Relativity Theory II (2019-20) -- PHY484S/1484S




Upgrading Newtonian gravity
Relativistic scalar gravity.
Building an action principle for the gravitational field
Building a tensor theory of gravity with the right Newtonian limit. Classical fields living in spacetime and covariant integration. Building an action principle for gravity.
Deriving Einstein's equations
Metric tensor variations. Ricci tensor variations. Surface terms. Einstein equations of motion.
Singularity theorems
When do timelike and null geodesics focus? Energy conditions.
Gravitational lensing
Deflection angles. Time delay. Magnification and multiple images. Experiment.
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.
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. (if time permits)
Analytic approximations and the super-horizon limit. Fluctuation evolution. Primordial gravitational waves. (if time permits)
Black hole thermodynamics
Laws of black hole mechanics. Hawking radiation.
Black hole information paradox. Baby introduction to string theory advances in understanding black hole entropy.
Gravity and extra dimensions
Visualizing extra dimensions and experimental limits. Electromagnetism and Newtonian gravity in higher dimensions. The Kaluza-Klein energy gap. KK reduction.
Alternative theories of gravity
Scalar-tensor theories. Examples. R-squared, Lovelock, Gauss Bonnet, f(R), etc.
Numerical relativity
ADM formalism and Hamiltonian approach to GR. Gauge choices.
Initial data. Adaptive mesh refinement and spectral methods. Applications


The grade you obtain in this course will be calculated from

Homework collaboration policy: students may discuss generalities of the assignment problems together, but must write up individual solutions in their own words. I do not want to see solutions copied from another student!

Lateness policy: Homeworks should be handed in to the professor in person or, if they are unavailable, to the Physics Undergraduate Office. Homeworks are due by 10:10am (the start of lecture) on the pertinent day. The lateness penalty is 5% per day up to a maximum of one week beyond the due date. See also the calendar of events and more details on deadline extensions and accessibility accommodations.

Writing requirements

Please do not hand in any graded work handwritten in pencil -- homeworks or final exams. Pencil can later be erased and changed during a grading dispute. If you write in pencil, you will not be allowed to request a regrade. Another reason to avoid pencil is that pencil written assignments are typically hard to read under artificial light, and their scans often have insufficient contrast. Please handwrite in black or blue pen (avoid red, for obvious reasons). Crossouts are fine, as long as they are clearly marked.

Accessibility is something I take seriously, because I am a person with disabilities myself. It is the main reason why I provide lecture notes online for all my courses. So 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 show all the key steps in your working in order to get full credit for a problem.

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.
Falsifying or altering any documentation required by the University, including doctor's notes.
Falsifying institutional documents or grades.

To remind you of these expectations, and help you avoid accidental offences, I will ask you to include a signed Academic Integrity Declaration with every assignment. If you do not include the statement, your work will not be graded.

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.

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.