Semi-Riemann Geometry and General Relativity
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TOC
1 The principal curvatures
1.1 Volume of a thickened hypersurface
1.2 The Gauss map and the Weingarten map
1.3 Proof of the volume formula
1.4 Gauss’s theorema egregium
1.4.1 First proof, using inertial coordinates
1.4.2 Second proof. The Brioschi formula
1.5 Problem set – Surfaces of revolution
2 Rules of calculus
2.1 Superalgebras
2.2 Differential forms
2.3 The d operator
2.4 Derivations
2.5 Pullback
2.6 Chain rule
2.7 Lie derivative
2.8 Weil’s formula
2.9 Integration
2.10 Stokes theorem
2.11 Lie derivatives of vector fields
2.12 Jacobi’s identity
2.13 Left invariant forms
2.14 The Maurer Cartan equations
2.15 Restriction to a subgroup
2.16 Frames
2.17 Euclidean frames
2.18 Frames adapted to a submanifold
2.19 Curves and surfaces – their structure equations
2.20 The sphere as an example
2.21 Ribbons
2.22 Developing a ribbon
2.23 Parallel transport along a ribbon
2.24 Surfaces in R3
3 Levi-Civita Connections
3.1 Definition of a linear connection on the tangent bundle
3.2 Christoffel symbols
3.3 Parallel transport
3.4 Geodesics
3.5 Covariant differential
3.6 Torsion
3.7 Curvature
3.8 Isometric connections
3.9 Levi-Civita’s theorem
3.10 Geodesics in orthogonal coordinates
3.11 Curvature identities
3.12 Sectional curvature
3.13 Ricci curvature
3.14 Bi-invariant metrics on a Lie group
3.15 Frame fields
3.16 Curvature tensors in a frame field
3.17 Frame fields and curvature forms
3.18 Cartan’s lemma
3.19 Orthogonal coordinates on a surface
3.20 The curvature of the Schwartzschild metric
3.21 Geodesics of the Schwartzschild metric
4 The bundle of frames
4.1 Connection and curvature forms in a frame field
4.2 Change of frame field
4.3 The bundle of frames
4.4 The connection form in a frame field as a pull-back
4.5 Gauss’ theorems
5 Connections on principal bundles
5.1 Submersions, fibrations, and connections
5.2 Principal bundles and invariant connections
5.3 Covariant differentials and covariant derivatives
6 Gauss’s lemma
6.1 The exponential map
6.2 Normal coordinates
6.3 The Euler field E and its image P
6.4 The normal frame field
6.5 Gauss’ lemma
6.6 Minimization of arc length
7 Special relativity
7.1 Two dimensional Lorentz transformations
7.2 Minkowski space
7.3 Scattering cross-section and mutual flux
8 Die Grundlagen der Physik
8.1 Preliminaries
8.2 Varying the metric and the connection
8.3 The structure of physical laws
8.4 The Hilbert “function”
8.5 Schrodinger’s equation as a passive equation
8.6 Harmonic maps
9 Submersions
9.1 Submersions
9.2 The fundamental tensors of a submersion
9.3 Curvature
9.4 Reductive homogeneous spaces
9.5 Schwarzschild as a warped product
9.6 Robertson Walker metrics
9.6.1 Cosmogeny and eschatology
10 Petrov types
10.1 Algebraic properties of the curvature tensor
10.2 Linear and antilinear maps
10.3 Complex conjugation and real forms
10.4 Structures on tensor products
10.5 Spinors and Minkowski space
10.6 Traceless curvatures
10.7 The polynomial algebra
10.8 Petrov types
10.9 Principal null directions
10.10Kerr-Schild metrics
11 Star
11.1 Definition of the star operator
11.3 The star operator on forms
11.4 Electromagnetism
11.4.1 Electrostatics
11.4.2 Magnetoquasistatics
11.4.3 The London equations
11.4.4 The London equations in relativistic form
11.4.5 Maxwell’s equations
11.4.6 Comparing Maxwell and London
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