Introduction to General Relativity

This course (Physics 3A03) is an introduction to Einstein’s Theory of General Relativity aimed at 3rd and 4th year undergraduates. The course contents are:

  1. Elements of Differential Geometry

    1.1 The Geometry of Surfaces
    1.2 More General Curved Space

  2. Special Relativity and Flat Spacetime

    2.1 Minkowski Spacetime
    2.2 Inertial Particle Motion
    2.3 Non-inertial Motion
    2.4 Conserved Quantities

  3. Weak Gravitational Fields

    3.1 Newtonian Gravity
    3.2 Gravity as Geometry
    3.3 Relativistic Effects in the Solar System

  4. Field Equations for Curved Space

    4.1 Gravity as Curvature
    4.2 Einstein’s Field Equations
    4.3 Rotationally Invariant Solutions

  5. Compact Stars and Black Holes

    5.1 Orbits
    5.2 Radial geodesics
    5.3 Singularities of the solution
    5.4 Black Holes and Event Horizons
    5.5 Quantum Effects Near Black Holes
    5.6 Stellar interiors

  6. Other Astrophysical Applications

    6.1 Gravitational Lensing
    6.2 Binary pulsars
    6.3 Astrophysical Black Holes

  7. Cosmology

    7.1 Kinematics of an Expanding Universe
    7.2 Distance vs Redshift
    7.3 Dynamics of an Expanding Universe
    7.4 The Present-Day Energy Content
    7.5 Earlier Epochs
    7.6 Hot Big Bang Cosmology


A handout with this information will be distributed at the first lecture (Monday January 7, 2013) and is also available in pdf format here.


Lectures meet Mondays from 14:30 to 16:20 and on Thursdays from 15:30 to 16:20 in HH 305. Attendance to the lectures is not compulsory, but if you come I ask you to pay attention and not disrupt the class with personal conversation. I will do what I can to ensure that you do not have to gnaw your own arm off to stay awake.


The course textbook is Gravity, An Introduction to Einstein’s General Relativity, by Jim Hartle, and has been ordered at the bookstore. I use this mostly for the assignments, and as an alternative point of view to my lecture notes (which my lectures will follow fairly closely).

Other texts which you may find useful as supplements are Sean Carroll’s Spacetime and Geometry; Steven Weinberg’s Gravitation and Cosmology; A First Course in General Relativity by Bernard Shutz; and Paul Dirac’s General Theory of Relativity. Hartle and Shutz’s books are aimed at undergraduate audiences, Weinberg’s and Dirac’s are timeless classics, and Carroll’s is a good modern introductory graduate text. While some of these will be aimed at a slightly higher level than the course, they will reward a bit of elbow grease with many insights.

Office Hours

Because I spend half my time at Perimeter Institute I may be hard to find in my office, so it is worth setting up any appointments in advance. I will reserve the time immediately after the class (starting at 16:30) if you would like to see me about anything, so either catch me in class or tell me there that you intend to meet me in my office. (I will not just hang about the office during that time unless I know students are coming by, so it is important to let me know in advance if you intend to stop in.) Otherwise, feel free to arrange another time with me on an individual basis. (I will make a point of being in my office for scheduled appointments, so if you do set up an appointment, please show up!)


The course TA is Matt Williams – willimr2 (at) – and he will hold office hours in ABB room 269D at times to be announced shortly.


The course work involves completing a weekly assignment. Like any field theory, General Relativity is a contact sport and so is only really learned by doing. It is very very strongly recommended to work the assignments even if you only audit the course.

Midterm Exam

A 90-minute midterm test will be held in class on Monday, March 4th 2013. The midterm provides the best possible practice for the final exam, so it would be silly not to write it. Those who do not write the midterm for whatever reason can avail themselves of Option C below. Be there or be square.

Final Exam

The Final Exam will be held during the April examination session.

Marking Scheme

The course marks are completely based on the weekly assignments, the midterm test and the final exam. The term mark will be computed from these according to whichever of the following formulae maximizes your final mark:

  • A) Assignment: 20%   Midterm: 30%  Final Exam: 50%
  • B) Assignment: 20%   Midterm: 15%  Final Exam: 65%
  • C) Assignment: 20%   Midterm:  0%   Final Exam: 80%

Additional Work and Supplemental Exam:

Additional work will NOT be available for students who might wish to improve their marks. The standard McMaster rules apply regarding the availability of supplemental exams.

Reading you your rights:

The Centre for Student Development offers free academic skill support (see for details).

Although hopefully it does not need saying, be warned that the University does not tolerate cheating, plagiarism and the like:


(see for more information).