Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
Visit our COVID-19 information website to learn how Warriors protect Warriors.
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.
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'll try to be always careful. The purpose is to prepare students for studies in quantum gravity, relativistic quantum information, black hole physics and cosmology.Quick summary of the contents:
In W20, I am planning to teach the follow-up course Quantum Field Theory for Cosmology, which was last taught in W18: AMATH872/PHYS875.
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.
Assigned viewing: 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.)
Here are the video recordings: PIRSA
If you are not enrolled here at UW but are using the videos and/or lecture notes please drop me a line, I'd just like to know. Thanks!
The grades will be based on an essay about the general relativity part of this course and a project on the cosmology part of this course, in equal parts.
Deadline for submitting both essay and project, combined in one PDF file not exceeding 20 pages, is 10:00am on 24 December 2019.
The essay topic is taken from the first half of the course. The topic will be: "Mathematical methods to capture shape".
Your task is to review the various mathematical methods that are being used to describe the "shape" of a Riemannian or Lorentzian manifold. Remember, there are methods to describe the shape of a manifold by using the fact that a nontrivial shape implies violations of Pythagoras' law. Then there are methods using deficiency angles or causal dynamical triangulations (look it up), and methods that use the nontriviality of parallel transport or the affine connection, the tetrad fields that diagonalize the metric-tensor, the overtone spectrum and spectral geometry (for compact Riemannian manifolds), the Weyl and Ricci curvature decomposition, the causal structure plus a conformal function description, and there are probably many others. Try to cover several ideas and approaches and feel free to include new ideas if some come to mind. The point is to explain some of those approaches and how they do or may relate to each other in your own words - possibly with a discussion of the possible pros and cons of these methods, i.e., with a discussion of which approach might be useful for which purpose. Focus on the main ideas and concepts rather than getting lost in details. You can choose to write much more on one or two approaches than on others. But in any case your essay should contain an overview over and a discussion of a range of methods.
The project topic is chosen from the second half of the course (in particular the lectures 15, 16, 17): "Numerical explorations of the dynamics of FRW cosmologies".
Your task is:
Briefly explain the origin of the three coupled equations that determine the time evolution of the scale factor, the energy density and the pressure in FRW cosmological models.
Main task: Study the past and future evolution of the night sky. Namely, explore the behavior of the flat FRW solutions numerically, i.e., numerically simulate the time evolution of the universe under different conditions. Here is a suggestion for how you may want to structure your explorations (you are free to structure your explorations differently):
For practice, and as a consistency check, I would suggest to first try to recover numerically the behavior in each cosmological epoch that we solve for analytically in class. Namely, make an assumption for a fixed value of the equation of state parameter, separately for the epochs (e.g., of radiation, dust and cosmological constant domination) that follow each other as the universe gets more and more diluted. Then solve the dynamical equations and plot the results for each epoch and compare with the analytically predicted behavior.
Then, make the equation of state parameter w vary in a prescribed way. To this end, assume that w is a given function w(rho) of the energy density, that you choose so that w(rho) runs successively between the values for radiation, dust and vacuum energy dominance as the universe gets diluted, i.e., as rho drops. Notice that w(rho) should not drop uniformly but should drop so as to reflect the presence of more or less clearly defined epochs connected by transitional periods.
One question you may want to explore whether the behavior of the dynamics is sensitive to how slowly or how abruptly w changes. For example, w may changes very quickly at a time when the matter in the universe undergoes a phase transition.
Here is another question you may want to address: No known form of matter, or of vacuum energy, or combination thereof, would lead to a value of w < -1. However, several experiments, including recent experimental data. "Leading dark energy theory incompatible with new measurement" on Scientific American indicate that the evolution of the universe is currently such that w is smaller -1, namely approximately -1.19. Explore numerically and/or analytically what the long term fate of the universe will be according to general relativity, if indeed w is and will stay at that value.
Another question that you may want to explore numerically and/or analytically is the question of whether there are choices of w(rho) such that a shrinking FRW universe would not collapse completely, i.e., so that the scale factor would get smaller only down to a finite smallest value and then grow again (a so-called bounce).
You may also want to consider how things change with K=1 or K=-1.
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.
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.
I wrote a little handbook for new graduate students which has more advice on communication skills, such as essay writing and talk presentations.
Format: introduction/motivation/methods/results/discussion/references.
Make sure you list your sources and the technology (software in particular) that you may be using.
At most about 10 pages.
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
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 promised 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 centralized within our Indigenous Initiatives Office.