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
11 - * The Petrov classication by the spinor method
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
Spinors are defined, their basic properties and relation to tensors are derived. The spinor image of the Weyl tensor is derived and it is shown that it is symmetric in all four of its spinor indices. From this, the classification of Weyl tensors equivalent to Petrov’s (by the Penrose method) is derived. The equivalence of these two approaches is proved. The third (Debever’s) method of classification of Weyl tensors is derived, and its equivalence to those of Petrov and Penrose is demonstrated. Extended hints for verifying the calculations (moved to the exercises section) are provided.
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- An Introduction to General Relativity and Cosmology , pp. 108 - 120Publisher: Cambridge University PressPrint publication year: 2024