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
Introduction into General Theory of Relativity
oferecido por
National Research University Higher School of Economics
Programa - O que você aprenderá com este curso
Semana
13 horas para concluir
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 vídeos (total de (Total 75 mín.) min), 1 teste
7 videos
Introduction1min
General covariance12min
Сonstant linear acceleration16min
Transition to the homogeneously accelerating reference frame (or system) in Minkowski space–time8min
Transition to the homogeneously accelerating reference frame in Minkowski space–time (part 2)13min
Geodesic equation8min
Christoffel symbols14min
Semana
24 horas para concluir
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 vídeos (total de (Total 124 mín.) min), 1 teste
9 videos
Tensors9min
Covariant differentiation15min
Parallel transport10min
Covariant differentiation(part 2)9min
Locally Minkowskian Reference System (LMRS)16min
Curvature or Riemann tensor15min
Properties of Riemann tensor13min
Tensors in flat space-time(part 1)21min
Tensors in flat space-time(part 2)13min
Semana
33 horas para concluir
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 vídeos (total de (Total 86 mín.) min), 1 teste
6 videos
Einstein equations19min
Matter energy–momentum (or stress-energy) tensor15min
Examples of matter actions17min
The least action (or minimal action) principle (part 1)11min
The least action principle (part 2)12min
Semana
43 horas para concluir
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 vídeos (total de (Total 55 mín.) min), 1 teste
5 videos
Schwarzschild solution(part 2)17min
Gravitational radius6min
Schwarzschild coordinates7min
Eddington–Finkelstein coordinates11min
Melhores avaliações
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.
Excellent course, and quite intensive mathematically. One will be well placed for a graduate level course on General relativity upon completing this.
Instrutores
Emil Akhmedov
Associate ProfessorFaculty of Mathematics
Sobre 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 communicamathematics, engineering, and more.
Learn more on www.hse.ru
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