4.6
85 ratings
29 reviews

#### 100% online

Start instantly and learn at your own schedule.

#### Approx. 50 hours to complete

Suggested: 12 weeks of study, 4-6 hours per week...

#### English

Subtitles: English

#### 100% online

Start instantly and learn at your own schedule.

#### Approx. 50 hours to complete

Suggested: 12 weeks of study, 4-6 hours per week...

#### English

Subtitles: English

### Syllabus - What you will learn from this course

Week
1
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....
7 videos (Total 75 min), 1 quiz
7 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
Week
2
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. ...
9 videos (Total 124 min), 1 quiz
9 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
Week
3
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....
6 videos (Total 86 min), 1 quiz
6 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
Week
4
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. ...
5 videos (Total 55 min), 1 quiz
5 videos
Schwarzschild solution(part 2)17m
Schwarzschild coordinates7m
Eddington–Finkelstein coordinates11m
4.6
29 Reviews

### Top Reviews

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.

By VUFeb 2nd 2017

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

## Instructor

### Emil Akhmedov

Associate Professor
Faculty of Mathematics

## 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...