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1. Course title

General Relativity II

2. Course code


Aikaisemmat leikkaavat opintojaksot 53736 Yleinen suhteellisuusteoria, 10 op ja PAP335 Yleinen suhteellisuusteoria, 10 op

3. Course status: optional

-Which degree programme is responsible for the course?
Master’s Programme in Particle Physics and Astrophysical Sciences

-Which module does the course belong to?
PAP3002 Advanced Studies in Particle Physics and Cosmology (optional for Study Track in Particle Physics and Cosmology)
TCM300 Advanced Studies in Theoretical and Computational Methods

-Is the course available to students from other degree programmes?

4. Course level (first-, second-, third-cycle/EQF levels 6, 7 and 8)

Master’s level, degree programmes in medicine, dentistry and veterinary medicine = secondcycle
degree/EQF level 7
Doctoral level = third-cycle (doctoral) degree/EQF level 8

-Does the course belong to basic, intermediate or advanced studies (cf. Government Decree
on University Degrees)?
Advanced studies

5. Recommended time/stage of studies for completion

This is an advanced course.

In terms of courses taught at the University of Helsinki, recommended prerequisite is General relativity I.

6. Term/teaching period when the course will be offered

Lectured every spring term in period IV.

7. Scope of the course in credits

5 cr

8. Teacher coordinating the course

Syksy Räsänen

9. Course learning outcomes

  • You will learn the physical and mathematical structure of the theory of general relativity.

  • You will learn how to do calculations in general relativity, including with black holes, linear perturbation theory, gravitational waves and a little bit also in cosmology.

10. Course completion methods

The course is offered in the form of contact teaching.
No attendance requirements.

The grade is based on the weekly exercises (1/3) and the exam (2/3). (Exception: for students who have taken the course before, the grade is based entirely on the exam.) You need about 45% of the maximum points to pass the course and about 25% to get the right to try to pass the course in a general exam. When retaking the exam, the exercise points are not counted towards the grade. It is only possible to retake a failed exam once without retaking the course. Not showing up for an exam without prior agreement counts as a failed attempt. If you pass the exam, you can try to raise your grade up to two times.

11. Prerequisites

-Description of the courses or modules that must be completed before taking this course or
what other prior learning is required

  • General relativity I

12. Recommended optional studies

-What other courses are recommended to be taken in addition to this course?
-Which other courses support the further development of the competence provided by this

  • Cosmology I and II
  • Specialised courses in cosmology, such as Cosmological Perturbation Theory.

13. Course content

-Description of the course content

  • Chapter 1: action formulation of general relativity

    Chapter 2: global structure of the Schwarzschild solution, black holes, Penrose diagram, brief overview of charged and rotating black holes and Hawking radiation

    Chapter 3: perturbation theory around Minkowski space, gauge transformations, gravitomagnetism, gravitational waves, generation of gravitational waves by a binary system, energy loss due to emission of gravitational waves

    Chapter 4: Killing vectors, symmetric spacetimes, FLRW spacetime, de Sitter space, anti-de Sitter space, Penrose diagrams

14. Recommended and required literature

-What kind of literature and other materials are read during the course (reading list)?
-Which works are set reading and which are recommended as supplementary reading?

  • The only required literature is the lecture notes.
  • Recommended supplementary reading includes one or more of the following:
    S.M. Carroll, Spacetime and Geometry (Addison Wesley 2004).
    M. Nakahara: Geometry, Topology and Physics (IOP Publishing 1990)
    S. Weinberg: Gravitation and Cosmology (Wiley 1972)
    C.W. Misner K.S. Thorne, J.A. Wheeler: Gravitation (Freeman 1973)
    R.M. Wald: General Relativity, (The University of Chicago Press 1984)
    B.F. Schutz: A First Course in General Relativity (Cambridge 1985)
    J. Foster and J.D. Nightingale: A Short Course in General Relativity, 2nd edition (Springer 1994, 1995).
    J.B. Hartle: Gravity - An Introduction to Einstein's General Relativity (Addison Wesley 2003)
    B. Schutz: Gravity from the Ground Up (Cambridge 2003)

15. Activities and teaching methods in support of learning

-See the competence map (
-Student activities
-Description of how the teacher’s activities are documented

16. Assessment practices and criteria, grading scale

-See the competence map (
-The assessment practices used are directly linked to the learning outcomes and teaching
methods of the course.

Course homepage

17. Teaching language


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