General Relativity for Cosmology AMATH875/PHYS786 in F2021

Tadpole Galaxy
  • Term: Fall 2021
  • Course codes: AMATH875/PHYS786
  • Instructor: Achim Kempf 
    Office hours: Tue+Thu 5:15pm. (i.e., after the interactive hours). Zoom Link 
  • TA: Hank Chen (hank.chen at uwaterloo).
    Office hours Tuesdays 12-1pm: Zoom Link
  • Prerequisites: A first course in General Relativity or consent of instructor
  • Time/venue of Interactive Hours (in place of lectures): Mondays and Thursdays, 4:00 - 5:20 pm, online. Zoom Link. 
    Video of the lectures has been pre-recorded at Perimeter Institute (see below).
  • First Interactive Hour: Thursday, 9 September 2021
  • ​Last Interactive Hour: Monday, 6 December 2021, 4pm

Picture on the right: This Hubble image lets us see half way through the age of the universe. In the "foreground" is the tadpole galaxy.



The following interactive hours needed to me moved:

  • The hour of 4pm, Monday 15 Nov. moves to 4pm Tuesday, 16 Nov. 
  • The hour of 4pm, Monday 29 Nov. moves to 4pm Tuesday, 30 Nov. (deadline for talk submissions moved by 24 hrs too)


This is an advanced graduate course which develops the math and physics of general relativity from scratch up to the highest level. The going will sometimes be steep but I try to be always careful. The purpose is to prepare for studies in quantum gravity, relativistic quantum information, black hole physics and cosmology. Quick summary of the contents:

  • Coordinate-free Differential Geometry, Weyl versus Ricci curvature versus Torsion, Vielbein Formalism, Spin-connections, Form-valued Tensors, Spectral Geometry, some Cohomology.
  • Derivations of General Relativity including as a Gauge Theory, Diffeomorphism Invariance vs. Symmetries, Bianchi Identities vs. Local and Global Conservation Laws.​
  • Penrose Diagrams for Black Holes and Cosmology, Types of Horizons, Energy Conditions and Singularity theorems, Properties and Classification of Exact Solutions.
  • Cosmology and Models of Cosmic Inflation​

In W22, I am planning to teach the follow-up course Quantum Field Theory for Cosmology AMATH872/PHYS785, which was last taught in W20.

Online format of this course

Unfortunately, our course this term is fully online. The format of our course is as follows:

  • The times originally scheduled for "lectures", namely Mon+Thu 4-5:20pm will not be lectures. Instead, Mon+Thu 4-5:20pm will be "Interactive Hours" on Zoom, to be explained below.  
  • Ahead of each "Interactive Hour", you are expected to view the recorded lecture, along with reading the lecture notes (links provided below). Also: 
    • Once every week, everyone submits to Crowdmark, by 11:59pm on Wednesdays (except reading week) a 1/2 to maximally 1 page summary of the take-home messages of the Monday and the Thursday lecture of this week.
    • Once every 4 interactive hours (depending on which of four groups you are in - see below) you present a question & answer, to be submitted (as a pdf file of slides) by 11:59pm on the day before the interactive hour.
      • Group 1: Ahmad, Amalia, Amir, Bruno. Presents Q&A: 23 Sep, 7 Oct, 28 Oct, 11 Nov, 25 Nov (every other Thursday, except for reading week)
      • Group 2: Dalila, Dan, Dorsa, Manu. Presents Q&A: 27 Sep, 18 Oct, 1 Nov, 15 Nov, 29 Nov (every other Monday except for reading week)
      • Group 3: Michael, Mohammad, Nate, Phil. Presents Q&A: 30 Sep, 21 Oct, 4 Nov, 18 Nov, 2 Dec (every other Thursday except for reading week)
      • Group 4: Robin, Tales, Tony, Xiaoran. Presents Q&A: 4 Oct, 25 Oct, 8 Nov, 22 Nov, 6 Dec (every other Monday except for reading week)
      • The questions/answer for an Interactive Hour are to be based on material from the present and the previous lecture.
  • In the Interactive Hour, the presentation of your exercise question and its solution should have a duration of 5 minutes plusminus 2 minutes. This will leave us time to discuss. The Interactive Hours will be recorded and posted on Learn (since there were problems with the Zoom cloud). 
  • In addition, an essay of up to 10 pages pdf is to be submitted to Crowdmark by 9pm, 23 Dec. 2021.
    • You can choose from three essay topics:
      1. On the mathematics and/or the physics of spectral geometry (get started, e.g., with Datchev and Hezari and papers by the instructor).
      2. General relativity as a gauge theory: how is it different from a conventional gauge theory such as QED or QCD? (get started , e.g., with this intro and its references.
      3. Summarise the range of currently favored models of cosmic inflation (start by finding relatively recent reviews of inflation) 
  • Grading scheme:
    • Essay: 50% weight.
    • Exercise questions & answers & presentation: 50% weight.
      • Of your 7 Question/Answer/Presentation in this course, only the top four will count.  
    • Weekly summaries: 0% weight.
      • Must be submitted every Wednesday 11:59pm for the Mon+Thu lecture
      • summaries will be graded 
      • minimum 75% average must be attained. 
    • Attendance at Interactive hours: 0% weight.
      • 85% attendance (17 out of 20) is expected (except for medical reasons etc).

