Master
2020/2021
Introduction into General Theory of Relativity
Type:
Elective course (Materials. Devices. Nanotechnology)
Area of studies:
Electronics and Nanoelectronics
Delivered by:
School of Electronic Engineering
When:
2 year, 3 module
Mode of studies:
distance learning
Instructors:
Renat Ikhsanov
Master’s programme:
Материалы. Приборы. Нанотехнологии
Language:
English
ECTS credits:
3
Contact hours:
2
Course Syllabus
Abstract
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 a student 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 a student will be able to solve basic standard problems of this theory.
Learning Objectives
- Objectives of mastering the discipline "Introduction into general theory of relativity": • give students an introduction into general theory of relativity. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity.
Expected Learning Outcomes
- Knowledge: Special Theory of Relativity.
- Skills: to do transformations to non-inertial reference systems in flat space-time.
- Possess: method of derivation of the geodesic equation for a general metric from the least action principle.
- Knowledge: - what is tensor in a general curved space-time; - what is connection, parallel transport and covariant differential; - what is Riemann tensor.
- Skills: to distinguish flat space-time in curved coordinates from curved space-times.
- Possess: methods of transformation of Christoffel symbols.
- Knowledge: - what is energy-momentum tensor for matter; - what is the basic generic properties of the Einstein equations.
- Skills: - to define Einstein equations from fundamental principles; - to derive the Einstein equations from the least action principle applied to the Einstein-Hilbert action.
- Possess: least action principle method in the simplest case of the scalar field in flat two-dimensional space-time.
- Knowledge: the Birkhoff theorem.
- Skills: - to solve the Einstein equations in the simplest settings; - to find the most famous solution of the Einstein equations, which is referred to as the Schwarzschild black hole.
- Possess: methods of solutions of the Einstein equations in the simplest settings.
- Knowledge: the Penrose-Carter diagram for flat space-time.
- Skills: to use the Penrose-Carter diagram for flat space-time.
- Possess: the Penrose-Carter diagram technique for the Schwarzschild black hole.
- Knowledge: Killing vectors and integrals of motion.
- Skills: to derive the explicit geodesic equation for the Schwarzschild space-time.
- Possess: methods of quantitative explanation of some basic properties of black holes.
- Knowledge: - perfect fluid energy-momentum tensor; - the Cosmic Censorship hypothesis and of the black hole No Hair Theorem.
- Skills: 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.
- Possess: fluid energy-momentum tensor technique.
- Knowledge: the basic properties of the black hole formation.
- Skills: to derive Oppenheimer-Snyder solution of the Einstein equations.
- Possess: Penrose-Carter diagram technique.
- Knowledge: the difference between energy-momentum conservation laws in the absence and in the presence of the dynamical gravity.
- Skills: to linearize the Einstein equations.
- Possess: the gravitational energy-momentum pseudo-tensor technique.
- Knowledge: properties of the exact shock gravitational wave solutions of the Einstein equations.
- Skills: to derive of the exact shock gravitational wave solutions of the Einstein equations.
- Possess: - the averaging over directions in space technique; - the retarded Green function technique.
- Knowledge: constant curvature three-dimensional homogeneous spaces concept.
- Skills: - to derive Friedman-Robertson-Walker cosmological solutions of the Einstein equations; - to derive the vacuum homogeneous but anisotropic cosmological Kasner solution.
- Knowledge: the geometric and causal properties of constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant.
- Skills: to derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant.
- Possess: Penrose-Carter diagrams technique.
Course Contents
- Topic 1. General Covariance.This lecture covers the basic notions of the Special Theory of Relativity and Minkwoskian coordinates in flat space-time. This module ends up with the derivation of the geodesic equation for a general metric from the least action principle. In this equation the Christoffel symbols are defined.
- Topic 2. Covariant differential and Riemann tensor.This lecture starts with the definition of what is tensor in a general curved space-time. Then the connection, parallel transport and covariant differential are we defined. The lecture shows that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. The lectures ends up with the definition of the Riemann tensor and the description of its properties.
- Topic 3. Einstein-Hilbert action and Einstein equations.This lecture starts 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 the Einstein equations are derived from the least action principle applied to the Einstein-Hilbert action.
- Topic 4. Schwarzschild solution.This lecture starts the study of the black hole type solutions. It explains how to solve the Einstein equations in the simplest settings. In this lecture one can also find perhaps the most famous solution of these equations, which is referred to as the Schwarzschild black hole. Lecture ends with the description of some properties of this Schwarzschild solution.
