Homepage summer semester 2020 Introduction into General Relativity


The lecture will cover (4+2)-hours per week, which means 4 hours of lectures and 2 hours of exercises. Regular attendence of the extercise classes is required. Each student is expected to work out individual solutions to the problem-sheets and present his/her solution upon request. At the end of semester each student should have presented, step by step, at least one complete solution in order to get the certificate of attendence (Studienleistung). The certificate of examination (Prüfungsleistung) is obtained after an examination covering all aspects of the course's content. This examination may be oral or written, depending on the number of students signing up.  

Modules / ECTS

Bachelor programme: Wahlbereich, Moderne Aspekte der Physik.
Masters programme: Fortgeschrittene Vertiefungsphase.
ECTS points: 8

Lectures (Start April 22. 2020)

Wednesday 16:15-17:45 hours (online)
Thursday 08:15-09:45 hours (online)

Exercises (Start April 24. 2020)

Friday 08:15-09:45 hours (online)
The problem sheets will be distributed as handouts on Fridays and discussed on the following Friday.  


Handwritten notes for each lecture as well as problem sheets will be distributed here. There you also find other material that you may find useful and/or interesting, like latest science news related to GR, as well as some other readings related to the history and and/or pedagogical aspects of GR. Check it out!

Course description

The lecture aims to develop the physical and mathematical foundations and techniques of General Relativity (GR). As GR interprets gravitation as an aspect of the geometry of spacetime, we will need more advanced techniques from differential geometry that are often not (yet) contained in the students' standard toolbox. Therefore we need to spend some time developing these tools. This we will mainly do in a separate lecture accompanying this one in order to not let the necessary mathematical explanations push aside the physics proper. Also, I will try to not only present the usual text-book scenery of "shiny truths", but also to introduce you to some of the "back-stage" aspects that will give you a more realistic impression of how scientific "insights" come about.

Our programme (preliminary)

  1. Recap of Newtonian concepts in gravity theory.
  2. Einstein's equivalence principle and some of its immediate consequences.
  3. Energy-momentum tensors (also called stress-energy tensors).
  4. Can we formulate a special relativistic theory of gravity? If so, what's wrong with it?
  5. Spacetime as a four-dimensional pseudo-Riemannian mannifold. Notions of derivatives (exterior, Lie, covariant). Connection, torsion, curvature and the uniqueness of Levi-Civita connection.
  6. Einstein's equations and the motion of test bodies.
  7. General principle according to which matter couples to gravity. Example: Electromagnetism.
  8. The linearised Einstein's equations, their gauge invariance, and the action they follow from. The Newtonian limit.
  9. Light propagation in static and weak gravitational fields and the generalised Fermat's principle. Gravitational lensing and its application for the detection of dark matter.
  10. Existence of gravitational waves (GW)and their properties. Examples of GW generation by rotating rigid bodies and gravitationally bound binary systems. Discussion of amplitude, frequency, polarisation, and luminosity.
  11. Solar-system tests of GR: light-deflection, Shapiro time-delay, and perihelion precession.
  12. The exterior Schwarzschild metric as the only sperically symmetric solutions to Einstein's equations without matter (vacuum solutions); Birkhoff's theorem.
  13. Motion of test particles (timelike geodesics) in the exterior Schwarzschild metric.
  14. Spherically-symmetric stars made of ideal fluid. The Tolman-Oppenheimer-Volkoff (TOV) equation. Explicit solution for the metric and the pressure in the incompressible case (interior Schwarzschild solution). The Buchdahl limit in this and the general case.
  15. The exterior Schwarzschild metric as a black hole (BH). Apparent and true curvature singularities and horizon structure. Alterative coordinates, freely falling observers and their fate at and after passing the horizon.
  16. Other BH solitions: Reissner-Nordström (charged BH) and Kerr (rotating BH).


Supplementary literature for the course, in particular a detailed script on differential geometry (in german) will be placed on the same page as for the problem sheets, which is here. In addition to that, we now list some textbooks on GR:

  1. Norbert Straumann: General Relativity (Second Edition), Springer Verlag (2013) Graduate Texts in Physics, 736 pages. Comment: A comprehensive textbook covering all aspects except cosmology. It also contains a concise discussion of differential-geometric tools.

  2. Wolfgang Rindler: Relativitätstheorie: Speziell, Allgemein und Kosmologisch, Wiley-VCH (2016), 530 pages. Comment: Reasonably priced, very good selection of topics, pedagogical approach, many exterices. Mathematical aspects are mostly suppressed.

  3. Robert M. Wald: General Relativity, University of Chicago Press (1984), 506 Seiten. Comment: A classic text.

  4. Charles W. Misner, Kip S. Thorne und John Archibald Wheeler: Gravitation, Princeton University Press (2017 Hardcover-Reprint der 1973 Ausgabe), 1328 pages. Comment: The classic text (much loved, sometimes hated).


Prof. Dr. Domenico Giulini