FY3452: Gravitation and Cosmology

V 2026
The course gives an introduction to classical field theory and general relativity including applications, in particular black holes, gravitational waves and cosmology.

Lecturer, time and place:

Michael Kachelrieß, email; office: D5-123
Lectures + Exercises: Thursday 14.15-16.00, C4-118
Lectures: Friday 10.15-12.00, C4-118.

Plan of the lectures (preliminary):

Week 2: Special relativity, Lagrangian mechanics and symmetries.
Week 3: Lagrangian mechanics and symmetries, basic differential geometry.
Week 4: Basic differential geometry, Killing vector fields.
Week 5: Schwarzschild solution and tests
Week 6: Schwarzschild black hole
Week 7: Kerr black holes
Week 8: Classical field theory
Week 9: Classical field theory, curvature.
Week 10: Einstein equations, GR hydrodynamics
Week 11: Home exam?
Week 12: Relativistic stars, Linearized gravity
Week 13: Linearized gravity, Gravitational waves
Week 14: Easter
Week 15: Gravitational waves
Week 16: cosmology, FRW models
Week 17: FRW models

Pensum

The pensum is defined by the content of the lectures. We will follow roughly my lecture notes (which will be updated during the course), see the link below. The number of excellent text-books on gravity is large: You should check out the library to find the optimal book(s) for you, depending on the degree of mathematical rigor you prefer and the focus of the various authors on different types of applications. Some recommend ones are
  • Carroll, Sean M.: Spacetime and Geometry: An Introduction to General Relativity, Pearson, 2003). For a free (early) draft version see Lecture Notes on General Relativity arXiv:gr-qc/9712019. [very clear, a bit more formal approach]
  • J.B. Hartle. Gravity: An Introduction to Einstein's General Relativity, Addison Wesely 2003.
    [The book used as pensum in previous years]
  • Hobson, M.P., Efstathiou, G.P., Lasenby, A.N.: General relativity: an introduction for physicists, Cambridge University Press 2006.
    [uses our conventions, same level as this course]
  • Landau, Lev D.; Lifshitz, Evgenij M.: Course of theoretical physics 2 - The classical theory of fields. Pergamon Press Oxford, 1975
    [Concise and clear description of the basics.]
  • Misner, Charles W.; Thorne, Kip S.; Wheeler, John A.: Gravitation. Freeman New York, 1998.
    [Entertaining and nice description of differential geometry -- but very lengthy...]
  • Schutz, Bernard F.: A first course in general relativity, Cambridge Univ. Press, 2004.
  • Stephani, H.: Relativity: an introduction to special and general relativity, Cambridge Univ. Press, 2004.
  • Robert M. Wald: General Relativity. University of Chicago Press 1986.
    [Uses a modern mathematical language]
  • Weinberg, Steven: Gravitation and cosmology, Wiley New York, 1972.
    [A classics. Many applications and useful connection to particle physics; but outdated concerning cosmology.]
  • Weyl, Hermann: Raum, Zeit, Materie, Springer Berlin, 1918; (Space, Time, Matter, Dover New York, 1952).
    [The classics.]

Lectures notes

You can download the script here, last update from Nov. 2025. If you find more errors, let me know. You can obtain a maximal bonus of 5% for finding errors in the lecture notes: 5% for the student finding most errors, 4% for second most, etc. Email errors to me.
Distribution of bonus points:
  • 5:
  • 4:
  • 3:
  • 2:
  • 1:

Material that can be omitted for the exam:


Corrections:

Language corrections are welcome, but not listed below
  • ...

Exercises

Week 3:
Week 4:
Week 5:
Week 6:
Week 7:
Week 9:
Week 10:
Week 11:
Week 12:
Week 13:

Software

You can find a python program for the calculation of Christoffel symbols, Riemann tensor, etc. here. Hartle provides several Mathematica notebooks.

Home exam

The home exam in week ?? will count 33% of the final mark.

Exam

The final exam of this course (counting 67%) takes place 1.6. Previous exams: from 2022, from 2023, from 2024, see also eksamensakiv.

Marks and solution