8.033: The physics of spacetime

In 2010, I took over lecturing 8.033, MIT's course on special relativity (SR). It's a bit of an odd element in our undergraduate curriculum, since SR is not really enough by itself to constitute a full semester-long course. I decided to focus on the idea that Lorentz symmetry is the guiding principle of the subject, and showed how by developing a notation that makes incorporating this symmetry "automagic," most of SR can be built up in a fairly straightforward way. The course concludes by discussing (very, very briefly) some of the concepts of general relativity (GR), showing how with the framework we developed for SR, GR doesn't require a very big conceptual leap. (Although, it does require a rather larger leap in terms of the kinds of calculations and computations that need to be done. As a consequence, 8.033 can only scratch the surface of this topic.)

Over the two years that I developed 8.033, I developed a decent set of handwritten notes on the subject, which multiple people have written asking about. I have accordingly decided to post them on my MIT webpage. Note that Fall 2011 is the last semester I taught this course. I went on sabbatical in Fall 2012, and Peter Fisher took over the lectures; he was followed by Tracy Slatyer. If I had taught the course more, I might have cleaned up the notes some more, perhaps even typed them up. Since I did not, these notes are the best I've got available. They follow the schedule and syllabus of Fall 2011, and likely do not synch up with the schedule of other semesters.

These notes are provided as is; I don't guarantee that they are free of typos or stupid mistakes. Feel free to send me corrections via email. I may post corrections when I have time, but don't hold your breath waiting!


1. Introduction: Concepts of relativity; Galilean transformations.

2. Galilean relativity and waves: How the wave equation behaves under a Galilean transformation; electromagnetic waves; failure to find the "ether" in which these waves live; constancy of the speed of light.

3. Lorentz transformations: Consequences of light's constant speed; time dilation and length contraction; the Lorentz transformation.

4. The geometry of spacetime: How to combine space and time into a single entity in a way that makes sense from the viewpoint of the Lorentz transformation. Invariant intervals; spacelike, timelike, and lightlike separations.

5. Geometric objects in spacetime: 4-vectors and their transformation properties. Introduction to the index notation. Kinematics of moving objects in SR.

6. More kinematics: Velocity addition rule. Conservation of energy and momentum; derivation of E = mc2.

7. 4-momentum: Unifying momentum and energy into a single spacetime object. Spacetime invariants more generally; 4-velocity.

8. Accelerated motion: The twin paradox, forces.

9. More on forces: Summary of electrodynamics; spacetime tensors.

10. More on tensors: The metric tensor; covariant and contravariant 4-vector components; how this notation allows us to construct invariants and examine how quantities transform between frames easily.

11. More transformation laws: Focus on transformation of electromagnetic fields.

12. Transformation of fields 2: Another way to understand the transformation of electromagnetic fields, focused on the way sources transform.

13. Interlude: Principle of least action, preparing for an introduction to general relativity.

14. Principle of least action: Fermat's principle, Lagrangian mechanics. Constants of the motion.

15. Motion in spacetime: Principle of extremal aging, geodesics. Introduction to field theories.

16. Relativistic field theories: Applying the principles we have learned to build a theory of interactions due to some fundamental field in manner consistent with the requirements of relativity.

17. Gravity 1: The challenges of incorporating relativity into a theory of gravity. First considerations on how to do this.

18. Gravity 2: Further details on making gravity relativistic; the principle of equivalence.

19. Geodesics in curvilinear coordinates: As preparation to modeling trajectories under gravity, first look at trajectories in special relativity, but in complicated coordinate systems.

20. Gravity 3: A cartoon-level overview of where spacetimes that describe gravity come from; some example solutions; motion in these spacetimes. Demonstration that this motion incorporates gravity. Non-Newtonian gravitational effects.

21. Strong gravity: Black holes.
21'. Strong gravity: Some details of the black holes lecture that, for some reason, are in a different PDF file.

22. Cosmology 1: The large-scale structure of our universe. Principles.
22'. Cosmology 1: Some details of the cosmology lecture that, for some reason, are in a different PDF file.

23. Cosmology 2: Our universe, at least as modern observations appear to indicate.