General Relativity, spring 2025
General Relativity I & II, spring 2025

PAP348 General Relativity I (5 op) (13.1.-27.2.)
PAP349 General Relativity II (5 op) (10.3.-29.4.)

Lecturer: Syksy Räsänen (Physicum C326)
Assistant: Ali Hassan (Physicum A314)

The lecturer and assistant can be reached at firstname.lastname at helsinki.fi. You can also drop by in our offices at any time.

General Relativity I

Lectures: Monday and Tuesday 14.15-16.00 at Exactum D122
Exercises: Thursday 12.15-14.00 at Exactum D122

First lecture: Monday January 13
Last lecture: Tuesday February 25
First exercise session: Thursday January 16
Last exercise session: Thursday February 27

The exam will be on Friday March 7 13.00-17.00 in Chemicum A110.

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

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 is not necessarily updated.)

General Relativity II

Lectures: Monday and Tuesday 14.15-16.00 at Physicum A315
Exercises: Thursday 12.15-14.00 at Exactum CK111

First lecture: Monday March 10
Last lecture: Tuesday April 19
First exercise session: Thursday March 13
Last exercise session: Thursday April 24

Because of the interperiod break, there are no lectures or exercises from March 3 to March 9.
Because of Easter, there are no lectures or exercises from April 17 to April 23.
Because of May Day, there are no lectures or exercises on May 1.

The exam will be on Wednesday May 7 13.00-17.00 in Exactum D123.

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

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 is not necessarily updated.)

Language: English

Exams and grades: 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.) There are 6 weekly exercises for each course. You need 45% of the maximum points to pass the course (grade 1) and 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.

Use of machine learning algorithms ("AI"): Doing homework exercises or exam problems (would be possible anyway only for take-home exams) with machine learning algorithms ("AI") is not allowed on the courses General relativity I and II. (General guidelines of the University of Helsinki on AI can be found here.) Prompting an AI to do them and passing the answers as your own constitutes cheating. More importantly, doing so means that you not learn how to do the calculations, which is the main method of learning general relativity. Hence, you will also have difficulty passing the exam. I advise against consulting AI even for general understanding of the topic, because of the lack of error control in the answers: they may seem convincing but be false or even nonsensical.

Contents

The first part covers the formalism of general relativity, with application to the Solar System. 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 the the Schwarzschild metric and the change in the perihelion of Mercury and bending of light by the Sun.

The second part goes through some applications, beginning with the action formulation, continuing with black holes, perturbation theory around Minkowski space and gravitational waves, finishing off with a bit of cosmology and maximally symmetric spacetimes.

General Relativity I

Chapter 1: review of symmetries in Newtonian mechanics, review of special relativity from the spacetime point of view, relativity principle in Newtonian mechanics and special relativity, electrodynamics in special relativity
Chapter 2: the equivalence principles, manifolds, tensors, the metric
Chapter 3: covariant derivative and connection, parallel transport, geodesics, curvature, Riemann tensor
Chapter 4: Einstein equation, geometrisation of Newtonian gravity, Newtonian limit
Chapter 5: Schwarzschild solution, precession of the perihelion of Mercury, bending of light by the Sun

General Relativity II

Chapter 6: action formulation of general relativity
Chapter 7: global structure of the Schwarzschild solution, black holes, brief overview of charged and rotating black holes and Hawking radiation
Chapter 8: 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 9: Killing vectors, symmetric spacetimes, FLRW spacetime, de Sitter space, anti-de Sitter space, Penrose diagrams

Prerequisites: Recommended background for General Relativity I includes mathematical methods, including non-Cartesian coordinate systems, coordinate transformations, linear algebra, vectors and tensors, Fourier transforms and partial differential equations. 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. Fysiikan matemaattiset menetelmät III is helpful but not necessary. General Relativity I is the prerequisite for General Relativity II.

Textbooks: The only required literature is the lecture notes. They have been influenced by Sean Carroll's book Spacetime and Geometry (Addison Wesley 2004). Carroll's lecture notes on which his book is based may also be useful; they are shorter and less polished than the book. Consulting other books will also probably be useful.

Three classic texts:

S. Weinberg: Gravitation and Cosmology (Wiley 1972)
C.W. Misner, K.S. Thorne, and 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:

J.B. Hartle: Gravity - An Introduction to Einstein's General Relativity (Addison Wesley 2003)
B. Schutz: Gravity from the Ground Up (Cambridge University Press 2003)
M.P. Hobson, G. Efstathiou, and A.N. Lasenby: General Relativity: An Introduction for Physicists (Cambdridge University Press 2006)
Mark Hindmarsh and Andrew Liddle: Introducing General Relativity (Wiley 2022)

Lecture notes

Lecture notes appear here before the lectures.

General Relativity I

Chapter 1: Special relativity
Chapter 2: Manifolds
Chapter 3: Curvature
Chapter 4: Gravitation
Chapter 5: The Schwarzschild solution

General Relativity II

Chapter 6: Action formulation
Chapter 7: Black holes
Chapter 8: Perturbation theory and gravitational waves
Chapter 9: Symmetries and cosmology

Homework problem sets

The homework problems appear here on Tuesday (at the latest). The solutions are returned via Moodle by the next Monday lecture.

General Relativity I

Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6

General Relativity II

Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6

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: April 23, 2025