Offered By

National Research University Higher School of Economics

About this Course

4.6

75 ratings

•

27 reviews

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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Approx. 49 hours to complete

Subtitles: English

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Approx. 49 hours to complete

Subtitles: English

Section

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....

7 videos (Total 75 min), 1 quiz

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

Section

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. ...

9 videos (Total 124 min), 1 quiz

Tensors9m

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

Section

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....

6 videos (Total 86 min), 1 quiz

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

Section

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. ...

5 videos (Total 55 min), 1 quiz

Schwarzschild solution(part 2)17m

Gravitational radius6m

Schwarzschild coordinates7m

Eddington–Finkelstein coordinates11m

4.6

By PP•Sep 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.

By VU•Feb 2nd 2017

Excellent course, and quite intensive mathematically. One will be well placed for a graduate level course on General relativity upon completing this.

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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