Master
2021/2022
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
Online hours:
46
Open to:
students of all HSE University campuses
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. It is expected that applicants to the discipline should be able to demonstrate knowledge of the following topics: general physics and higher mathematics (calculus, linear algebra, ordinary differential equations) at least in the scope of a technical university program
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: - perfect fluid energy-momentum tensor; - the Cosmic Censorship hypothesis and of the black hole No Hair Theorem.
- Knowledge: - what is energy-momentum tensor for matter; - what is the basic generic properties of the Einstein equations.
- Knowledge: - what is tensor in a general curved space-time; - what is connection, parallel transport and covariant differential; - what is Riemann tensor.
- Knowledge: constant curvature three-dimensional homogeneous spaces concept.
- Knowledge: Killing vectors and integrals of motion.
- Knowledge: properties of the exact shock gravitational wave solutions of the Einstein equations.
- Knowledge: Special Theory of Relativity.
- Knowledge: the basic properties of the black hole formation.
- Knowledge: the Birkhoff theorem.
- Knowledge: the difference between energy-momentum conservation laws in the absence and in the presence of the dynamical gravity.
- 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.
- Knowledge: the Penrose-Carter diagram for flat space-time.
- Possess: - the averaging over directions in space technique; - the retarded Green function technique.
- Possess: - the averaging over directions in space technique; - the retarded Green function technique.
- Possess: fluid energy-momentum tensor technique.
- Possess: least action principle method in the simplest case of the scalar field in flat two-dimensional space-time.
- Possess: method of derivation of the geodesic equation for a general metric from the least action principle.
- Possess: methods of quantitative explanation of some basic properties of black holes.
- Possess: methods of solutions of the Einstein equations in the simplest settings.
- Possess: methods of transformation of Christoffel symbols.
- Possess: Penrose-Carter diagram technique.
- Possess: Penrose-Carter diagrams technique.
- Possess: the gravitational energy-momentum pseudo-tensor technique.
- Possess: the Penrose-Carter diagram technique for the Schwarzschild black hole.
- Skills: - to define Einstein equations from fundamental principles; - to derive the Einstein equations from the least action principle applied to the Einstein-Hilbert action.
- Skills: - to derive Friedman-Robertson-Walker cosmological solutions of the Einstein equations; - to derive the vacuum homogeneous but anisotropic cosmological Kasner solution.
- 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.
- Skills: to derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant.
- Skills: to derive of the exact shock gravitational wave solutions of the Einstein equations.
- Skills: to derive Oppenheimer-Snyder solution of the Einstein equations.
- Skills: to derive the explicit geodesic equation for the Schwarzschild space-time.
- 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.
- Skills: to distinguish flat space-time in curved coordinates from curved space-times.
- Skills: to do transformations to non-inertial reference systems in flat space-time.
- Skills: to linearize the Einstein equations.
- Skills: to use the Penrose-Carter diagram for flat space-time.
Course Contents
- Topic 1. General Covariance.
- Topic 2. Covariant differential and Riemann tensor.
- Topic 3. Einstein-Hilbert action and Einstein equations.
- Topic 4. Schwarzschild solution.
- Topic 5. Penrose-Carter diagrams.
- Topic 6. Classical tests of General Theory of Relativity.
- Topic 7. Interior solution and Kerr's solution.
- Topic 8. Collapse into black hole.
- Topic 9. Gravitational waves.
- Topic 10. Gravitational radiation.
- Topic 11. Friedman-Robertson-Walker cosmology.
- Topic 12. Cosmological solutions with non-zero cosmological constant.
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 - ISBN: 5-9221-0056-4 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/2236
- Степаньянц, К. В. Классическая теория поля : учебное пособие / К. В. Степаньянц. — Москва : ФИЗМАТЛИТ, 2009. — 544 с. — ISBN 978-5-9221-1082-2. — Текст : электронный // Лань : электронно-библиотечная система. — URL: https://e.lanbook.com/book/2328 (дата обращения: 00.00.0000). — Режим доступа: для авториз. пользователей.
- 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