Informações sobre o curso
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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Advanced Level

Nível avançado

Clock

Approx. 49 hours to complete

Sugerido: 12 weeks of study, 4-6 hours per week
Comment Dots

English

Legendas: English
Globe

cursos 100% online

Comece imediatamente e aprenda em seu próprio cronograma.
Advanced Level

Nível avançado

Clock

Approx. 49 hours to complete

Sugerido: 12 weeks of study, 4-6 hours per week
Comment Dots

English

Legendas: English

Syllabus - What you will learn from this course

1

Section
Clock
3 hours to complete

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-time. We show that non-inertial observers correspond to curved coordinate systems in flat space-time. 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....
Reading
7 videos (Total 75 min), 1 quiz
Video7 videos
General covariance12m
Сonstant linear acceleration16m
Transition to the homogeneously accelerating reference frame (or system) in Minkowski space–time8m
Transition to the homogeneously accelerating reference frame in Minkowski space–time (part 2)13m
Geodesic equation8m
Christoffel symbols14m

2

Section
Clock
4 hours to complete

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 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 space-time in curved coordinates from curved space-times. For this module we provide complementary video to help students to recall properties of tensors in flat space-time. ...
Reading
9 videos (Total 124 min), 1 quiz
Video9 videos
Covariant differentiation15m
Parallel transport10m
Covariant differentiation(part 2)9m
Locally Minkowskian Reference System (LMRS)16m
Curvature or Riemann tensor15m
Properties of Riemann tensor13m
Tensors in flat space-time(part 1)21m
Tensors in flat space-time(part 2)13m

3

Section
Clock
3 hours to complete

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 Relativity is played by the metric tensor of space-time. Then we derive the Einstein equations from the least action principle applied to the Einstein-Hilbert action. Also we define the energy-momentum 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 energy-momentum 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 two-dimensional space-time....
Reading
6 videos (Total 86 min), 1 quiz
Video6 videos
Einstein equations19m
Matter energy–momentum (or stress-energy) tensor15m
Examples of matter actions17m
The least action (or minimal action) principle (part 1)11m
The least action principle (part 2)12m

4

Section
Clock
3 hours to complete

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 space-time. ...
Reading
5 videos (Total 55 min), 1 quiz
Video5 videos
Schwarzschild solution(part 2)17m
Gravitational radius6m
Schwarzschild coordinates7m
Eddington–Finkelstein coordinates11m

5

Section
Clock
3 hours to complete

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 the use of these coordinates we define Penrose-Carter diagram for the Schwarzschild black hole. This diagram allows us to qualitatively understand the fundamental properties of the black hole....
Reading
4 videos (Total 59 min), 1 quiz
Video4 videos
Kruskal–Szekeres coordinates14m
Penrose–Carter diagram for the Schwarzschild black hole10m
Penrose–Carter diagram for the Schwarzschild black hole (part 2)16m

6

Section
Clock
3 hours to complete

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 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 space-time....
Reading
5 videos (Total 62 min), 1 quiz
Video5 videos
Test particle motion on Schwarzschild black hole background11m
Test particle motion on Schwarzschild black hole background(part 2)8m
Mercury perihelion precession24m
Light ray deviation in the vicinity of the Sun8m

7

Section
Clock
3 hours to complete

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 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....
Reading
4 videos (Total 66 min), 1 quiz
Video4 videos
Interior solution.16m
Interior solution (part 2)13m
Kerr’s rotating black hole21m

8

Section
Clock
3 hours to complete

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 origin of the Hawking radiation and of the basic properties of the black hole formation....
Reading
4 videos (Total 70 min), 1 quiz
Video4 videos
Penrose–Carter diagram for the Oppenheimer– Snyder collapsing solution18m
Hints on Hawking radiation14m
Black hole horizon creation12m

9

Section
Clock
3 hours to complete

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. Then we continue with the linearized approximation to the Einstein equations which allows us to clarify the meaning of the pseudo-tensor. We end up this module with the derivation of the free monochromatic gravitational waves and of their energy-momentum pseudo-tensor. These waves are solutions of the Einstein equations in the linearized approximation....
Reading
3 videos (Total 61 min), 1 quiz
Video3 videos
Weak field approximation17m
Free gravitational waves19m

10

Section
Clock
4 hours to complete

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. ...
Reading
6 videos (Total 97 min), 1 quiz
Video6 videos
Intensity of the radiation10m
Shock gravitational wave or Penrose parallel plane wave10m
Properties of the Penrose parallel plane wave10m
Green functions28m
Averages13m

11

Section
Clock
3 hours to complete

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. We end up this module with the derivation of the vacuum homogeneous but anisotropic cosmological Kasner solution....
Reading
4 videos (Total 67 min), 1 quiz
Video4 videos
Friedmann–Robertson–Walker metric15m
Homogeneous isotropic cosmological solutions18m
Anisotropic Kasner cosmological solution15m

12

Section
Clock
4 hours to complete

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 provide coordinate systems which cover various patches of these space-times. ...
Reading
9 videos (Total 92 min), 1 quiz
Video9 videos
Geometry of the de Sitter space–time11m
Metric of the de Sitter space–time9m
Penrose–Carter diagram for the de Sitter space–time10m
Poincare' patches for the de Sitter space–time16m
Geometry of the anti–de Sitter space–time7m
Metric of the anti–de Sitter space–time9m
Penrose–Carter diagram of the anti–de Sitter space–time8m
Poincare' patch of the anti–de Sitter space–time6m
4.6

Top Reviews

By MSMar 23rd 2017

One of the bests ways of understanding the GR ant it's mathematical's tools. The apresentation of the class ( The "blackboard" ) is very impressive and helps the student in calculations . Thank You!

By PPSep 24th 2017

Best Course for Physics Enthusiasts. It is a must for those who are interested in theoretical or mathematical physics. I really enjoyed the course though it was tough.

Instructor

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Emil Akhmedov

Associate Professor

About National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru...

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