Syllabus
* Introduction.
Theory of gravitation in physical picture of the world. Evolution of ideas on space, time and gravitation. Outline of main starting-points, predictions and applications of general theory of relativity. Reminding the formalism of special relativity (lecture NOFY023).
* Starting principles and their immediate applications.
Principle of equivalence, its various formulations and relevant experiments. Principle of general covariance ('general relativity'). Parallel transport, affine connection and Christoffel symbols. Equation of geodesic and its Newtonian limit. Time dilation and frequency shift in gravitational field, Newtonian limit: static case and orbiting-satellite case. Covariant derivative: introducing covariant and absolute derivative, rewriting the parallel-transport and geodesic equations.
* Curvature.
Riemann curvature tensor, its symmetries, geometrical and physical meaning (non-integrability of parallel transport, equation of geodesic deviation). Bianchi identities. Ricci tensor and curvature scalar.
* Energy-momentum tensor and conservation laws.
Energy-momentum tensor of charged incoherent dust and of (its) EM field. Ideal fluid: conservation laws, Euler equation of motion and equation of continuity; conditions of hydrostatic equilibrium.
* Einstein's gravitational law.
Motivation. Derivation of Einstein equations on the basis of the Riemann-tensor uniqueness, Bianchi identities, conservation laws and Newtonian limit of the theory. The question of cosmological constant. Properties of Einstein equations. Principle of minimal coupling.
* Lie derivative and space-time symmetries.
Vector field and its flow, natural concept of scalar and vector transport and its coordinate expression. Lie derivative and space-time isometries. Basic properties of Killing vector fields.
* Schwarzschild solution of Einstein equations.
Metric of spherically symmetric spacetime, Birkhoff's theorem. Basic features of Schwarzschild metric, Schwarzschild black hole - horizon, singularity. Motion of free spinless test particles in the Schwarzschild field - constants of motion, effective potentials, capture and escape of particles; comparison with Keplerian motion in Newtonian central gravitational field.
* Relativistic cosmology.
Basic observational data on universe as a whole - distribution of mass, Hubble formula, relict radiation, 'big bang'. Description of 'cosmic fluid', homogeneity and isotropy of the universe and introduction of comoving coordinates. Spatial geometry on hypersurfaces of homogeneity and Friedmann-Lemaitre-Robertson-Walker metric. Roles of matter and radiation. Einstein equations and basic cosmological models - qualitative discussion. Friedmann cosmological models. Cosmology in terms of 'Omega-factors'.
Annotation
First semester of the course of general relativity and its applications in astrophysics and cosmology. Introduction to general relativity: principle of equivalence and principle of general covariance, parallel transport and geodesics, gravitational shift of frequency; curvature, energy-momentum tensor and Einstein's gravitational law.
Schwarzschild solution of the Einstein equations, the notion of black hole. Homogeneous and isotropic cosmological models.
For the bachelor study of physics, mainly for students who plan to graduate in theoretical physics or astronomy and astrophysics.