Overview
 Platform:
 Coursera
 Provider:
 Higher School of Economics
 Length:
 12 weeks
 Effort:
 46 hours a week
 Language:
 English
 Credentials:
 Paid Certificate Available
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: Lectures on General Relativity (lecturer: E.Akhmedov)
Syllabus
General Covariance
To start with, we recall the basic notions of the Special Theory of Relativity. We explain that Minkwoskian coordinates in flat spacetime correspond to inertial observers. Then we continue with transformations to noninertial reference systems in flat spacetime. We show that noninertial observers correspond to curved coordinate systems in flat spacetime. In particular, we describe in grate details Rindler coordinates that correspond to eternally homogeneously accelerating observers. This shows that our Nature allows many different types of metrics, not necessarily coincident with the Euclidian or Minkwoskain ones. We explain what means general covariance. We end up this module with the derivation of the geodesic equation for a general metric from the least action principle. In this equation we define the Christoffel symbols.
Covariant differential and Riemann tensor
We start with the definition of what is tensor in a general curved spacetime. Then we define what is connection, parallel transport and covariant differential. We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. We end up with the definition of the Riemann tensor and the description of its properties. We explain how Riemann tensor allows to distinguish flat spacetime in curved coordinates from curved spacetimes. For this module we provide complementary video to help students to recall properties of tensors in flat spacetime.
EinsteinHilbert 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 Relativity is played by the metric tensor of spacetime. Then we derive the Einstein equations from the least action principle applied to the EinsteinHilbert action. Also we define the energymomentum tensor for matter and show that it obeys a conservation law. We describe the basic generic properties of the Einstein equations. We end up this module with some examples of energymomentum tensors for different sorts of matter fields or bodies and particles.To help understanding this module we provide complementary video with the explanation of the least action principle in the simplest case of the scalar field in flat twodimensional spacetime.
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 formulate the Birkhoff theorem. We end this module with the description of some properties of this Schwarzschild solution. We provide different types of coordinate systems for such a curved spacetime.
PenroseCarter diagrams
We start with the definition of the PenroseCarter diagram for flat spacetime. On this example we explain the uses of such diagrams. Then we continue with the definition of the KruskalSzekeres coordinates which cover the entire black hole spacetime. With the use of these coordinates we define PenroseCarter diagram for the Schwarzschild black hole. This diagram allows us to qualitatively understand the fundamental properties of the black hole.
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 spacetime. We derive the explicit geodesic equation for this spacetime. This equation provides a quantitative explanation of some basic properties of black holes. We use the geodesic equation to explain the precession of the Mercury perihelion and of the light deviation in curved spacetime.
Interior solution and Kerr's solution
We start with the definition of the so called perfect fluid energymomentum 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 the General Theory of Relativity. Then we continue with a brief description of the Kerr solution, which corresponds to the rotating black hole. We end up this module with a brief description of the Cosmic Censorship hypothesis and of the black hole No Hair Theorem.
Collapse into black hole
We start with the derivation of the OppenheimerSnyder solution of the Einstein equations, which describes the collapse of a star into black hole. We derive the PenroseCarter diagram for this solution. We end up this module with a brief description of the origin of the Hawking radiation and of the basic properties of the black hole formation.
Gravitational waves
With this module we start our study of gravitational waves. We explain the important difference between energymomentum conservation laws in the absence and in the presence of the dynamical gravity. We define the gravitational energymomentum pseudotensor. Then we continue with the linearized approximation to the Einstein equations which allows us to clarify the meaning of the pseudotensor. We end up this module with the derivation of the free monochromatic gravitational waves and of their energymomentum pseudotensor. These waves are solutions of the Einstein equations in the linearized approximation.
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 to understand this module we provide two complementary videos. One with the explanations how to perform the averaging over directions in space. And the other video is with the derivation of the retarded Green function.
FriedmanRobertsonWalker cosmology
With this module we start our discussion of the cosmological solutions. We define constant curvature threedimensional homogeneous spaces. Then we derive FriedmanRobertsonWalker cosmological solutions of the Einstein equations. We describe their properties. We end up this module with the derivation of the vacuum homogeneous but anisotropic cosmological Kasner solution.
Cosmological solutions with nonzero cosmological constant
In this module we derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with nonzero cosmological constant. We describe the geometric and causal properties of such spacetimes and provide their PenroseCarter diagrams. We provide coordinate systems which cover various patches of these spacetimes.
Taught by
Emil Akhmedov

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Coursera Introduction into General Theory of Relativity
Higher School of Economics via Coursera
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