Introduction to General Relativity. Pensum.

1. Curvilinear coordinates
   1. Einstein's equivalence principle
   2. Curvilinear coordinates. Covariant differentiation and Christoffel symbols DA^a=dA^a+Γ^a_bcA^bdx^c. Geodesic as a "no-acceleration" trajectory Du^a=0.
   3. Metric tensor ds²=g_abdx^adx^b. Geodesics from the variational principle δ∫ds=0. Connection between Christoffel symbols and the metric tensor Γ_abc=¹/₂(g_ab,c−g_bc,d+g_cd,b).
   4. Motion of a particle in a gravitational field. Maxwell equations in the presence of a gravitaional field F^ab_;a=4πj^b. Motion of a particle in the presence of both gravitational and electromagnetic field, [(Du^a)/(ds)]=F^abu_b.
2. Gravitational field equations
   1. Riemann curvature tensor R^a_bcd. Properties of the curvature tensor.
   2. Invariant volume element √{−g}dΩ.
   3. Hilbert's action for the gravitation field S=−[1/(2κ)]∫R√{−g}dΩ.
   4. Energy-momentum tensor T_ab of the matter.
   5. Einstein's equations for the gravitational field R_ab−¹/₂Rg_ab=κT_ab.
3. Solutions of the field equations
   1. Newton's limit of the Einstein's equations.
   2. Weak gravitational waves.
   3. Centrally symmetric field. Schwarzschild metric.
   4. Gravitational collapse. Lemaitre metric.
   5. Mercury perihelon advance. Bending of light. Gravitational red shift.
4. Relativistic cosmology
   1. Uniform and isotropic universe and its geometry.
   2. Friedman's equation. Friedman-Lemaitre metric.
   3. Red shift and the Hubble constant.