Pages for registered students: Spring term 2025

Introduction meeting

The introduction meeting takes place at the same time as the first lecture.

If you cannot access the pages for registered students and want to take this course, contact the teacher.

Homework/exercises

There will be 5 sets of exercises. They contain 4-5 problems each of which one, randomly chosen, will be graded. This way you can earn up to 5 points that count towards the score on the take-home exam. The exercise sessions are the solutions being presented and discussed by the students. The exercises are not required but strongly recommended. There are several reasons for this way of doing them:

Doing the homework problems is important, it is very easy to sit in the lectures and think you understand everything. Homework is one way to test for yourself if you do.
Discussing and collaborating (not copying) with other students is an important part of the learning process. Since we do not want to get into discussions of whether you copied solutions, the points given are not so much that you will be able to pass the course simply on this but enough to encourage you to do them.
Having a teacher or teaching assistant simply present solutions makes for very passive exercise sessions. In addition, presenting stuff will be useful for you in further life so we encourage it very much. Having you, and not only the top students, present solutions allows us to judge places where students have misunderstood things. A last one is that having been forced to present things on the whiteboard makes you appreciate what a teacher has to do as well.

Examination

There are three parts

Written take home exam
Oral exam to test understanding (a list of typical questions will be made available during the course)
Oral presentation of a special topic (suggested topics will be made available during the course)

Planned schedule: check for precise dates in the link given above to the present course instance

Presentations: Typically early May, topics assigned at least one week earlier, wishes to be given about two weeks earlier
Take home exam: one week, typically just after mid May
Oral exam: typically last week of term normally in teacher's office
Reexam will take place in the second half of August. Contact me in good time if you want to do this, before the 10th of August.

Lecture content

Note that this is only approximately

Introductory meeting and Special relativity
Special relativity: geometrical view and vector analysis
Tensor analysis in special relativity
Equivalence principle and preparing for curvature
Preparing for curvature, Curved manifolds
Curved manifolds; derivation Riemann tensor
Curved Manifolds, Energy momentum tensor
Optional: geodesic equation from shortest distance
Energy-Momentum Tensor, Physics in curved space times
Einstein Field Equations, Weak field equations
Spherically symmetric metrics, spherically symmetric solutions and stars
Orbits, perihelion shift and deflection of light
A java applet to play with orbits or Wolfram demonstrations of orbits
An alternative solution to the deflection of light equation
Black Holes
Hawking Radiation, Gravitational Radiation
Gravitational radiation, discovery
Cosmology
Cosmology: advanced topics, Presentation

Literature

The course book is: B.F. Schutz, A first course in general relativity, Third edition, Cambridge University Press, 2022. Note Third edition. Some extra notes will be handed out for advanced topics. The slides from the lectures are available from the canvas page. Misprints for the first printing (pre 2011) of the second edition. Some misprints in second printing

A book covering essentially the same topics but with a more American style (boxes, different tracks, more storylike in places etc.) which is a recommended second read if you have difficulties with some topics in the course book is Thomas A. Moore, A General Relativity Workbook, University Science Books, 2013, ISBN 978-1-891389-82-5. There are unfortunately some differences in notation with the course book.

Possibly useful extra literature

There are very many books on general relativity. Below is a selection that I have found useful and/or interesting. Note the final part with books suggested as useful by other students (which I did not check out myself but clearly someone has found helpful)

Somewhat higher level:

Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity , Addison-Wesley, 2004, ISBN 0-8053-8732-3.
Sean Carrol: Lecture notes on general relativity, arXiv:gr-qc/9712019
James B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Addison Wesley, 2002, ISBN 0-8053-8662-9.
M.P. Hobson, G.P. Efstathiou and A.N. Lasenby, General relativity: an introduction for physicists, Cambridge University Press, 2006
A. Zee, Einstein Gravity in a nutshell, Princeton University Press, 2013. This book uses the lagrangian formalism throughout, if you know this then a useful complement for those who want to know more and it goes on to a number of very advanced topics.