- Topic 5. Penrose-Carter diagrams.This lecture starts with the definition of the Penrose-Carter diagram for flat space-time. On this example the lecturer explains the uses of such diagrams. Then he continues with the definition of the Kruskal-Szekeres coordinates which cover the entire black hole space-time. With the use of these coordinates he defines Penrose-Carter diagram for the Schwarzschild black hole. This diagram allows one to qualitatively understand the fundamental properties of the black hole.
- Topic 6. Classical tests of General Theory of Relativity.The lecture starts with the definition of Killing vectors and integrals of motion, which allows one to provide conserving quantities for a particle motion in Schwarzschild space-time. The lecturer derives the explicit geodesic equation for this space-time. This equation provides a quantitative explanation of some basic properties of black holes.
- Topic 7. Interior solution and Kerr's solution.The lecture starts with the definition of the so called perfect fluid energy-momentum tensor and with the description of its properties. This tensor is used 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. The lecture ends up with a brief description of the Cosmic Censorship hypothesis and of the black hole No Hair Theorem.
- Topic 8. Collapse into black hole.The lecture starts with the derivation of the Oppenheimer-Snyder solution of the Einstein equations, which describes the collapse of a star into black hole. The lecturer starts derivation the Penrose-Carter diagram for this solution. This module ends up with a brief description of the origin of the Hawking radiation and of the basic properties of the black hole formation.
- Topic 9. Gravitational waves.The lecture explains the important difference between energy-momentum conservation laws in the absence and in the presence of the dynamical gravity. The lecturer defines the gravitational energy-momentum pseudo-tensor. Then he continues with the linearized approximation to the Einstein equations which allows us to clarify the meaning of the pseudo-tensor. He ends 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.
- Topic 10. Gravitational radiation.The lecture shows how moving massive bodies create gravitational waves in the linearized approximation. Then it continues with the derivation of the exact shock gravitational wave solutions of the Einstein equations.
- Topic 11. Friedman-Robertson-Walker cosmology.This module starts our discussion of the cosmological solutions. The lecturer defines constant curvature three-dimensional homogeneous spaces. Then he derives Friedman-Robertson-Walker cosmological solutions of the Einstein equations. He describes their properties. He ends up this module with the derivation of the vacuum homogeneous but anisotropic cosmological Kasner solution.
- Topic 12. Cosmological solutions with non-zero cosmological constant.In this module the lecturer derives constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant. He describes the geometric and causal properties of such space-times and provides their Penrose-Carter diagrams. He provides coordinate systems which cover various patches of these space-times.
Assessment Elements
- Экзамен (тест)If a student misses the exam because of some valid reason, s/he receives «absence» grade. The grade for the course is calculated on the course page on the basis of the student’s number of points that are awarded to the student for answering questions of the proposed tests. Контрольные работы и экзамен по курсу проводятся в письменной форме на платформе Coursera (https://www.coursera.org/learn/general-relativity). Во время написания контрольных и экзаменационных работ студентам запрещено: общаться с кем-либо, пользоваться конспектами и подсказками. Кратковременным нарушением связи во время контрольной работы или экзамена считается нарушение связи менее часа. Долговременным нарушением связи считается нарушение связи в течение часа и более. При долговременном нарушении связи студент не может продолжить участие в контрольной или экзамене. Процедура пересдачи аналогична процедуре сдачи.
- Самостоятельная работа
Bibliography
Recommended Core Bibliography
- Date, G. (2015). General Relativity : Basics and Beyond. Boca Raton, FL: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=906063
- Ландау Л.Д., Лифшиц Е.М. - Теоретическая физика. Т.2. Теория поля - Издательство "Физматлит" - 2006 - 536с. - ISBN: 5-9221-0056-4 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/2236
- Степаньянц К.В. - Классическая теория поля - Издательство "Физматлит" - 2009 - 544с. - ISBN: 978-5-9221-1082-2 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/2328
- Теоретическая физика. Т.2: Теория поля, , 2003
Recommended Additional Bibliography
- Вергелес С. Н. - ТЕОРЕТИЧЕСКАЯ ФИЗИКА. ОБЩАЯ ТЕОРИЯ ОТНОСИТЕЛЬНОСТИ 2-е изд., испр. и доп. Учебник для бакалавриата и магистратуры - М.:Издательство Юрайт - 2019 - 190с. - ISBN: 978-5-534-03243-7 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/teoreticheskaya-fizika-obschaya-teoriya-otnositelnosti-437658