Relativity and Cosmology I

14756.2015 Winter term 2022/23

Prof. Dr. Claus Kiefer

This course will be given in a hybrid format. The lectures are available online here. For convenience we will republish the material on this page weekly on Monday and Wednesday.

In addition, there will be a weekly zoom meeting for questions and discussions. This is planned for Wednesdays 14:00-15:30, start October 18. Details will be given by e-mail.

Master core course within the Primary Area of Specialization GR-QFT

Black hole concept drawing


Mirrors Section Release
Vimeo YouTube 0. Overview Sep 12, 2022
Vimeo YouTube 1. Introduction: A brief history of spacetime Oct 10
Vimeo YouTube 2. Newtonian gravity. I.
Vimeo YouTube 2. Newtonian gravity. II. Oct 12
Vimeo YouTube 3. Special relativity
3.1. Poincaré transformation
Vimeo YouTube 3.2. Spacetime diagrams and the relativity of simultaneity Oct 17
Vimeo YouTube 3.3. Time dilation and Lorentz contraction
Vimeo YouTube 3.4. Spacetime and four-vectors Oct 19
Vimeo YouTube 3.5. Special relativistic mechanics
Vimeo YouTube 4. Gravity as geometry
4.1. The twin problem in special relativity
4.2. Gravitational redshift
Vimeo YouTube 4.3. Motion in a gravitational field Oct 26
Vimeo YouTube 4.4. Newtonian limit (heuristic derivation)
Vimeo YouTube 4.5. Curved spaces
Vimeo YouTube 5. Differential geometry I: Classic formulation
5.1. Differentiable manifolds
5.2. Vector and tensor fields
5.3. The metric tensor
Nov 2
Vimeo YouTube 5.4. Tensor densities
Vimeo YouTube 5.5. Riemann normal coordinates Nov 7
Vimeo YouTube 5.6. Tensor analysis
Vimeo YouTube 5.7. The Riemann curvature tensor Nov 9
Vimeo YouTube 5.8. Further discussions of the Riemann tensor
5.9. The Weyl tensor
Vimeo YouTube 5.10. Integration
5.11. Invariant tensor fields and Lie derivatives
Nov 16
Vimeo YouTube 5.12. Killing vectors
5.13. Fermi–Walker transport
Nov 21
Vimeo YouTube 6. Einstein's field equations and their linear approximation
6.1. The field equations
Nov 28
Vimeo YouTube 6.2. The linear approximation and the Newtonian limit Dec 5
Vimeo YouTube 6.3. Field equations from a variational principle Dec 7
Vimeo YouTube 6.4. Some important examples Dec 12
Vimeo YouTube 6.5. Conservation laws
6.6. General remarks on the structure of the theory
Dec 21
Vimeo YouTube 7. Gravitational waves
7.1. Propagation of gravitational waves
Jan 10
Vimeo YouTube 7.1. Propagation of gravitational waves (cont'd)
7.2. The energy in gravitational wave
Jan 12
Vimeo YouTube 7.3. Emission of gravitational waves Jan 16
Vimeo YouTube 7.3. Emission of gravitational waves (cont'd)
7.4. Detection of gravitational waves
Jan 18
Vimeo YouTube 8. The Schwarzschild solution Jan 23
Vimeo YouTube Summary and outlook Jan 25

Lecture notes

A script of the lecture notes for the first six chapters can be found here. The password will be sent by email.
Script is under construction; additional sections will be added as soon as they are available.

Exercise classes

Exercise classes will be given on Thursdays. Please show up on October 20 at the time of your choice (14:00 or 16:00). Perhaps we have to redistribute the participants in case that too many people have chosen one class (and too little the other classes). For those of you who haven't attended this week's class, please join the second group (16:00) as there are free spots there.

The solutions of the exercise sheets can be found here. The password will be sent by email.


Oct 20

Time and place:   

Exercise class I: 14:00–15:30
Tutor: Tatevik Vardanyan (; seminar room (old building)
Exercise class II: 16:00–17:30
Tutor: Mauricio Ortiz Torres (; seminar room (old building)
Exercise class III: 14:00–15:30
Tutor: Blanca Hergueta (; room 0.02 (new building)

Exercise sheets

New exercise sheets will be released each Thursday and can be downloaded below. The exercises have to be submitted one week later in the class (one solution sheet per a group of up to two students). The exercises will be discussed on the specified date on the sheet and below.

   Exercise Release Submit Discuss Remark
   Sheet 1 Oct 20, 2022 Oct 27 Nov 3
   Sheet 2 Oct 27 Nov 3 Nov 10
   Sheet 3 Nov 3 Nov 10 Nov 17
   Sheet 4 Nov 10 Nov 17 Nov 24
   Sheet 5 Nov 17 Nov 24 Dec 1
   Sheet 6 Nov 24 Dec 1 Dec 8
   Sheet 7 Dec 1 Dec 8 Dec 15
   Sheet 8 Dec 8 Dec 15 Jan 12, 2023 ver 1.02
   Sheet 9 Dec 15 Jan 12 Jan 19
   Sheet 10 Jan 12 Jan 19 Jan 26 ver 1.01
   Sheet 11 Jan 19 Jan 26 Feb 2

Criteria for passing the course

  • Active participation in the exercise class.
  • At least 50 % of the credit points from the exercises.


Further reading

Contents of the lecture
  1. Introduction
  2. Newtonian Gravity
  3. Special relativity
  4. Gravity as geometry
  5. Differential geometry I: Classic formulation
  6. Einstein's field equations and their linear approximation
  7. Gravitational waves
  8. The Schwarzschild solution
  9. Geodesics in the Schwarzschild solution

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