The Lectures: Links to the Videos and Lecture Notes

Here are the Video Recordings of the Lectures: PIRSA  (recorded in F19 at Perimeter Institute)

Here are the Lecture Notes:

Lecture 1: Overview. Differentiable manifolds.   This lecture is to be viewed and read before the first Interactive Hour at 5:30pm on Thursday Sep. 9th. No submission of a summary or Q&A is needed for Lecture 1.

Lecture 2: Algebra of function germs. Algebraic definition of tangent space.

Lecture 3: Physical and geometric definitions of tangent space. Tangent bundle and its sections.

Lecture 4: Algebra of differential forms. (Anti-) derivations. Exterior derivative. Cohomology. Categories.

Lecture 5: Inner derivation. Lie derivative on forms and on general tensors. Infinitesimal diffeomorphisms.

Lecture 6: Orientability and volume. Divergence. Integration. Stokes', Green's and Gauss' theorems.

Lecture 7: Hodge *. Hilbert Space Λ. Co-derivative. d'Alembertian. Klein Gordon and Maxwell equations.

Lecture 8: Affine connection. Absolute covariant derivative. Autoparallels and parallel transport.

Lecture 9: Torsion. Geodesics. Riemann normal coordinates. Curvature. Ricci identity. Bianchi identities.

Lecture 10: Levi-Civita connection. D. Moving frames. Tensor-valued p-forms. Cartan Structure equations.

Lecture 11: Riemannian structure. Spectra of Laplacians on p-forms. Infinitesimal spectral geometry.

Lecture 12: Euler Lagrange equations. Klein Gordon, Electromagnetic & Dirac equations in curved spacetime.

Suggested additional viewing without need for a Summary and Q&A: Planck scale physics and spectral geometry

Lecture 13: Energy momentum tensor. Killing vectors. Tab;b = 0. Tab for scalar fields and perfect fluids.

Lecture 14: Static and stationary spacetimes. Weak and dominant energy conditions. Induced gravity action.

Lecture 15: General relativity in the tetrad formulation. GR as a gauge theory.

Lecture 16: Using the tetrad formalism to obtain cosmological solutions. Friedman equation.

Lecture 17:  Cosmological epochs. Flatness and horizon problems. Inflation. Quantum origin of structure.

Lecture 18: Curvature in terms of light cones and conformal factor. Conformal equivalence. Penrose diagrams.

Lecture 19: Lie algebra of Killing vector fields. Segre classification. Weyl curvature. Petrov classification.

Lecture 20: Time orientability. Chronological and causal future and past.

Lecture 21: Stable vs. strong causality. Cauchy horizons. Incomplete geodesics. Singularities. 

Lecture 22: Strong energy condition. Raychaudhury equation. Singularity theorems. Cosmological singularities.

Lecture 23: Black holes: coordinate systems, trapped surfaces, apparent and event horizons, singularities.

Lecture 24: What replaces gravitational potential energy? ADM and SB momentum and angular momentum.

Lecture 25: Curvature from quantum noise. (Results of a collaboration with previous students of this course.)


Deadline for submitting the essay in one PDF file not exceeding 10 pages, is 10:00am on 15 December 2021.

Essay topics: TBA.

General advice on what is expected in the essay:

  • Format: title and abstract page/motivation/main parts/summary (or conclusions)/bibliography.
  • Bibliography: Again, list all of your sources explicitly. Of course you can use Wikipedia but you should not cite it - because it can change from day to day and because as it not (yet) reliable enough to meet scientific standards. Instead, cite books and papers that you may have found via Wikipedia. Also, it is good style to list items in the bibliography in that sequence in which they are first referred to in the text.
  • At most about 10 pages.
  • An essay should be a review of existing literature on a given topic. The sources can be textbooks, lecture notes 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 website 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. 

  • In your essay, there is of course no need to explain very basic concepts. If you cover advanced concepts that we covered in the course, they should be at least briefly introduced. Any material beyond the lecture notes should be properly introduced and explained. 

  • A good essay describes. An excellent essay explains.

  • 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 in you! Don't worry, you are not expected to solve the problem of quantum gravity here. Just show that you are thinking for yourself.

  • Don't forget to state all your sources clearly, also for the project. This also includes the software that you use. If you use a specialized software package for GR, mention this. In this case the expectations are somewhat higher of course.

  • I wrote a little handbook for new graduate students which has more advice on communication skills, such as essay writing and talk presentations.  


We will be using mainly material from the following three texts:

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

Note: These three texts are available at the Davis Library.

Recommended general references are also:

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

We will cover Sakharov's "induced gravity" argument. Read the original (very short) paper here:

Here is a link to very nice lecture notes on Real Analysis (PMATH351) (including an introduction to topology) by my colleague Laurent Marcoux: here

Here are links to general online reviews:

You may also wish to have a look at these free only resources: A collection of links