General Relativity for Cosmology AMATH875/PHYS786 in F2023

Tadpole Galaxy

  • Term: Fall 2023
  • Course codes: AMATH875/PHYS786
  • Instructor: Achim Kempf 
  • Instructor Office hours: Wednesdays 12:30-1:30pm, MC6334, starting Sep 20
    Exceptions: Office hours of Sep 27 and Oct 4 will be 10:30-11:30am in MC6334
  • TA: Dorsa Sadat Khajouei (d2sadathosseinikhajouei at
  • Prerequisites: A first course in General Relativity or consent of instructor
  • The Lectures

    Regular Time & Venue: 
    Tue + Thu 2-3:20pm, in the Alice Room at Perimeter Institute 

    * The class of Thu. Sep 21 will be in the Bob room.
    * The class of Tue. Oct. 17 will be in the Space room.

    First lecture: Thu Sep 7
    Reading week: no lectures Tue Oct 10 and Thu Oct 12.
    Last lecture: Tue Dec 5

    The lectures will be live on Zoom: here and recorded on PIRSA: here

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. 



Health precautions (permanent):

  • If you have cold/flu/covid symptoms, don't come to class. Do the right thing, which is taking care of yourself and getting healthy first. 
    • For videos of the lectures scroll down to the bottom of this page.
    • All important updates will always be posted here.


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 W24, I will teach the follow-up course Quantum Field Theory for Cosmology AMATH872/PHYS785, which was last taught in W22.


  • Assignments:
    • For each lecture, except lecture 1, write a summary of at most one page. This leaves not much space for details, which is OK because the point of the assignments is to demonstrate understanding of the flow of the material. Make sure to write in full sentences, i.e., do not just list items.
    • Each Monday (except for reading week) at noon, upload via Crowdmark, a single file with 2 pages of pdf, one page for each of the prior week's lectures. The assignments need to be typeset.
    • To give you an idea of what's expected in a summary, here is an Example Summary of Lecture 1. While Lecture 1 didn't contain equations, in your summaries for the the later lectures, the most important equations should be displayed.  
    • While the assignments will be graded, these grades will not enter the final grade for the course. But a minimum average of 75% in the assignments is needed to pass the course.
  • The grade for the course will be based on an essay.
    • Deadline: 11:30pm on Friday Dec. 15, 2023. Submit via Crowdmark.
    • Essay topic. You can choose from these three topics for your essay: 
      • Summary of spectral geometric results
        Start your research for this essay with reviews on spectral geometry, such as the review by Datchev and Hezari, or also, e.g. here. See also, e.g., part II of the textbook by Mukhanov and Winitzki for an important type of applications of spectral geometric methods, in that case to quantum field theory.
      • Review of applications of spectral geometric ideas to quantum gravity
        This essay topics covers work of mine and others. For example, see here, here, here, here, here, here and here.
      • Overview of models of inflationary cosmology
        The essay should be a broad overview of what inflationary models have been considered and are being considered, with a discussion of pros and cons of models, for example, the pros and cons choices of inflaton potentials, or the pros and cons of single field vs multi-field inflation, or the experimental status of competing models. Focus on the general relativity part, not the quantum part.

Advice on how to prepare the essay: 

  • Format: title and abstract page/motivation/main parts/summary (or conclusions)/bibliography.
  • Bibliography: List all of your sources explicitly. Of course you can use Wikipedia and the new Bing but you should not cite it - because it can change from day to day and because they are not (yet) reliable enough to meet scientific standards. Instead, cite books and papers that you may have found via Wikipedia or the Bing or similar tools. Also, it is good style to list items in the bibliography in that sequence in which they are first referred to in the text.
  • Essay length: strictly at most 12 pages.
  • An essay should be a review of existing literature on the 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 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 campus (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. This also includes the software that you use. 

  • ChatGPT and GPT4 or similar tools: 

    • Yes, do use AI tools if you find them useful. 
    • Don't trust them. Check their output very carefully. This includes content as well as references.
    • No need to specify whether or not you used AI tools. 
    • As a consequence of the free availability of the AI tools, the expectation today is that the submitted essays have impeccable writing:
      • The writing should be clear and concise 
      • There should be virtually no typos or grammatical errors.  

More Advice (also e.g. on how to prepare a presentation):

  • See my handbook for new graduate students which has detailed essential advice also on giving presentations (and other topics). The art of giving presentations is extremely important, it can make or break careers. Study this carefully. 


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

  • If you want to look ahead already, here are videos and lecture notes of a prior teaching of this course: 

    Video Recordings of the lectures of a pre-pandemic teaching of this course are here: PIRSA 

    Here are the Lecture Notes:

    Lecture 1: Overview. Differentiable manifolds.   

    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: 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.)


    Tadpole galaxy