Part I The General Theory of Relativity
1 Introduction
2 Physics in External Gravitational Fields
2.1 Characteristic Properties of Gravitation
2.1.1 Strength of the Gravitational Interaction
2.1.2 Universality of Free Fall
2.1.3 Equivalence Principle
2.1.4 Gravitational Red- and Blueshifts
2.2 Special Relativity and Gravitation
2.2.1 Gravitational Redshift and Special Relativity
2.2.2 Global Inertial Systems Cannot Be Realized in the Presence of Gravitational Fields
2.2.3 Gravitational Deflection of Light Rays
2.2.4 Theories of Gravity in Flat Spacetime
2.2.5 Exercises
2.3 Spacetime as a Lorentzian Manifold
2.4 Non-gravitational Laws in External Gravitational Fields
2.4.1 Motion of a Test Body in a Gravitational Field
2.4.2 World Lines of Light Rays
2.4.3 Exercises
2.4.4 Energy and Momentum "Conservation" in the Presence of an External Gravitational Field
2.4.5 Exercises
2.4.6 Electrodynamics
2.4.7 Exercises
2.5 The Newtonian Limit
2.5.1 Exercises
2.6 The Redshift in a Stationary Gravitational Field
2.7 Fermat's Principle for Static Gravitational Fields
2.8 Geometric Optics in Gravitational Fields
2.8.1 Exercises
2.9 Stationary and Static Spacetimes
2.9.1 Killing Equation
2.9.2 The Redshift Revisited
2.10 Spin Precession and Fermi Transport
2.10.1 Spin Precession in a Gravitational Field
2.10.2 Thomas Precession
2.10.3 Fermi Transport
2.10.4 The Physical Difference Between Static and Stationary Fields
2.10.5 Spin Rotation in a Stationary Field
2.10.6 Adapted Coordinate Systems for Accelerated Observers .
2.10.7 Motion of a Test Body
2.10.8 Exercises
2.11 General Relativistic Ideal Magnetohydrodynamics
2.11.1 Exercises
3 Einstein's Field Equations
3.1 Physical Meaning of the Curvature Tensor
3.1.1 Comparison with Newtonian Theory
3.1.2 Exercises
3.2 The Gravitational Field Equations
3.2.1 Heuristic "Derivation" of the Field Equations
3.2.2 The Question of Uniqueness
3.2.3 Newtonian Limit, Interpretation of the Constants A and X
3.2.4 On the Cosmological Constant A
3.2.5 The Einstein-Fokker Theory
3.2.6 Exercises
3.3 Lagrangian Formalism
3.3.1 Canonical Measure on a Pseudo-Riemannian Manifold
3.3.2 The Einstein-Hilbert Action
3.3.3 Reduced Bianchi Identity and General Invariance
3.3.4 Energy-Momentum Tensor in a Lagrangian Field Theory
3.3.5 Analogy with Electrodynamics
3.3.6 Meaning of the Equation □ · T = 0
3.3.7 The Equations of Motion and □ · T = 0
3.3.8 Variational Principle for the Coupled System
3.3.9 Exercises
3.4 Non-localizability of the Gravitational Energy
3.5 On Covariance and Invariance
3.5.1 Note on Unimodular Gravity
3.6 The Tetrad Formalism
3.6.1 Variation of Tetrad Fields
3.6.2 The Einstein-Hilbert Action
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Part Ⅱ Applications of General Relativity
Part Ⅲ Differential Geometry
