General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015

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- Week 1 -
**General Covariance**

To start with, we recall the basic notions of the Special Theory of Relativity. We explain that Minkwoskian coordinates in flat space-time correspond to inertial observers. Then we continue with transformations to non-inertial reference systems in flat space-t... - Week 2 -
**Covariant differential and Riemann tensor**

We start with the definition of what is tensor in a general curved space-time. Then we define what is connection, parallel transport and covariant differential. We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesi... - Week 3 -
**Einstein-Hilbert action and Einstein equations**

We start with the explanation of how one can define Einstein equations from fundamental principles. Such as general covariance, least action principle and the proper choice of dynamical variables. Namely, the role of the latter in the General Theory of Relativ... - Week 4 -
**Schwarzschild solution**

With this module we start our study of the black hole type solutions. We explain how to solve the Einstein equations in the simplest settings. We find perhaps the most famous solution of these equations, which is referred to as the Schwarzschild black hole. We... - Week 5 -
**Penrose-Carter diagrams**

We start with the definition of the Penrose-Carter diagram for flat space-time. On this example we explain the uses of such diagrams. Then we continue with the definition of the Kruskal-Szekeres coordinates which cover the entire black hole space-time. With th... - Week 6 -
**Classical tests of General Theory of Relativity**

We start with the definition of Killing vectors and integrals of motion, which allow one to provide conserving quantities for a particle motion in Schwarzschild space-time. We derive the explicit geodesic equation for this space-time. This equation provides a ... - Week 7 -
**Interior solution and Kerr's solution**

We start with the definition of the so called perfect fluid energy-momentum tensor and with the description of its properties. We use this tensor to derive the so called interior solution of the Einstein equations, which provides a simple model of a star in th... - Week 8 -
**Collapse into black hole**

We start with the derivation of the Oppenheimer-Snyder solution of the Einstein equations, which describes the collapse of a star into black hole. We derive the Penrose-Carter diagram for this solution. We end up this module with a brief description of the ori... - Week 9 -
**Gravitational waves**

With this module we start our study of gravitational waves. We explain the important difference between energy-momentum conservation laws in the absence and in the presence of the dynamical gravity. We define the gravitational energy-momentum pseudo-tensor. Th... - Week 10 -
**Gravitational radiation**

In this module we show how moving massive bodies create gravitational waves in the linearized approximation. Then we continue with the derivation of the exact shock gravitational wave solutions of the Einstein equations. We describe their properties. To help t... - Week 11 -
**Friedman-Robertson-Walker cosmology**

With this module we start our discussion of the cosmological solutions. We define constant curvature three-dimensional homogeneous spaces. Then we derive Friedman-Robertson-Walker cosmological solutions of the Einstein equations. We describe their properties. ... - Week 12 -
**Cosmological solutions with non-zero cosmological constant**

In this module we derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant. We describe the geometric and causal properties of such space-times and provide their Penrose-Carter diagrams. We ...

**Emil Akhmedov**, Associate Professor

Faculty of Mathematics

Coursera est une entreprise numérique proposant des formation en ligne ouverte à tous fondée par les professeurs d'informatique Andrew Ng et Daphne Koller de l'université Stanford, située à Mountain View, Californie.

Ce qui la différencie le plus des autres plateformes MOOC, c'est qu'elle travaille qu'avec les meilleures universités et organisations mondiales et diffuse leurs contenus sur le web.

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