Relativity Theory I (2017-18) -- PHY483F/1483F
This course will introduce General Relativity from a physics point of view, emphasizing a modern perspective. I will begin with a review and rephrasing of special relativity, including the case of constant acceleration. Then I will motivate curved spacetime, the covariant derivative, and the Christoffel connection. I will derive the equation for a geodesic, the analogue of a straight line in curved spacetime. From there, I will develop the Riemann curvature tensor, and show how it encodes useful physics like tidal forces and geodesic deviation. Then I will introduce the famous Einstein equations and show how they can be used to study equations for stars and black holes. I will showcase some of the experimental successes of GR, before finishing up with a description of the origin of Einstein's equations starting from an action principle and a brief introduction to cosmology. (For more details, please see the syllabus.)
Classes are held on Mondays and Thursdays from 11:10-12:00 and tutorials are held on Fridays from 13:10-14:00, both in MP134. My regular weekly office hours are held on Tuesdays from 16:15-17:30 in MP1118.
Tips for understanding lectures
I frequently update the lecture notes to make them clearer and to remove typos, after questions that I get (a) in class, (b) after class in the corridor, and (c) during office hours. So if some point in the lecture notes didn't seem clear the first time you tried reading about it, try reloading the notes to see if there is an upgraded explanation. You can tell if I have updated the notes because I put the date stamp for each version on the front page.
It is often easier to get confident with the material if you read the lecture notes before class. I never cover more than 6 pages per lecture, so if you read the next 6 pages beyond where we finished last time, you should find classes easier. I welcome questions in class, as long as asking them is likely to benefit other students as well as you. If your question is more individual, I am happy to answer it in office hours.
These will be posted in reverse chronological order, to make the latest ones easiest to find.
- R21Sep: Homework 1 is posted. You can already do Q1 and Q2; by this time next week we will have covered everything you need for Q3 as well. Start on Q1 and Q2 now, and feel free to make full use of my office hours to ask any question you like about HW1.
- W20Sep: This is a note for all students, but especially new graduate students who may not yet be familiar with all UofT procedures. Remember: the primary mechanism UofT uses to send you course-related email announcements is your UTmail+ (@utoronto.ca) account, *not* your departmental account. Make sure you have your email app configured to send and receive UTmail+ as well as departmental email. Also, note that you need to learn how to use Blackboard, our UofT courseware, in order to access online tutorial notes from the TA. Log in to portal.utoronto.ca to get started.
- M18Sep: Today I began by finishing the discussion of the action principle for the relativistic point particle that I started last time. Then I moved to introducing some important new four-vectors. First, I showed how the partial derivative four-vector naturally arises as a covariant vector with a downstairs index. Then I introduced the four-vector gauge potential of electromagnetism and showed how the electric and magnetic fields are embedded in the gauge field strength tensor, which naturally arises with two downstairs indices and is antisymmetric. I also introduced the 4-vector current and showed how Maxwell's equations can be written in much more succinct form in relativistic notation. (You will flesh out the details of how this works in HW1 problem 2.) Then I described how to understand the case of constant relativistic acceleration in special relativity, and mentioned that you can use the formulae I derived to solve the Twin Paradox. In brief: the physics resolution of the
paradox is that yes, the accelerating twin does age less quickly than their homebody twin! (You will flesh out the details of this in HW1 problem 1.)
- R14Sep: Today, I started with the invariant interval and the definition of the light cone. Then I introduced four-vectors and 4D Minkowski spacetime. In particular, I introduced the Minkowski metric tensor, which is used to raise and lower indices, or to take the inner product between two 4-vectors. I noted that in this course I will be using the mostly minus signature convention for the Minkowski metric. Then I introduced the general definition for a tensor: something that transforms in a particular well-defined way under coordinate transformations. The general utility of tensors in physics is that they are the objects we use to write down dynamical equations, like generalizations of Newton's 2nd Law. I gave examples of the momentum 4-vector, and the 4-velocity and the 4-acceleration vectors for massive particles. In any tensor equation we might write, the number and type of indices must match on the left and right hand side, like in the equation relating the 4-momentum of a point particle to its mass (a relativistic invariant) and its 4-velocity. I finished up by discussing action principles for point particles.
- M11Sep: Today, I introduced the concepts of contravariant and covariant vectors in 3D Euclidean space, which you can think of as like column vectors and row vectors respectively. I explained that rotations have the important property of leaving the norm of a 3-vector invariant, and I showed how rotations affect the components of 3-vectors. Next, I introduced index notation and the Einstein summation convention, and showed that they provide a simpler and more compact way of representing multiplication of matrices and vectors. Then I introduced the Kronecker delta tensor, which is used to raise and lower vector indices or to take the inner (dot) product between two vectors in Euclidean 3-space. I also introduced the permutation tensor, which can be used in 3D to take the outer (cross) product of two vectors. Then I showed how Lorentz transformations look a lot more natural in the rapidity parametrization.
- F08Sep: The TA has brought to my attention the fact that the room for our tutorial has been changed to MP134. I really apologize for the confusion!
- R07Sep: Most of what I discussed today was organizational matters -- the syllabus, when homeworks and the midterm will be, textbooks, prerequisites, deadline and accommodation policies, etc. Please read through all the pages on this website to familiarize yourself with their contents, especially the syllabus and calendar of events. Office hours for the prof. were settled to be on Tuesdays from 16:15-17:30.
- R03Aug: The first day of class will be Thursday 7th September at 11am in MP134, and I look forward to meeting you all. Bring your calendars: we will decide on my office hours together.