notes

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index

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Catalogue of Spacetimes by Mueller and Grave in 2009

1 Introduction and Notation

1.1 Notation

1.2 General remarks

1.3 Basic objects of a metric

1.4 Natural local tetrad and initial conditions for geodesics

  • 1.4.1 Orthonormality condition
  • 1.4.2 Tetrad transformations
  • 1.4.3 Ricci rotation-, connection-, and structure coefficients
  • 1.4.4 Riemann-, Ricci-, and Weyl-tensor with respect to a local tetrad
  • 1.4.5 Null or timelike directions
  • 1.4.6 Local tetrad for diagonal metrics
  • 1.4.7 Local tetrad for stationary axisymmetric spacetimes

1.5 Newman-Penrose tetrad and spin-coefficients

1.6 Coordinate relations

  • 1.6.1 Spherical and Cartesian coordinates
  • 1.6.2 Cylindrical and Cartesian coordinates

1.7 Embedding diagram

1.8 Equations of motion and transport equations

  • 1.8.1 Geodesic equation
  • 1.8.2 Fermi-Walker transport
  • 1.8.3 Parallel transport
  • 1.8.4 Euler-Lagrange formalism
  • 1.8.5 Hamilton formalism

1.9 Units

1.10 Tools

  • 1.10.1 Maple/GRTensorII
  • 1.10.2 Mathematica

2 Spacetimes

2.1 Minkowski

  • 2.1.1 Cartesian coordinates
  • 2.1.2 Cylindrical coordinates
  • 2.1.3 Spherical coordinates
  • 2.1.4 Conform-compactified coordinates
  • 2.1.5 Rotating coordinates
  • 2.1.6 Rindler coordinates

2.2 Schwarzschild spacetime

  • 2.2.1 Schwarzschild coordinates
  • 2.2.2 Schwarzschild in pseudo-Cartesian coordinates
  • 2.2.3 Isotropic coordinates
  • 2.2.4 Eddington-Finkelstein
  • 2.2.5 Kruskal-Szekeres
  • 2.2.6 Tortoise coordinates
  • 2.2.7 Painlevé-Gullstrand
  • 2.2.8 Israel coordinates

2.3 Alcubierre Warp

2.4 Barriola-Vilenkin monopol

2.5 Bertotti-Kasner

2.6 Bessel gravitational wave

  • 2.6.1 Cylindrical coordinates
  • 2.6.2 Cartesian coordinates

2.7 Cosmic string in Schwarzschild spacetime

2.8 Ernst spacetime

2.9 Friedman-Robertson-Walker

  • 2.9.1 Form 1
  • 2.9.2 Form 2
  • 2.9.3 Form 3

2.10 Gödel Universe

  • 2.10.1 Cylindrical coordinates
  • 2.10.2 Scaled cylindrical coordinates

2.11 Halilsoy standing wave

2.12 Janis-Newman-Winicour

2.13 Kasner

2.14 Kerr

  • 2.14.1 Boyer-Lindquist coordinates

2.15 Kottler spacetime

2.16 Morris-Thorne

2.17 Oppenheimer-Snyder collapse

  • 2.17.1 Outer metric
  • 2.17.2 Inner metric

2.18 Petrov-Type D – Levi-Civita spacetimes

  • 2.18.1 Case AI
  • 2.18.2 Case AII
  • 2.18.3 Case AIII
  • 2.18.4 Case BI
  • 2.18.5 Case BII
  • 2.18.6 Case BIII
  • 2.18.7 Case C

2.19 Plane gravitational wave

2.20 Reissner-Nordstrøm

2.21 de Sitter spacetime

  • 2.21.1 Standard coordinates
  • 2.21.2 Conformally Einstein coordinates
  • 2.21.3 Conformally flat coordinates
  • 2.21.4 Static coordinates
  • 2.21.5 Lemaître-Robertson form
  • 2.21.6 Cartesian coordinates

2.22 Straight spinning string

2.23 Sultana-Dyer spacetime

2.24 TaubNUT

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