Introduction into General Theory of Relativity

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About this course: 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 o…

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When you enroll for courses through Coursera you get to choose for a paid plan or for a free plan

  • Free plan: No certicification and/or audit only. You will have access to all course materials except graded items.
  • Paid plan: Commit to earning a Certificate—it's a trusted, shareable way to showcase your new skills.

About this course: 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

Created by:  Higher School of Economics
  • Taught by:  Emil Akhmedov, Associate Professor

    Faculty of Mathematics
Level Advanced Commitment 12 weeks of study, 4-6 hours per week Language English How To Pass Pass all graded assignments to complete the course. User Ratings 4.7 stars Average User Rating 4.7See what learners said Задания курса

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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

Syllabus


WEEK 1


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 expand


  1. Video: Introduction
  2. Video: General covariance
  3. Video: Сonstant linear acceleration
  4. Video: Transition to the homogeneously accelerating reference frame (or system) in Minkowski space–time
  5. Video: Transition to the homogeneously accelerating reference frame in Minkowski space–time (part 2)
  6. Video: Geodesic equation
  7. Video: Christoffel symbols

Graded: Problems 1

WEEK 2


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 expand


  1. Video: Tensors
  2. Video: Covariant differentiation
  3. Video: Parallel transport
  4. Video: Covariant differentiation(part 2)
  5. Video: Locally Minkowskian Reference System (LMRS)
  6. Video: Curvature or Riemann tensor
  7. Video: Properties of Riemann tensor
  8. Video: Tensors in flat space-time(part 1)
  9. Video: Tensors in flat space-time(part 2)

Graded: Problems 2

WEEK 3


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 expand


  1. Video: Einstein–Hilbert action
  2. Video: Einstein equations
  3. Video: Matter energy–momentum (or stress-energy) tensor
  4. Video: Examples of matter actions
  5. Video: The least action (or minimal action) principle (part 1)
  6. Video: The least action principle (part 2)

Graded: Problems 3

WEEK 4


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 expand


  1. Video: Schwarzschild solution
  2. Video: Schwarzschild solution(part 2)
  3. Video: Gravitational radius
  4. Video: Schwarzschild coordinates
  5. Video: Eddington–Finkelstein coordinates

Graded: Problems 4

WEEK 5


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.


4 videos expand


  1. Video: Penrose–Carter diagrams
  2. Video: Kruskal–Szekeres coordinates
  3. Video: Penrose–Carter diagram for the Schwarzschild black hole
  4. Video: Penrose–Carter diagram for the Schwarzschild black hole (part 2)

Graded: Problems 5

WEEK 6


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.


5 videos expand


  1. Video: Killing vectors and conservation laws
  2. Video: Test particle motion on Schwarzschild black hole background
  3. Video: Test particle motion on Schwarzschild black hole background(part 2)
  4. Video: Mercury perihelion precession
  5. Video: Light ray deviation in the vicinity of the Sun

Graded: Problems 6

WEEK 7


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.


4 videos expand


  1. Video: Energy–momentum tensor of a perfect relativistic fluid
  2. Video: Interior solution.
  3. Video: Interior solution (part 2)
  4. Video: Kerr’s rotating black hole

Graded: Problems 7

WEEK 8


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.


4 videos expand


  1. Video: Oppenheimer–Snyder collapse
  2. Video: Penrose–Carter diagram for the Oppenheimer– Snyder collapsing solution
  3. Video: Hints on Hawking radiation
  4. Video: Black hole horizon creation

Graded: Problems 8

WEEK 9


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.


3 videos expand


  1. Video: Energy–momentum pseudo–tensor for gravity
  2. Video: Weak field approximation
  3. Video: Free gravitational waves

Graded: Problems 9

WEEK 10


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.


6 videos expand


  1. Video: Gravitational radiation by moving massive bodies
  2. Video: Intensity of the radiation
  3. Video: Shock gravitational wave or Penrose parallel plane wave
  4. Video: Properties of the Penrose parallel plane wave
  5. Video: Green functions
  6. Video: Averages

Graded: Problems 10

WEEK 11


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.


4 videos expand


  1. Video: Homogeneous three–dimensional spaces
  2. Video: Friedmann–Robertson–Walker metric
  3. Video: Homogeneous isotropic cosmological solutions
  4. Video: Anisotropic Kasner cosmological solution

Graded: Problems 11

WEEK 12


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.


9 videos expand


  1. Video: De Sitter and anti-de Sitter
  2. Video: Geometry of the de Sitter space–time
  3. Video: Metric of the de Sitter space–time
  4. Video: Penrose–Carter diagram for the de Sitter space–time
  5. Video: Poincare' patches for the de Sitter space–time
  6. Video: Geometry of the anti–de Sitter space–time
  7. Video: Metric of the anti–de Sitter space–time
  8. Video: Penrose–Carter diagram of the anti–de Sitter space–time
  9. Video: Poincare' patch of the anti–de Sitter space–time

Graded: Problems 12

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