## An Introduction to General RelativityThis textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text. |

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### Contents

Introduction | 1 |

Vectors and tensors in flat threespace | 10 |

Aspects of special relativistic geometry | 24 |

Tensor analysis on manifolds | 46 |

Covariant differentiation | 53 |

Properties of Riemann tensor | 57 |

Riemannian geometry | 62 |

The Lie derivative | 67 |

The Schwarzschild solution | 104 |

Gravitational redshift and time dilation | 109 |

The geodesic equation for the Schwarzschild solution | 112 |

Classical tests | 117 |

The extended Schwarzschild solution | 126 |

Black holes and gravitational collapse | 133 |

Interior solutions | 137 |

The Kerr solution | 142 |

### Common terms and phrases

affine parameter assumption Bianchi type black hole calculation commutator components connection conservation constant contravariant vector contravariant vector field coordinate patch cosmology covariant vector curvature curve deduce defined density differentiation Einstein's equations Einstein's theory electromagnetic emitted energy Exercises for chapter expression field equations Figure fluid follows formula four-momentum four-velocity function fundamental observers future-pointing geodesic deviation geodesic equation geometry given gravitational field homogeneous hypersurfaces isometry group isotropic Kerr solution Killing vectors Lie algebra light cone Lorentz manifold mass matrix Maxwell's metric Minkowski motion notation null geodesics obtain orbit orthogonal overlap regions particle perihelion photon Rabcd radial radius red-shift relation relativistic Ricci tensor Riemann tensor Riemannian satisfies scalar field Schwarzschild solution Show sin2 singularity space space-time special relativity spherically symmetric star Suppose surface tangent vector tensor field theorem time-like torsion-free transformation transition vacuum vanishes vector field velocity Weyl tensor zero