metrics in this catalogue are implemented in the Motion4D-library[MG09] and can be visu- alized using the GeodesicViewer[MG10]. Except for the Minkowski and Schwarzschild spacetimes, the metrics are sorted by their names. 1.1 Notation The notation we use in this catalogue is as follows: Indices: Coordinate indices are represented either by Greek letters or by coordinate names. Tetrad indices are indicated by Latin letters or coordinate names in brackets. Einstein sum convention: When an index appears twice in a single term, once as lower index and once as upper index, we build the sum over all indices: 3 m m zm z ∑ zm z : (1.1.1) ≡ m=0 m Vectors: A coordinate vector in x direction is represented as ¶xm ¶m . For arbitrary vectors, we use ≡ m boldface symbols. Hence, a vector a in coordinate representation reads a = a ¶m . m Derivatives: Partial derivatives are indicated by a comma, ¶y=¶x ¶m y y m , whereas covariant ≡ ≡ ; derivatives are indicated by a semicolon, ∇y = y;m . Symmetrization and Antisymmetrization brackets: 1 1 a b = am
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