General Relativity, spring 2017

53736 Yleinen suhteellisuusteoria (10 op) (16.1.-5.5.)

Lecturer: Syksy Räsänen, C326
Assistant: Eemeli Tomberg, A313

The lecturer and assistants can be reached at firstname.lastname(at)

Lectures: Monday 14-16 (Physicum A315), Tuesday 12-14 (Physicum A315)
Exercises: Wednesday 16-18 (Physicum E206)

Help session (the assistant at your service) each Friday 14-16 in room A313.

Because of the interperiod break, there are no lectures or exercises from March 6 to March 12.
Because of Easter, there are no lectures or exercises from April 13 to April 19. Because of May Day, there are no lectures on May 1.

First lecture: Monday January 16
First exercise session: Thursday January 19

First exam: Tuesday March 7 at 13.00-17.00 in Physicum E205 (Results)

Second exam: Monday May 8 at 13.00-17.00 in Physicum D112 (Results and grades)

Language of instruction: English

Register to the course in WebOodi. (Only registered students get their homework graded and receive course announcements by e-mail.)

Contents: Special relativity review. Vector and tensor fields. Manifolds and differential geometry. Field equations and curvature. Black holes. Perturbation theory. Gravitational waves. Basics of cosmology.

Exams and grades: The grade is based both on the weekly exercises (1/3) and on the two exams (1/3 and 1/3). (Exception: for students who have taken the course before, the grade is based entirely on the exams.) 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 department exam (this has to be done before the course is lectured again; registration for the department exam is done on WebOodi). When retaking the exam, the exercise points are not counted. It is only possible to retake the exam once without retaking the course. (I.e. there is only one 'free' retake ever.) Not showing up for an exam without prior arrangement counts as a failed attempt. The first and second exams cannot be retaken individually.

Exercises: The homework problems are out on Mondays on this page. You are supposed to do the homework and return them into a cardboard box marked with a text "General Relativity" (in 2nd floor A corridor) for grading by the following Monday lecture, unless otherwise announced.

Textbook: S.M. Carroll, Spacetime and Geometry (Addison Wesley 2004). This book is in the reference library. It is not necessary to buy the book; the lecture notes given below cover the content of this course and you could also read Carroll's lecture notes (on which his book is based; the lecture notes are good too, but they are shorter and less polished than the book) which are freely available on the net.

Another nice and clear book is General Relativity: An Introduction for Physicists by M.P. Hobson, G. Efstathiou and A.N.Lasenby (Cambdridge 2006).

Other literature

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 (in order to introduce some of the tools needed in general relativity). The students are assumed to know special relativity.

The recommended background includes mathematical methods (curvilinear coordinate systems, coordinate transformations, linear algebra, vectors and tensors, Fourier transforms, partial differential equations), classical mechanics (including the variational principle), special relativity and electrodynamics. Differential geometry (as well as some of the other math needed) will be reviewed in the course, so previous knowledge of it is not necessary.

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, taught on the course Fysiikan matemaattiset menetelmät III, is the language of general relativity, so having taken FYMM III is helpful. However, the required tools of differential geometry will be introduced in the course, so FYMM III is not a necessary prerequisite.

Lecture notes

These are the old lecture notes (courtesy of Hannu Kurki-Suonio, with some brief additional notes by Syksy Räsänen). There may be additions or changes as the course progresses.

Chapter 0: Introduction to General Relativity
Chapter 1: Review of Special Relativity
Chapter 2: Manifolds
Chapter 3: Curvature (Notes on Newtonian gravity)
Chapter 4: Gravitation (Notes on the Newtonian limit)
Chapter 5: Schwarzschild Solution (Notes on the perihelion of Mercury)
Chapter 6: Black Holes
Chapter 7: Perturbation Theory and Gravitational Radiation (Notes on energy loss due to gravitational waves)
Chapter 8: Cosmology (Notes on Killing vectors)

Homework problem sets

(Problem sets appear here on Mondays 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

Collection of equations (available in the exams)
A wiki dictionary of terms in general relativity and cosmology in Finnish.
Last updated: May 23, 2017