General Relativity for Physicists
Book: Landau and Lifshitz "The Classical Theory of Fields" plus a few modern extensions
1. Kinematics (space tells matter how to move - a particle in a gravitational field)
1.0. The equivalence principle (motivation for describing the gravitational field as a space-time geometry)
1.1. Curved coordinates (basic mathematical tools, vectors, one-forms, tensors etc, introduction of the notation)
1.2. Trajectories in gravitational field (principle of least action leads to geodesics, mathematics of geodesics, covariant derivative, parallel transport, Newtonian limit)
1.3. Electromagnetic field (interpretation of geometry as a medium, transformation optics)
1.4. Quantum fields (Unruh effect, EPR correlations in the quantum vacuum)
2. Dynamics (matter tells space how to curve - Einstein's equations)
2.0. Curvature (intrinsic, extrinsic)
2.1. Riemann's curvature tensor (geometrical definition from parallel transport along closed loops, Foucault's pendulum, geodesic deviation and tidal forces, mathematical properties of the curvature tensor, Ricci tensor and curvature scalar)
2.2. The action of the gravitational field (motivation for Einstein-Hilbert action)
2.3. Energy-momentum tensor (derivation of the energy-momentum tensor from the principle of least action, examples electromagnetic field and astro dust)
2.4. Einstein's equations (derivation from Einstein-Hilbert action)
2.5. Energy-momentum conservation (proof from Einstein's equations, consistency with equations of motions)
3. Examples
3.0. Newtonian gravity (Einstein's equations in the limit of weak fields)
3.1. Post-Newtonian corrections
3.2. Gravitational waves (Maxwellian limit, small perturbations of the metric, properties of gravitational waves, experiments for detection)
3.3. Spherical symmetric gravitational field (Schwarzschild solution in several coordinate systems, black hole)
3.4. Planetary motion (solution of Hamilton-Jacobi equation, Newtonian limit: Kepler's ellipses, perihelion precession, gravitational lensing, fall into black hole)
3.5 Hawking radiation
3.6. Cosmology (Robertson-Walker metric, Friedmann equations, dark energy)