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Homepage / Education / Master’s course

    Master course “Physics”

    Specialization “Gravitation, cosmology and relativistic astrophysics”

    Internal (full-time) tuition

    Graduates of Master’s course are awarded state diplomas “Master of Science in Physics”




    Curriculum                                                                                                                                                       Information for master course entrants



    Classical theory of gravitation (36 hours). The subject is an introduction to general relativity. Riemannian geometry and tensor analysis (metrics, geodesic curves, covariant derivative, curvature tensor). Gravitational field equations. Energy-momentum tensor. Exact solutions of Einstein-Hilbert’s equations (Schwarzschild and Friedmann solutions). Classical effects of general relativity (perihelion shift, relativistic deflection of light, gravitational red shift).


    General astronomy (36 hours). Spherical astronomy (coordinate systems, time measurement). Solar system (the Sun, planets, satellites). Stars ( structure, spectrum, luminosity). The Galaxy (structure, motion of stars). Extragalactic astronomy (galactic types and their spatial distribution, active galaxy nuclei). Elements of Cosmology (Copernicus principle, model of Hot Universe).


    History and methodology of physics (35 hours). General information. Prehistory of physics. Age of Antiquity. The Middle Ages. The Renaissance. Physics coming into being science. Classical physics. Modern physics. Basic concepts and notions of physics.


    Current problems of physics (12 hours) Advances in classical theory of gravitation (wormholes, multidimensional models, branes, cosmological models with phantom and ekpyrotic matter). Advances in quantum gravity (quantum geometry, superstring theory). Application of hypercomplex numbers to field theory, relativity and quantum mechanics. Advances in quantum mechanics (quantum nonlocality, quantum teleportation).


    Introduction to classical field theory (36 hours). Fundamental fields (scalar, spinor and vector ones). Field equations (Klein-Gordon, Dirac, Maxwell equations). Lagrangian selection criterion. Conservation laws (Noether`s theorem). Global and local gauge invariance. Interacting fields. Abelian and non-Abelian gauge fields. Young-Mills field.


    Mathematician methods in theory of gravity (24 hours). The subject covers the theory of smooth manifolds and aimed at investigating modern gravitational models. Elements of analysis on normalized spaces (sets and mapping, Frechet’s derivative). Smooth manifolds (maps and atlases, category notion). Tangential spaces (differential, vector field). Exterior forms.


    Relativistic astrophysics and cosmology (48 hours). The subject is devoted to the fundamentals of theoretical and relativistic astrophysics, observational and theoretical cosmology. Theory of radiative transfer in stellar atmospheres. Gas nabulae. Background radiation. Final stages of stellar evolution. Extragalactic astronomy. Gravitational waves. Cosmological models. Cosmological scenarios.

    Physical cosmology. Anisotropic cosmological models. Rotation origin problem in astronomy.


    Quantum gravity (48 hours). The subject contains different approaches to quantization in gravity theory and their application to physics of black holes and cosmology. Classification of quantized schemes in gravity (Zelmanov’s cube). Quantum mechanics of a charge in a centrally symmetric gravitational field (nonrelativistic case with regard for DeWitt’s self-force). Electromagnetic and gravitational radiations of graviatoms (quantum systems around primordial black holes). Quantum geometrodynamics (Wheeler-DeWitt’s equation). Quantum cosmology (the Universe is birth as a tunneling). Quantum field theory in curved space-time (Hawking and Unruh effects, particles creation in the Early Universe).


    Multidimensional models in theory of gravitation (36 hours). Smooth manifolds with a metric (metric, covariant derivative, curvature tensor, geodesic equations). Multidimensional cosmologic models (multidimensional generalization of Kasner and de Sitter solutions, accelerated expansion of the Universe and gravitational constant variation). Models with branes (form fields and scalar fields, supergravity).


    Advanced theoretical physics (72 hours). The subject is devoted to analytical mechanics, special relativity, hydrogasdynamics, microscopic electrodynamics, continuum electrodynamics and quantum mechanics, entering into the state examination programme on specialization “Gravitation, cosmology and relativistic astrophysics” and being of value for astronomical applications (celestial mechanics, relativistic astrophysics, quantum cosmology, etc.)  


    Philosophical problems of natural sciences (36 hours). Physical picture of the world as an integral image of  Nature. Conceptual structure of modern physics.


    Computer technologies in science and education. Overview of computer technologies application  to physical research (mathematical  computing, modelling, process management and etc.).  Usage of PC and most popular operating systems (Windows, Linux). PC software aimed at scientific computing. Further perspective of IT technology development.


    Stellar evolution, galaxy dynamics, interstellar medium physics (12 hours). Interstellar medium (cloudy structure, Strömgren zones, gravitational instability). Protostars (Jeans length, Jeans mass, fragmentation under molecular cloud compression, protostar tracks). Stellar equilibrium (hydrostatic equlibrium and energy balance, Emden’s equation, luminosity dependence on the mass, temperature at the star centre). Stellar dynamics (star orbits in Galaxy, statistic distribution of stars, rotation curves, spiral structure).


    Cosmic electrogasdynamics (12 hours). The subject covers fundamental problems of magnetohydrodynamics, gasdynamics and plasma physics of astrophysical  objects. Electrogasdynamical processes in astrophysics (shock waves in supernova explosions, solar flares, cosmic ray acceleration, accretion onto compact astrophysical objects). Magnetohydrodynemics (freezing-in magnetic force lines, forceless field).  Waves in plasma (Alfvén waves, acoustic and magnetoacoustic waves, plasma oscillations).


    Algebra and geometry of space-time (12 hours). Spinor and twistor structures (spinor and isotropic vectors, field equations in spinor formalism). Quaternions and physical geometry (biquaternion theory of relativity, relation between biquaternions and twistors). Quaternion analysis and algebradynamics (Eikonal equation, particles as singularities of the quaternion field).  Concept of complex space-time (complex time and quantum uncertainty).


    Introduction to black hole and wormhole physics (12 hours). Static spherical symmetry. Kruskal metric. Carter-Penrose diagrams. Scalar-tensor theory. Horizons, singularities and wormholes. No-hair and no-go theorems. Phantom fields and regular configurations. Regular black holes.


    The Selection Committee: phone for enquiries 8 915 071 65 14


    All publications of  “Gravitation and Cosmology”   1995-2007  и  2008-2009