Introduction to General Relativity and Cosmology 1st Edition by J Plebanski, A Krasinski ISBN 052185623X 9780521856232
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ISBN 10: 052185623X
ISBN 13: 9780521856232
Author: J Plebanski, A Krasinski
Introduction to General Relativity and Cosmology 1st Table of contents:
1. How the theory of relativity came into being (a brief historical sketch)
1.1 Special versus general relativity
1.2 Space and inertia in Newtonian physics
1.3 Newton’s theory and the orbits of planets
1.4 The basic assumptions of general relativity
Part I: Elements of Differential Geometry
2. A short sketch of 2-dimensional differential geometry
2.1 Constructing parallel straight lines in a flat space
2.2 Generalisation of the notion of parallelism to curved surfaces
3. Tensors, tensor densities
3.1 What are tensors good for?
3.2 Differentiable manifolds
3.3 Scalars
3.4 Contravariant vectors
3.5 Covariant vectors
3.6 Tensors of second rank
3.7 Tensor densities
3.8 Tensor densities of arbitrary rank
3.9 Algebraic properties of tensor densities
3.10 Mappings between manifolds
3.11 The Levi-Civita symbol
3.12 Multidimensional Kronecker deltas
3.13 Examples of applications of the Levi-Civita symbol and multidimensional Kronecker delta
3.14 Exercises
4. Covariant derivatives
4.1 Differentiation of tensors
4.2 Axioms of the covariant derivative
4.3 A field of bases on a manifold and scalar components of tensors
4.4 The affine connection
4.5 The explicit formula for the covariant derivative of tensor density fields
4.6 Exercises
5. Parallel transport and geodesic lines
5.1 Parallel transport
5.2 Geodesic lines
5.3 Exercises
6. The curvature of a manifold; flat manifolds
6.1 The commutator of second covariant derivatives
6.2 The commutator of directional covariant derivatives
6.3 The relation between curvature and parallel transport
6.4 Covariantly constant fields of vector bases
6.5 A torsion-free flat manifold
6.6 Parallel transport in a flat manifold
6.7 Geodesic deviation
6.8 Algebraic and differential identities obeyed by the curvature tensor
6.9 Exercises
7. Riemannian geometry
7.1 The metric tensor
7.2 Riemann spaces
7.3 The signature of a metric, degenerate metrics
7.4 Christoffel symbols
7.5 The curvature of a Riemann space
7.6 Flat Riemann spaces
7.7 Subspaces of a Riemann space
7.8 Flat Riemann spaces that are globally non-Euclidean
7.9 The Riemann curvature versus the normal curvature of a surface
7.10 The geodesic line as the line of extremal distance
7.11 Mappings between Riemann spaces
7.12 Conformally related Riemann spaces
7.13 Conformal curvature
7.14 Timelike, null and spacelike intervals in a 4-dimensional spacetime
7.15 Embeddings of Riemann spaces in Riemann spaces of higher dimension
7.16 The Petrov classification
7.17 Exercises
8. Symmetries of Riemann spaces, invariance of tensors
8.1 Symmetry transformations
8.2 The Killing equations
8.3 The connection between generators and the invariance transformations
8.4 Finding the Killing vector fields
8.5 Invariance of other tensor fields
8.6 The Lie derivative
8.7 The algebra of Killing vector fields
8.8 Surface-forming vector fields
8.9 Spherically symmetric 4-dimensional Riemann spaces
8.10 * Conformal Killing fields and their finite basis
8.11 * The maximal dimension of an invariance group
8.12 Exercises
9. Methods to calculate the curvature quickly – Cartan forms and algebraic computer programs
9.1 The basis of differential forms
9.2 The connection forms
9.3 The Riemann tensor
9.4 Using computers to calculate the curvature
9.5 Exercises
10. The spatially homogeneous Bianchi type spacetimes
10.1 The Bianchi classification of 3-dimensional Lie algebras
10.2 The dimension of the group versus the dimension of the orbit
10.3 Action of a group on a manifold
10.4 Groups acting transitively, homogeneous spaces
10.5 Invariant vector fields
10.6 The metrics of the Bianchi-type spacetimes
10.7 The isotropic Bianchi-type (Robertson–Walker) spacetimes
10.8 Exercises
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Tags: J Plebanski, A Krasinski, General Relativity