Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Special relativity
- 2 Vector analysis in special relativity
- 3 Tensor analysis in special relativity
- 4 Perfect fluids in special relativity
- 5 Preface to curvature
- 6 Curved manifolds
- 7 Physics in a curved spacetime
- 8 The Einstein field equations
- 9 Gravitational radiation
- 10 Spherical solutions for stars
- 11 Schwarzschild geometry and black holes
- 12 Cosmology
- Appendix A Summary of linear algebra
- References
- Index
6 - Curved manifolds
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Special relativity
- 2 Vector analysis in special relativity
- 3 Tensor analysis in special relativity
- 4 Perfect fluids in special relativity
- 5 Preface to curvature
- 6 Curved manifolds
- 7 Physics in a curved spacetime
- 8 The Einstein field equations
- 9 Gravitational radiation
- 10 Spherical solutions for stars
- 11 Schwarzschild geometry and black holes
- 12 Cosmology
- Appendix A Summary of linear algebra
- References
- Index
Summary
Differentiable manifolds and tensors
The mathematical concept of a curved space begins (but does not end) with the idea of a manifold. A manifold is essentially a continuous space which looks locally like Euclidean space. To the concept of a manifold is added the idea of curvature itself. The introduction of curvature into a manifold will be the subject of subsequent sections. First we study the idea of a manifold, which we can regard as just a fancy word for ‘space’.
Manifolds
The surface of a sphere is a manifold. So is any m-dimensional ‘hyperplane’ in an n-dimensional Euclidean space (m ≤ n). More abstractly, the set of all rigid rotations of Cartesian coordinates in three-dimensional Euclidean space will be shown below to be a manifold. Basically, a manifold is any set that can be continuously parametrized. The number of independent parameters is the dimension of the manifold, and the parameters themselves are the coordinates of the manifold. Consider the examples just mentioned. The surface of a sphere is ‘parametrized’ by two coordinates θ and φ. The m-dimensional ‘hyperplane’ has m Cartesian coordinates, and the set of all rotations can be parametrized by the three ‘Euler angles’, which in effect give the direction of the axis of rotation (two parameters for this) and the amount of rotation (one parameter). So the set of rotations is a manifold: each point is a particular rotation, and the coordinates are the three parameters. It is a three-dimensional manifold.
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- A First Course in General Relativity , pp. 142 - 170Publisher: Cambridge University PressPrint publication year: 2009