Book contents
- Frontmatter
- Contents
- The scope of this text
- Preface to the second edition
- Acknowledgments
- 1 How the theory of relativity came into being (a brief historical sketch)
- Part I Elements of differential geometry
- 2 A short sketch of 2-dimensional differential geometry
- 3 Tensors, tensor densities
- 4 Covariant derivatives
- 5 Parallel transport and geodesic lines
- 6 The curvature of a manifold; at manifolds
- 7 Riemannian geometry
- 8 Symmetries of Riemann spaces, invariance of tensors
- 9 Methods to calculate the curvature quickly: differential forms and algebraic computer programs
- 10 The spatially homogeneous Bianchi-type spacetimes
- 11 * The Petrov classication by the spinor method
- Part II The theory of gravitation
- References
- Index
3 - Tensors, tensor densities
from Part I - Elements of differential geometry
Published online by Cambridge University Press: 30 May 2024
Book contents
- Frontmatter
- Contents
- The scope of this text
- Preface to the second edition
- Acknowledgments
- 1 How the theory of relativity came into being (a brief historical sketch)
- Part I Elements of differential geometry
- 2 A short sketch of 2-dimensional differential geometry
- 3 Tensors, tensor densities
- 4 Covariant derivatives
- 5 Parallel transport and geodesic lines
- 6 The curvature of a manifold; at manifolds
- 7 Riemannian geometry
- 8 Symmetries of Riemann spaces, invariance of tensors
- 9 Methods to calculate the curvature quickly: differential forms and algebraic computer programs
- 10 The spatially homogeneous Bianchi-type spacetimes
- 11 * The Petrov classication by the spinor method
- Part II The theory of gravitation
- References
- Index
Summary
Tensors and tensor densities of arbitrary rank on arbitrary differentiable manifolds are defined and described. The difference between covariant and contravariant vectors is explained and illustrated with examples. Mappings between manifolds and the associated mappings of tensors are discussed, and it is shown that coordinate transformations are examples of such mappings. The Levi-Civita symbols and multidimensional Kronecker deltas are defined, and their usefulness in calculations involving determinants is demonstrated.
Keywords
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- An Introduction to General Relativity and Cosmology , pp. 12 - 23Publisher: Cambridge University PressPrint publication year: 2024