An Introduction to General Relativity

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Cambridge University Press, 1990 - Mathematics - 183 pages
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This 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

Geodesics
71
Geodesic deviation
76
Differential forms
85
The transition from Newtonian theory
92
Einsteins field equations
96
The slowmotion approximation
101
Homogeneous and isotropic threespaces
151
Cosmology kinematics
155
Cosmology dynamics
163
Anisotropic cosmologies
170
Index
180
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About the author (1990)

Hughston, Merrill Lynch International, London, and King's College, London.

Tod, University of Oxford.

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