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 course of the next 10 years as other faculty taught the class, the curriculum (which had never been clearly stated in any of our department's documentation) adiabatically drifted as people adjusted slightly to emphasize topics that they particularly like. Though each shift wasn't too large, the cumulative impact of these changes was to make a course that had become quite a bit more challenging than 8.033 (which is primarily targeted at first-semester sophomores just starting the major) was originally intended to be.
A careful study by the department agreed that 8.033 needed to be adjusted, and proposed a new curriculum. Part of this planned curriculum was to move some of the advanced material on general relativity into a new course to be taught over IAP, leaving more time to carefully develop the special relativistic foundations. The notes below are how I implemented this curriculum in the Fall 2021 semester. (Should you be interested, my old notes are still available here.)
Please note that 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!
2. Electromagnetic radiation: Maxwell's equations and electromagnetic waves. The speed of light; the search for a "rest frame" for light, and our failure to find it.
3. Lorentz transformations: Consequences of light's constant speed; time dilation and length contraction; the Lorentz transformation.
4. Spacetime: How to combine space and time into a single entity in a way that make sense from the viewpoint of the Lorentz transformation. The invariant intervals, our first example of a spacetime invariant. Spacelike, timelike, and lightlike (also known as null) separations.
5. Introduction to 4-vectors: The spacetime displacement vector and its transformation properties. Vector components and index vectors. Introduction to the index notation.
6. Kinematics: Velocity addition rule. Conservation of momentum, relativistic kinetic energy. The derivation of E = mc2.
7. 4-momentum and 4-velocity: Unifying momentum and energy into a single spacetime object; an energy-momentum invariant. More on 4-vectors in general, particularly the scalar product of 4-vectors. 4-velocity, and its relation to the rather more familiar 3-velocity.
8. Using 4-momentum: Important applications of this concept --- the energy measurement by an observer; collisions and decays; the center of momentum frame; scattering; Doppler effect and aberation of propagating light. Emphasis on how to exploit invariant relationships to expedite calculations.
9. More mathematical structure: The spacetime metric tensor; dual vectors; tensors more generally.
10. Describing bulk matter (as opposed to pointlike particles): the number current 4-vector; its close relation to the charge-current 4-vector; the stress-energy tensor.
11. Electromagnetism I: How to organize the electric and magnetic field to insure that the forces they produce are clearly Lorentz covariant. Brief discussion of forces and accelerations; the Faraday field tensor and the Lorentz force law. Transformation of the electric and magnetic fields.
12. Electromagnetism 2: The electromagnetic field equations. The dual Faraday tensor; automatic conservation of source. Electromagnetic invariants. Gauge freedom and potentials.
13. Acceleration and forces: 4-acceleration and the momentarily comoving reference frame. A uniformly accelerated observer and their motion. Forces: 4-forces versus 3-forces. How 3-forces transform between reference frames.
14. Prelude to gravity: The geometry of spacetime according to a uniformly accelerated observer. Rindler spacetime and Rindler coordinates. The de-synchronization of separated but uniformly accelerated clocks. (Note, some of the material presented here is more advanced than the core of 8.033.)
15. Introduction to gravity: The principle of maximum aging. Considerations on relativistic gravity; the action of gravity on light.
16. The calculus of variations: Introduction to Euler's equation; application to the principle of maximum aging. Brief discussion of Lagrangian mechanics, and how we use this in relativity.
17. The principle of equivalence: Farewell to global Lorentz frames; considerations on locally Lorentz frames. Considerations on a relativisic theory of gravity; qualitative sketch of the Einstein field equation.
18. Consequences of relativistic gravity 1: Some important solutions to the Einstein field equation; using those solutions. The Newtonian limit; qualitative discussion of perihlion precession.
19. Consequences of relativistic gravity 2: Strong gravity; examining motion of light and material bodies in the Schwarzschild spacetime. Seemingly perverse behavior in the very strong-field of this spacetime.
20. Consequences of relativistic gravity 3: Exploring strong gravity. Resolving the seemingly perverse behavior of the previous lecture by focusing on what is being measured, and the meaning of different clocks' measurements. The event horizon and black holes. Orbits of black holes; their non-Newtonian behavior.
The next two lectures present material that goes beyond the core of 8.033, discussing recent measurements and observations that relate to material we have discussed this term.
21. Data on strong gravity: How measurements in very strong spacetimes have confirmed the predictions of general relativity and given us confidence that Einstein's theory describes gravity very, very accurately.
22. Cosmology: How application of general relativity to the large-scale structure of our universe works, but leaves us with deep questions, and a lot of mysteries still to be unraveled.