PHY484S

Relativity Theory II (2020-21) -- PHY484S/1484S

Syllabus

Topics

Singularity theorems
When do timelike and null geodesics focus rather than spread out? Energy conditions.
Isotropic, homogeneous FRW cosmology
Isotropy, homogeneity. Coordinate systems. FRW metric. Geodesics in FRW.
Dynamical Einstein equations of motion for FRW. Components of cosmological fluids. Energy-momentum tensors.
Multi-component cosmological fluid solutions. Cosmological redshift. Recovering Hubble's Law. Distances in FRW. Age of universe.
Inflation
Introduction. Particle and event horizons. The role of the strong energy condition. What kind of scalar field can drive inflation? Slow roll approximation.
The amount of inflation. What kind of field is the inflation? How does inflation start? How does it end? Basic models of inflation.
First 3 minutes
What happened when? When do we have local thermal equilibrium? Equilibrium thermodynamics for cosmological fluids. Effective number of relativistic species.
Entropy. Beyond equilibrium. Dark matter relics. Big bang nucleosynthesis.
Cosmological perturbations
SVT decomposition. Perturbations and basic aspects of gauge invariance.
Evolution of curvature perturbations. Initial conditions from inflation.
Power spectra of curvature and matter-density perturbations. Theory vs CMB observations.
Newtonian fluids and the Jeans instability. More details on relativistic perturbations and gauge invariance.
Analytic approximations and the super-horizon limit. Fluctuation evolution. Primordial gravitational waves.
Deriving Einstein's equations
Upgrading Newtonian gravity. Relativistic scalar gravity and its experimental failures. Building a tensor theory of gravity.
The Einstein-Hilbert action. Classical fields, covariant integration, and the divergence theorem. Building an action for tensor gravity.
Deriving the field equations. Metric tensor variations. Ricci tensor variations. Surface terms. Einstein's field equations.
Alternative theories of gravity
Extra dimensions of space, experimental size limits. EM and Newtonian gravity in higher D. Kaluza-Klein reduction.
Scalar-tensor theories. R-squared, Lovelock, Gauss Bonnet, f(R), etc. Supergravity, UV issues.
Black hole thermodynamics
Laws of black hole mechanics. Hawking radiation. Black hole information paradox.
Modern advances in understanding black hole entropy using strings and D-branes.
Introduction to the AdS/CFT Correspondence.
Gravitational lensing
Behaviour of light in gravitational fields. Deflection angles. Time delay. Magnification and multiple images.
Numerical relativity
ADM formalism and Hamiltonian approach to GR. Gauge choices.
Initial data. Adaptive mesh refinement and spectral methods. Applications.

Grading

Weighting
The final grade you obtain in this course will be computed from two major components:-
  1. four Homework assignments, each worth 15%, due on R04Feb, M22Feb, M22Mar, M05Apr;
  2. an individual Final Project, with a written report due on R08Apr worth 20% and a half-hour oral presentation plus Q&A during exam period worth 20%.
Homework collaboration policy
You are welcome to discuss general concepts involved in assignment problems with each other, but you must solve the problems yourself and write up individual solutions in your own words.
Lateness policy
Homeworks are due by 10:10am (the start of lecture) on the specified due dates. The lateness penalty is 5% per day up to a maximum of one week beyond the due date.
To help alleviate academic pressure at the busiest times of the semester, each student gets a bank account of seven Grace Days, which allows handing in homeworks late without penalty. You may use all seven on one assignment, or spread them out differently. My only rule is that you must inform the TA by email that you will be using a specific number of Grace Days at least 24 hours before the original due date.
Handing in
Homeworks must be handed in electronically in PDF format on Quercus. If you do your homeworks on paper, you must scan or photograph them to a PDF file with decent contrast and a file size that is not too large. No other formats will be accepted.
If you prefer to prepare your homework assignments in LaTeX, that is just as good as handwriting from my perspective. Regardless of the format you use, you must clearly show the steps in your working in order to get full credit for a problem.

Accessibility accommodations

Students with diverse learning styles and needs are welcome in this course. In particular, if you have a disability/health consideration that may require accommodations, please feel free to approach me as well as Accessibility Services. I am disabled myself, and very motivated to be decent to students managing physical and/or mental health disabilities. Here are some further important details on deadline extensions and accessibility accommodations.

