FY3452: Gravitation and Cosmology

V 2025
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: Monday 08.15-10.00, C4-118
Lectures + Exercises: Thursday 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: Relativistic stars, Linearized gravity
Week 12: Linearized gravity, Gravitational waves
Week 13: Gravitational waves, cosmology
Week 14: Home exam
Week 15: FRW models
Week 16: Easter
Week 17: no lecture

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 Jan. 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:
  • 2:
  • 1:

Material that can be omitted for the exam:


Corrections:

Language corrections are welcome, but not listed below
  • p.22, after Eq, (2.1): \(q\) should be \(q^i\) (JNSC)
  • p.28, before Eq, (2.40): \(l=L/m\) should be \(l=L/\mu\) (ASR)
  • p.35, Eq. (3.40): first term \(y\partial_z z\partial_y\) should be \(y\partial_z z\partial_x\) (LMD)
  • p.41, Eq. (3.56): first term \(g_{\mu \nu, \lambda}\) should be \(g_{\lambda\nu, \mu} \) (MJH)
  • p.43, before Eq. (3.56): \(g^{\kappa\mu}\) should be \(g^{\kappa\lambda}\) (JNSC)
  • p.43, Eq. (3.64): delete the, indices at the differentials \(dx^n\) (JNSC)
  • p.52, Eq, (4.27+28): include the usual minus sign into the definition of the potential, i.e. set \( \Phi=-GM/r\) (MK)
  • p.53, Eq, (4.31): delete the superfluous \( \sin^2\theta\) (RS)
  • p.62, Eq, (4.76): change \(\epsilon\) into \(\omega\) (RS)
  • p.130, Eq, (9.21): change \(\sin^2\theta\) into \(\sin\theta\) (ASR)

g^(kappa lambda) inside the red square. Page 41, right above eqn 3.56.

Exercises

Week 3: exercises 1 and solutions.
Week 4: exercises 2 and solutions.
Week 5: exercises 3 and solutions.
Week 6: exercises 4 and solutions.
Week 7: exercises 5 and solutions.
Week 9: exercises 6 and solutions.
Week 10: exercises 7 and solutions.
Week 11: exercises 8

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 14 will count 33% of the final mark. The exam will be available Thursday, 27.03. after the lectures. Or you can download the home exam here after 12.15. Solutions should be handed in Monday 07.04., latest 10.00, via Inspera. You can find the home exam from 2020 and 2023 here and here.

Exam

The final exam of this course (counting 67%) takes place

Marks and solution