The principle of equivalence: Realization of the principle of equivalence in terms of the four-dimensional metric. Equations of geodesics and their limit for speed much lower than the speed of light. Identification of the 00 component of the metric field with the Newtonian gravitational potential.

Relativistic Kinematics: Synchronizing clocks in curved space-time measurements of distances and times. Space-time symmetries and Killing vectors. The energy-momentum tensor in special and general relativity. The Pound-Rebka experiment and the direct testing of the equivalence principle.

Part II: Einstein's equations and their consequences. Einstein's equations for the gravitational field. Variational formulation and Hilbert-Palatini action. Bianchi identity.

Exact solutions: the study of Einstein's equations in the vacuum in the presence of symmetries and their exact solutions. The spherical symmetry case and the Schwarzschild solution. Axial symmetry and the Kerr solution.

Weak field and gravitational waves: Linearization of Einstein's equations. Solutions of the linearized equations and their interpretation as gravitational waves. " Ownership of "gravitational waves" and experimental methods for their detection. Quadrupole formula to calculate the intensity of the emission of gravitational waves. Laboratory tests: The experimental tests of the classical Einstein's field equations: precession of the perihelion of Mercury; gravitational deflection of light; indirect evidence for gravitational waves from the observations on the Pulsar PSR 1913 +16.

Relativistic cosmology: The Olbers' Paradox. Homogeneous spaces and metric of Friedman-Robertson-Walker. Hubble's law. The Cosmological term of Einstein's equations. Shifting the frequency of the radiation in cosmology and the standard model of the universe. The cosmic expansion and the problem of the missing matter density.

The initial value problem: Einstein's equations in the 3+1 formalism and their Hamiltonian structure. Using the 3 +1 formalism for the numerical solution of Einstein's equation (outline)