Similar level:

John Dirk Walecka, Introduction to General Relativity, World Scientific, 2007
Benjamin Crowell, General Relativity, http://www.lightandmatter.com/genrel/

Somewhat lower level:

Bernard Schutz, Gravity from the Ground Up : An Introductory Guide to Gravity and General Relativity, Cambridge University Press, 2003, ISBN 0-5214-5506-5.
Edwin F. Taylor and John Archibald Wheeler, Exploring Black Holes: Introduction to General Relativity, Addison Wesley, 2000, ISBN 0-2013-8423-X.

The (advanced) classics:

Charles W. Misner, Kip S. Thorne, and John A. Wheeler, Gravitation, W. H. Freeman & Company, 1973
Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, 1972

More philosopically oriented (note that this can get you confused since it also talks a lot about alternatives, so first understand the course, then read this):

Kevin Brown, Reflections on Relativity, MathPages, 2010, Exists also online http://mathpages.com/rr/rrtoc.htm

Youtube contains many lectures on general relativity. Try searching for any of the following

"general relativity" will give you much more than you want
"General Relativity by Leonard Susskind" Both lower level and higher level lectures exist.
This channel is from The WE-Heraeus International Winter School on Gravity and Light
One student in spring23 recommended the series https://www.youtube.com/playlist?list=PLJHszsWbB6hqlw73QjgZcFh4DrkQLSCQa

General relativity courses with lecture notes exist all over the place. Examples are:

Matthias Blau, University of Berne 900+ pages, has both good introductory and more advanced parts (but uses Lagrangians for some parts)
Lectures by David Tong, Cambridge University

Books suggested by earlier students:

- Foster and Nightingale, A Short Course in General Relativity, Springer, 2006 (accessible online at Lund University)

Solutions to the exercises

Use these intelligently, i.e. first do a proper attempt to solve the problem yourself.

Solutions to a number of the exercises can be found in the appendix of the first edition of the book. A similar set of solutions for the second edition is accessible via Schutz's homepage as

FirstCourseGR2.Solutions.1_0.pdf

Other resources for the exercises:

There is a book that has the solution to about half the problems in the course book and a large number of additional problems as well. This is

Robert B. Scott, A Student's Manual for "A first course in general relativity,", Cambridge University Press, 2016, ISBN 978-1-107-63857-0.

More info via Robert Scott's notes for self study
Solutions to Problems in General Relativity (for the first edition)
There are many more sites with solutions, but be careful not everything on the web is correct.

My own solutions to the homework problems will appear in the restricted pages after the hand-in date

Lecturer

Johan Bijnens, phone 046-2220447

Course evaluation

To help us improve our courses we want you to complete a course evaluation after the examination.
The results from earlier years:

- vt06, vt07, vt08, vt09: Course not given as lectures, vt10, vt11, vt12, vt13, vt14, vt15, vt16, vt17, vt18, vt19, vt20, vt21, vt22, vt23

Academic honesty, gender equality and equal opportunity

Academic honesty, gender equality and equal opportunity for all are important for us. More information can be found in the department's guidelines.

Links

The wikipedia link collection for General Relativity might be useful.

An overview of cosmology and related results is the Universe from the WMAP collaboration.

Some others I found nice are:

About black holes
Film on gravitational waves
The General Relativity Tutorial
http://www.einstein-online.info/elementary.html
Very basic relativity
Physics of the universe
Some Nobel prizes in Physics related to things in this course, this page contains also low-level explanations:
- 1993 The binary pulsar
- 2006 Cosmic microwave background
- 2011 Cosmic expansion
- 2017 Gravitational waves
- 2019 Cosmology
- 2020 Black holes: singularities and at the center of the Milky Way

Research links: some representative examples are:

- Professional preprint server
- Relativity bookmarks
- LIGO The US gravitational wave observatory
- Living Reviews in Relativity A series of review articles on various subjects, advanced but in overview understandable with the knowledge of the course

This page can be found at

https://canvas.education.lu.se/courses/2869/pages/general-information-for-the-ongoing-slash-next-course-instance