General Theory of Relativity

Course guarantor:

Boris Tomášik

Lecturer:

Boris Tomášik

Tutor:

Tomáš Poledníček, Boris Tomášik

Supervisor:

Department of Physics

Synopsis:

The goal is to learn the basics of General Relativity theory as well as its applications, mainly in cosmology. The students will get acquainted with the starting points of General Relativity. The course includes the explanation of necessary mathematics: differential geometry. Classic results are derived, like the precession of Mercury, gravitational frequency shift and gravitational bending of light. The participants learn about Schwarzschild metrics and its solution leading to black holes. In the application part the Friedman-Robertson-Walker metrics is introduced and dynamics of the Universe is discussed.

Requirements:

Resources

Moodle page of the course: https://moodle-vyuka.cvut.cz/course/view.php?id=4396

Organisation

All materials will be provided in the moodle page. The cycle in which each topic is delivered can be summarized shortly: pre-recorded video-lecture online JITT quiz lecture with the discussion of the topic recitation, problem solving homework.

Each lecture is pre-recorded and made available about one week before the Discussion. Students must watch the lecture and it is recommended to read the corresponding chapter from the course literature, which will be indicated in moodle.

Students are required to pass the Just-In-Time-Teaching (JITT) test by the deadline (ca 24 hours before the Lecture). The test is evaluated and contributes to the final grade. Every student must formulate a question about the topic, which will be answered (or discussed) in the Lecture.

The Lecture is scheduled on Thursday at 10:00. Presence of students is expected.

The lecture is followed by recitation in the scheduled time. Students should participate in the recitation. (To stimulate the attendance, flash quiz may be issued unexpectedly at random time during the recitation. It contributes to the final grade.)

Homework will be given on moodle. Solutions must be handed in to the Assistant or uploaded in due time to be graded. The homework can be discussed with the teachers before the due date (and this is recommended in case of problems.)

Grading

Grading is based on points collected throughout the whole semester and at the final exam. Points can be collected from

1. JITT tests. Expected 30 points. 3 points per test. Must be submitted in moodle on time.

2. Homeworks. Expected 30 points. Must be submitted on time.

3. Flash quizzes. (may be unexpected and may also not be used at all) Maximum 10 points.

4. Midterm test. 50 points. Midterm test can be repeated, provided that all homeworks are uploaded by the time of the first midterm test. Request for repeated test must be submitted to instructor within 72 hours after the evaluation of the first test was announced.

5. Final test. 50 points. Final test can be repeated, provided that all homeworks are uploaded by the time of the first final test. Request for repeated test must be submitted to instructor within 72 hours after the evaluation of the first test was announced.

6. Exam. 150 points. (Exam is compulsory even if the student has collected enough points.)

Zápočet: all JITT tests submitted on time & minimum 50% of all points for 1+2+3+4+5.

Exam grade (Zkouška): based on the ratio of the obtained points to the maximum possible points R.

A: R0.9; B: R0.8; C: R0.7; D: R0.6; E: R0.5; Fx: R<0.5.

Syllabus of lectures:

1.Starting points of general relativity, Einstein equivalence principle, Locally Inertial Frames

2.Curved spaces in more dimensions, Riemann spaces, metrics, tensors, geodetics

3.Riemann tensor, energy-momentum tensor, Einstein equations

4.Schwarzschild metrics, Birkhoff theorem, geodesics and photon trajectory in Schwarzschild metrics

5.Experimental tests of general relativity: gravitational frequency shift, precession of Mercury, gravitational light bending, gravitational lenses

6.Black holes

7.Rotating black holes

8.Gravitational waves

9.Friedman-Robertson-Walker metrics

10.Cosmological models

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] M.P. Hobson, G.P. Efstathiou, A.N. Lasenby: General Relativity, An Introduction for Physicists, Cambridge UP, Cambridge, 2006

[2] R.H. Price: General Relativity Primer, American Journal of Physics 50 (1982) 300

Recommended references:

[3] W. Rindler: Relativity (Special, General, and Cosmological), 2 edition, Oxford University Press, New York, 2006

[4] L. Ryder: Introduction to General Relativity, Cambridge UP, Cambridge, 2009

[5] A. Zee, Einstein Gravity in a Nutshell, Princeton University Press, 2013

[6] K. Kuchař, Základy obecné teorie relativity, Academia, Praha, 1968

[7] G. tHooft: Introduction to General Relativity, Rinton Press, Princeton, 2001

Note:

Further information:

https://moodle-vyuka.cvut.cz/course/view.php?id=4396

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Jaderná a částicová fyzika (compulsory elective course)
• Nuclear and Particle Physics (compulsory elective course)
• Nuclear and Particle Physics (compulsory elective course)