Equity, Diversity, and Inclusivity

At the University of Toronto, we strive to be an equitable and inclusive community, rich with diversity, protecting the human rights of all persons, and based upon understanding and mutual respect for the dignity and worth of every person. We seek to ensure to the greatest extent possible that all students enjoy the opportunity to participate as they see fit in the full range of activities that the University offers, and to achieve their full potential as members of the University community.

Our support for equity is grounded in an institution-wide commitment to achieving a working, teaching, and learning environment that is free of discrimination and harassment as defined in the Ontario Human Rights Code. In striving to become an equitable community, we will also work to eliminate, reduce or mitigate the adverse effects of any barriers to full participation in University life that we find, including physical, environmental, attitudinal, communication or technological.

Our teaching, scholarship and other activities take place in the context of a highly diverse society. Reflecting this diversity in our own community is uniquely valuable to the University as it contributes to the diversification of ideas and perspectives and thereby enriches our scholarship, teaching and other activities. We will proactively seek to increase diversity among our community members, and it is our aim to have a student body and teaching and administrative staffs that mirror the diversity of the pool of potential qualified applicants for those positions.

We believe that excellence flourishes in an environment that embraces the broadest range of people, that helps them to achieve their full potential, that facilitates the free expression of their diverse perspectives through respectful discourse, and in which high standards are maintained for students and staff alike. An equitable and inclusive learning environment creates the conditions for our student body to maximize their creativity and their contributions, thereby supporting excellence in all dimensions of the institution.

Academic Integrity

Academic integrity is fundamental to learning and scholarship at the University of Toronto. Participating honestly, respectfully, responsibly, and fairly in this academic community ensures that the UofT degree that you earn will be valued as a true indication of your individual academic achievement, and will continue to receive the respect and recognition it deserves.

Students are expected to know what constitutes academic integrity: Familiarize yourself with the University of Toronto's Code of Behaviour on Academic Matters. It is the rule book for academic behaviour at the UofT. Potential offences include, but are not limited to:

In papers and assignments:
Using someone else's ideas or words without appropriate acknowledgement.
Copying material word-for-word from a source (including lecture and study group notes) and not placing the words within quotation marks.
Submitting your own work in more than one course without the permission of the instructor.
Making up sources or facts.
Including references to sources that you did not use.
Obtaining or providing unauthorized assistance on any assignment including (a) working in groups on assignments that are supposed to be individual work, (b) having someone rewrite or add material to your work while editing.
Lending your work to a classmate who submits it as his/her own without your permission.
On tests and exams:
Using or possessing any unauthorized aid, including a cell phone.
Looking at someone else's answers.
Letting someone else look at your answers.
Misrepresenting your identity.
Submitting an altered test for re-grading.
Misrepresentation:
Falsifying or altering any documentation required by the University, including doctor's notes.
Falsifying institutional documents or grades.

The University of Toronto treats cases of academic misconduct very seriously. All suspected cases of academic dishonesty will be investigated following the procedures outlined in the Code. The consequences for academic misconduct can be severe, including a failure in the course and a notation on your transcript. If you have any questions about what is or is not permitted in this course, please do not hesitate to contact me. If you have questions about appropriate research and citation methods, seek out additional information from me, or from other available campus resources like the U of T Writing Website. If you are experiencing personal challenges that are having an impact on your academic work, please speak to me or seek the advice of your college registrar.

Notice of video recording and sharing

During pandemic conditions, I am teaching online-only to safeguard our lives and our health. Lectures in this course, including student participation, will be recorded on video, and will be available to students for viewing remotely in any timezone. Video links will be provided via announcements on Quercus a few hours after each class. Please note that course videos and materials belong to the instructor, the University, and/or other source depending on the specific facts of each situation, and are protected by copyright. In this course, you are permitted to download session videos and other materials provided via Quercus for your own academic use, but you should not copy, share, or use them for any other purpose without the explicit permission of the instructor.

Licence terms for course materials on this website

In the spirit of academic openness, I share all my course materials except class recordings on my own website for free, rather than securing them behind a tuition firewall. I started off by sharing lecture notes online soon after I began teaching at UofT in 2000, as an automatic accessibility accommodation so that students who could not take notes in class did not have to ask for a note taker. I hope that sharing my course materials online is also useful to other UofT instructors wondering what I cover in my courses and how, and perhaps even to interested parties elsewhere. All my stuff is offered under a specific kind of licence which is less restrictive than traditional copyright but still has conditions: a Creative Commons Attribution-NonCommercial-NoDerivs Canada 2.5 licence. Please read the terms carefully.