Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
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):
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:
In W24, I will teach the follow-up course Quantum Field Theory for Cosmology AMATH872/PHYS785, which was last taught in W22.
Advice on how to prepare the essay:
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:
More Advice (also e.g. on how to prepare a presentation):
We will be using mainly material from the following three texts:
Note: These three texts are available at the Davis Library.
Recommended general references are also:
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
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. T^{ab}_{;b} = 0. T^{ab} 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.)
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
PDF files require Adobe Acrobat Reader
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is co-ordinated within the Office of Indigenous Relations.