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
9 - Methods to calculate the curvature quickly: differential forms and algebraic computer programs
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
It is shown that the Riemann tensor can be calculated in a simpler way when the metric is represented by a basis of differential forms. The formulae for the basis components of the Christoffel symbols (called Ricci rotation coefficients) and of the Riemann tensor are derived. A still-easier way to calculate the Riemann tensor, by using algebraic computer programs, is briefly advertised.
- Type
- Chapter
- Information
- An Introduction to General Relativity and Cosmology , pp. 89 - 93Publisher: Cambridge University PressPrint publication year: 2024