AMATH 875, PHYS 786

General Relativity for Cosmology

Fall 2009
AMATH 875 / PHYS 786
Instructor: A. Kempf,  (MC6071, ext. 5462)
Prerequisite: Introductory general relativity, e.g., AMATH 675.
Time and venue: Mon + Thu 4-5:20pm, Bob room at PI
Video-linked to:

     *  Univ. of Waterloo, Room MC6091
     *  Univ. of Guelph, Room Rozanski106.
     *  McMaster Univ., Room ABB131
Office hours: by arrangement
Literature: Detailed lecture notes will be freely available electronically. Also, see the texts listed below.  


This course begins by introducing the differential geometry of Lorentzian manifolds from scratch and then builds up quickly to the advanced framework in terms of differential forms and the vielbein formalism. These methods are then used to define general relativity, also as a gauge theory. We then study some of general relativity's deeper properties, such as the formalism of spinors, and aspects of the causal structure and singularities. One key goal is to lay the foundations for students who wish to proceed to studies in quantum gravity. We then apply general relativity to cosmological models and to cosmological perturbation theory. Thereby, we are covering the theory of cosmic inflation which is very successful in predicting, in particular, the properties of the cosmic microwave background radiation.

New schedule for the lectures:

Mon Oct 26: 4-5:20pm (unchanged)
Thu Oct 29:  4-6:50pm (doubled session)
Mon Nov 2: 4-5:20pm (unchanged)
Thu Nov 5:  4-6:50pm (doubled session)
Mon Nov 9: 4-5:20pm (unchanged)
Thu Nov 12:  4-6:50pm (doubled session)
Mon Nov 16: 4-5:20pm (unchanged)
Thu Nov 19:  4-6:50pm (doubled session)

Lecture Notes:

Continually posted here in pdf format: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22

Note: If you have read the lecture notes or viewed the recordings without having being enrolled in the course, would you mind sending me an email please? I'd just like to know. Thanks!

The video recordings of the lectures are posted here.

You may also be interested in viewing the colloquium that I gave at Perimeter Institute on Sep.16, 2009:
"Spacetime can be simultaneously continuous and discrete, in the same way that information can".
For the recording, see here.


Grades:    60% Essay, 40% Final

Thursday, Dec. 10th, 4-5:15pm, PI Bob room.

due as a pdf file via email by 0.01am on Saturday., Dec. 19th.

Eligible essay topics:

  • From `Can one hear the shape of a drum?' to the spectral geometry of Riemannian manifolds without boundaries
  • The relationship between the Dirac and the Laplace operator in curved spacetime
  • Review of general relativity as a gauge theory
  • How is general relativity different in dimensions other than 1+3?
  • Arguments for and against the presence of torsion
  • Hamiltonian formulation of general relativity
  • Models of cosmic inflation (types of potential, and/or multi-field models)


What is expected in an essay:

An essay should be a review of existing literature on a given topic. The sources can be textbooks or review articles or original articles or some of each. All and everything that is used needs to be cited. Most articles are now available online and for example "Google Scholar" can get you there quickly. Try for example searching for a few key words along with the words "Review" or "Introduction". Most electronic journals require a subscription, which the university library usually has. For the license to be recognized you may need to browse either from a university computer (the domain is what counts) or you log into the library web site from home and go to an electronic journal through the library's electronic journal search engine. 

In the essay, your task is to show that you have understood and critically reflected upon the material by making it your own. You make it your own by coming up with an original way for presenting the material that you are bringing together. Try to give it your own angle or spin. Wherever possible, try to put things into a larger context. Sometimes (hopefully very rarely) it may be necessary to stick quite closely to a source, e.g., when a calculation is to be presented and the source does it in a way that is just hard to improve upon. In this case, you can make it your own for example by filling in a few steps in the calculation that the author omitted. In this case, it is important that you point out at that place that you do so. Filling in steps obviously proves that you understood that calculation.

No original research is expected. But, you are encouraged to make educated speculations about what interesting things could be done in this area. You have been a regurgitating undergraduate for a long time. This is an opportunity to show that you still have some creativity left! Don't worry, you are not expected to solve the problem of quantum gravity here. Just show that you are thinking for yourself.

Essay format:

  • Length: 15-20 pages, pdf format
  • Format: title+abstract page / motivation / main parts / summary (or conclusions) / bibliography
  • It is very important that you refer to your sources explicitly, i.e. completeness of the bibliography is very important. List items in the sequence in which you are referring to them in the text.
  • Deadline: 0.01am, Saturday Dec 19th, via email.

Plagiarism alert:
Sorry to bother you with this disclaimer since for most of you that's not a concern at all, of course. But universities (correctly) take this issue very seriously and so I need to point this out: Yes, nowadays it is easy to cut and paste together an essay. But, when this is suspected, it is actually very easy to check for it. Did you know that when you put a few consecutive words of a text in quotation marks and google for them the source will generally pop right up?


We are using mainly material from the following three texts:

  1. N. Straumann, General Relativity with Applications to Astrophysics, Springer (2004)
  2. J. Stewart, Advanced General Relativity, Cambridge (1991)
  3. S. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge (1973)

These three texts are available on 1-day course reserve at the Davis Library (along with a hardcopy of the lecture notes, which have the library code: UWD1490).

Recommended general references are also:

    1. Scott Dodelson, Modern Cosmology, Academic Press, San Diego, (2003)
    2. A.R. Liddle, D.H. Lyth, Cosmological Inflation and Large-Scale Structure, CUP (2000)
    3. G.F.R. Ellis and J. Wainwright, Dynamical Systems in Cosmology, CUP (1997)
    4. R. M. Wald, General Relativity, University of Chicago Press (1984)
    5. H. Stephani, General Relativity, Cambridge University Press (CUP) (1982)

We will cover Sakharov's "induced gravity" argument. You can find the original (very short) paper here: Sakharov, and this review.

Here are links to general online reviews:

    1. Through the SLAC library at Stanford: here
    2. Living Reviews in Relativity: here

You may also wish to have a look at these free only resources:

  1. A collection of links
  2. Straumann (2005)


This course is one in a group of four related graduate courses whose curricula have been coordinated so as to optimally complement another:

  • AMATH875/PHYS786, General Relativity for Cosmology, (this course) taught F09, F11 etc.
  • AMATH872/PHYS785, Introduction to Quantum Field Theory for Cosmology, taught W08, W10 etc. 
  • PHYS784/AMATH874,  Advanced Techniques in General Relativity and Applications to Black Holes (and gravitational waves), taught W08, W10, etc.
  • PHYS703,AMATH873, Quantum Field Theory (for high energy physics), taught W09, W11 etc.

These courses can be taken in arbitrary sequence and no course is a pre- or anti- requisite for another.


Last Modified:  Friday 1 October 2009