Questions for the exam

1. Cartan's formalism of representation-valued forms: tensors, tensor densities, pseudotensors. Examples of tensor-valued forms appearing in General Relativity.
2. Covariant differentiation. Linear connections. Torsion and curvature.
3. Covariant differential. Ricci formula.
4. Bianchi identities.
5. Koszul notation and his axioms for the connections. Torsion and curvature in this language.
6. Parallel transport and selfparallels.
7. Geodesics. Christoffel symbols. Levi-Civita connection.
8. Determination of a connection in terms of its torsion and nonmetricity.
9. Riemann tensor and its symmetries.
10. Vanishing of the Riemann tensor. Local form of the metric having vanishing Riemann tensor.
11. Decomposition of the Riemann tensor: the Weyl, the Schouten and the Ricci tensors. Ricci scalar. The numbers of independent components for these objects.
12. Contracted Bianchi identities. The Einstein tensor.
13. Formulation of General Relativity. The minimal coupling principle and the Newtonian limit of equations of motion for a free particle.
14. Einstein's field equations and the Einstein Universe.
15. Linearization of Einstein's equations.
16. Determination of the universal constant on the right hand side of Einstein's equations from the Newtonian limit.
17. Tidal forces and the Jacobi equation.
18. Gravitational waves in linearized theory. Evolution of a ball of dust in the field of a monochromatic gravitational wave.
19. Killing fields and the Killing equation. Maximal group of symmetries and the spaces of constant curvature. DeSitter and anti-DeSitter spaces.
20. Stationary, static and spherically symmetric gravitational fields. The most general form of a spherically symmetric Lorentzian 4-metric.
21. Spherically symmetric gravitational fields in vacuum. The Schwarzschild metric.
22. Maximal analytic extension of the Schwarzschild metric. The Kruskal diagram.
23. Radial free fall in the Schwarzschild spacetime.
24. Equations of motion of a test particle in the Schwarzschild spacetime.
25. Perihelion advance of planets orbiting around Schwarzschild black hole.
26. Deflection of light near a Schwarzschild black hole.
27. Mathematical formulation of spatial homogeneity. Bianchi models.
28. Mathematical fomulation of spatial isotropy. Robertson-Walker metric.
29. Friedman-Lemaitre-Robertson-Walker cosmologies. Friedman equations with and without cosmological constant. Discussion of solutions. Possible topologies.
30. Measurable cosmological parameters: the Hubble constant and Omega.