General Relativity, spring 2022
General Relativity, spring 2022

PAP335 Yleinen suhteellisuusteoria (10 op) (17.1.-4.5.)

Lecturer: Syksy Räsänen (Physicum C326)
Assistant: Kasper Siilin

The lecturer and assistant can be reached at firstname.lastname at You can also drop in at the lecturer's office any time. (Most likely he'll be there on Mondays, Tuesdays and Thursdays.)

Lectures: Monday 14.15-16.00 at Physicum E207, Tuesday 14.15-16.00 at Exactum CK112
Exercises: Wednesday 16.15-18.00 at Physicum D208

Help session (assistant's office hours) each Friday 14-15 in Physicum's cafe.

Because of the interperiod break, there are no lectures or exercises from March 7 to March 13.
Because of Easter, there are no lectures or exercises from April 14 to April 20.

First lecture: Monday January 17
Last lecture: Tuesday May 3
First exercise session: Wednesday January 19
Last exercise session: Wednesday May 4

The course exam will be from noon Monday May 16 until noon Monday May 23. The problems will be sent and the solutions returned via Moodle. Details will be given via email and on the Moodle page. Please email the lecturer in case of any questions.

Students have to register for the course on Sisu. Registered students automatically have access to the course Moodle page.

Exercises are returned on Mondays by 14.15 via Moodle. There is a conversation area on the Moodle page, and you can always send email to the lecturer and assistant.

This page has the up-to-date correct information about the course. Updates will also be sent to registered students via email. (The automatically generated course webpage here is not updated.)

Contents: This is a first course in general relativity. The first part of the course covers the formalism. It begins with a review of special relativity, and goes on to discuss manifolds, curvature, the relation between matter and curvature (i.e. equations of motion for the spacetime geometry), the Newtonian limit and finally the action formulation. The second part goes through some applications, beginning with the Schwarzschild metric and the perihelion of Mercury and bending of light by the Sun, continuing with black holes, perturbation theory around Minkowski space, gravitational waves and finishing off with a bit of cosmology and maximally symmetric spacetimes.

Language of instruction: English

Exams and grades: The grade is based on the weekly exercises (1/3) and the exams (2/3). (Exception: for students who have taken the course before, the grade is based entirely on the exam.) There are 13 weekly exercises, and there are either one or two exams, to be decided during the course. You need about 45% of the maximum points to pass the course (grade 1) 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.

Textbooks: The course does not follow any textbook. Sean Carroll's Spacetime and Geometry (Addison Wesley 2004) may be handy. Carroll's lecture notes on which his book is based may also he useful; they are shorter and less polished than the book. Another nice and clear book is General Relativity: An Introduction for Physicists by M.P. Hobson, G. Efstathiou and A.N. Lasenby (Cambdridge 2006).

Three classic texts:

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)

Two good short textbooks that do not cover the whole course, but which are easy to read:

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).

More recent books, with a different approach:

J.B. Hartle: Gravity - An Introduction to Einstein's General Relativity (Addison Wesley 2003)
B. Schutz: Gravity from the Ground Up (Cambridge 2003)


The course begins with a review of special relativity to introduce some of the concepts and notation needed in general relativity. Knowledge of special relativity is assumed.

Recommended background includes mathematical methods (curvilinear coordinate systems, coordinate transformations, linear algebra, vectors and tensors, Fourier transform, partial differential equations), classical mechanics (including the variational principle), special relativity and electrodynamics.

In terms of courses taught at the University of Helsinki, recommended prerequisites are Matemaattiset apuneuvot I ja II, Fysiikan matemaattiset menetelmät Ib, Fysiikan matemaattiset menetelmät IIa, Suhteellisuusteorian perusteet, Mekaniikka and Elektrodynamiikka. Differential geometry is the language of general relativity, so having taken FYMM III is helpful, but not required, as the necessary material will be covered in the course.

Lecture notes

Lecture notes appear here before the lectures.
Chapter 1: Special relativity
Chapter 2: Manifolds
Chapter 3: Curvature
Chapter 4: Gravity
Chapter 5: The Schwarzschild solution
Chapter 6: Black holes
Chapter 7: Perturbation theory and gravitational waves
Chapter 8: Symmetries and cosmology

Homework problem sets

Problem sets appear here on Tuesdays at the latest.

Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6
Homework 7
Homework 8
Homework 9
Homework 10
Homework 11
Homework 12
Homework 13

A collection of equations that may be helpful. It will be available in the exams.
A dictionary of terms in general relativity and cosmology from English to Finnish.
Last updated: May 3, 2022