FY3452: Gravitation and Cosmology

V 2024
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: D5-126
Lectures: Monday 12.15-14.00, 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: Basic differential geometry.
Week 4: Basic differential geometry.
Week 5: Schwarzschild solution and tests, no Thursday lecture
Week 6: Schwarzschild solution, tests, black hole
Week 7: Kerr black holes
Week 8: Classical field theory
Week 9: Classical field theory, curvature.
Week 10: Curvature, Einstein equations
Week 11: Home exam
Week 12: Linearized gravity,
Week 13: Easter
Week 14: only Thursday lecture: gravitational waves
Week 15: Monday: Guest lecture numcerical GR hydrodynamics, slides; Cosmology
Week 16: Cosmology, FRW models
Week 17: FRW models, Structure formation, inflation

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 April, 2024. 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.

Corrections:

Language corrections are welcome, but not listed below
  • After Eq. (1.20), indices in the 2.nd and 3.rd expression should corrected (OeA, EBK)
  • p.28, in Eq. (2.43): \(l/r^3\) should be \(l^2/r^3\) (SS)
  • p.38, in example 3.2: the index of last unit vector should be up, \(e^\sigma\) (MK)
  • p.43, in Eq. (3.64): the volume elements should be \(d^n x\) (MK)
  • p.55, in Eq. (4.55): add \(C\) to the last term (SS)
  • p.55, in Eq. (4.57): \(\alpha\sin\phi\) should be \(\alpha\phi\sin\phi\) (SS)
  • p.57, in Eq. (4.71): \(r\) should be \(r_0\) (SS)
  • p.57, in Eq. (4.72): first factor \( 1/(1-2M/r) \) should be \( (1-2M/r)\) (SS)
  • p.61, in Eq. (5.3): \( \theta_E \) should be \( \theta_E^2\) (SS)
  • p.64, first line: \(x=\beta/ \theta_E \) should be \( x=\theta_E/\beta\) (SS)
  • p.64, in Eq. (5.11): prefactor should be \(1/4\) (SS)
  • p.64, in Eq. (512): \( \pm \) should be \(+\) (SS)
  • p.76, after Eq. (6.41): \(u^\alpha=(1,0,0,0)\) should be \(u^\alpha=(u^t,0,0,0)\) (MK)
  • p.76, in Eq. (6.43): \(-a\cos\theta\) should be \(-a^2\cos^2\theta\) (SS)
  • p.97, add the mass term in Eq. (7.72) (EBK)
  • p.136, first line: add "...setting g=1 appropriate for Minkowski space.." (BD).
  • --- errors above should be corrected ---

Exercises

Week 3: exercises 1 and solutions.
Week 4: exercises 2 and solutions.
Week 5: exercises 3 and solutions; some more exercises 3b and solutions.
Week 7: exercises 4 and solutions.
Week 8: exercises 5 and solutions.
Week 9: exercises 6 and solutions.
Week 10: exercises 7
Week 12: exercises 8 and solutions.
Week 15: exercises 9 and solutions.
Week 16: exercises 10

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 one-week home exam in the middle of the semester will count 33% of the final mark. The exercises will be available Thursday, 07.03. after the lectures. Or you can download the home exam here after 12.15. Solutions should be handed in Monday 18.03., latest 12.00, in my mailbox (D5-166), the lectures or by email. For (concise) solutions see For solutions see here.
You can find the home exam from 2023 here.

Exam

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

Marks and solution

Reference group

consists of