8.962: General relativity by Professor Scott A. Hughes

MIT has a one semester course in general relativity, which I have taught several times. My personal spin on this topic is to spend half of the course focused on the formal mathematical framework (not getting too sophisticated, just making sure to carefully develop differential geometry well enough that all the important curvature tensors can be fully understood), and then spending the second half on applications (primarily astrophysical, since that is my personal expertise and research concentration). This schedule works well at MIT chronologically, since the dividing line corresponds almost exactly to the usual MIT Spring Break.

One could easily do the course in different ways: some lecturers may choose to emphasize more mathematical tools (for example, my version very sadly skips over differential forms, and leaves out many important tools for doing operations on manifolds); or, one can choose to emphasize different applications with less of an astrophysical focus. This version of 8.962 should be taken as reflecting this one lecturer's interests and (to be frank) biases regarding the most critical elements of the core. A core guiding principle for me is that 8.962 is a course intended for graduate students. Graduate students should be mature enough to do in-depth self study of facets of a subject that are of particular interest to them. In a single semester course, certain choices must be made; the choices which went into 8.962 work well for me, and leave students well placed to go much further in depth as their needs dictate.

This webpage presents the handwritten notes I have developed and used the last few times that I lectured the course. It should be strongly emphasized that these notes are due for an overhaul. Some of these pages were originally written up in 2007. In the intervening years, I have sometimes come up with better ways to present this material, and even found a few errors. In some cases, I provide typewritten addenda that correct or expand on topics discussed in some of the notes; these addenda are linked below along with the main note files.

A major reason for providing these notes now is that OpenCourseWare is publishing a new set of lectures that I recorded in Spring 2020 as I was teaching this course. (This turned out to be an awkward semester thanks to the COVID-19 pandemic. If our luck is good, you may be reading this and thinking "Ah, I remember that minor blip to society, what a droll set of months that turned out to be." Or perhaps our luck turned out bad, and you are an archaeologist from a distant star system sifting through Earth's ruins wondering what kind of people we were. Personally, I think the bulk of us were pretty good, but we had an unfortunate tendency to make some rather poor collective decisions.) The lectures and some of the material I provided to OCW reference these handwritten lecture notes, so I am providing them as they currently stand to interested readers. Please note that the handwritten notes and the recorded lectures do not sync up precisely: sometimes lectures ran long and I did not cover material in the notes, sometimes I skipped over things and left the notes for students to read on their own. The correspondence is pretty good, though.

It is not out of the question that I will clean these notes up (perhaps even typesetting them and turning them into a small book) at some point in the future. As I type this webpage, we are still dealing with the pandemic, so it is hard for me to plan much farther than a few days into the future. In the meantime, these notes are provided as is; I don't guarantee that they are free of typos or stupid mistakes. If you find something that appears to be egregiously wrong, please let me know. Chances are I already know, but haven't had the bandwidth to correct it yet. If it's something really bad, I will do my utmost to post a correction when I have time. Please don't hold your breath waiting!

0. Lecture guide: An outline and brief synopsis of all the lectures listed here.

1. Introduction: Geometric viewpoint on physics.

2. More on geometry: Vectors and tensors; derivatives of tensors; number flux and conservation laws.

3. More on tensors: Contractor of tensors; the dual nature of vectors and their associated 1-form.

4. Geometry and physics: Volumes and volume elements; how to go between differential and integral formulations of conservation laws. Electrodynamics in geometric language. Introduction to the stress-energy tensor.

5. More geometry and physics: Physical meaning of the stress-energy tensor. Prelude to curvature: special relativity and tensor analyses in curvilinear coordinates. Derivatives and the Christoffel symbol.

6. The principle of equivalence: The use of freely-falling frames as a generalization of inertial frames; different forms of the equivalence principle.

7. The principle of equivalence, continued: Using coordinate transformations to make the spacetime "as flat as possible" in the neighborhood of an event; what we learn from this exercise. Derivatives on curved manifolds, and notions of transport. Parallel transport.

8. More transport: Lie transport, the Lie derivative. Symmetries of spacetime and Killing's equation. Tensor densities; applications of the metric determinant.

9. Kinematics of point bodies: Geodesic trajectories; symmetries of spacetime and conserved quantities.

10. Curvature tensors: The Riemann curvature. (Lecture also covers the Newtonian limit.)

11. More curvature: The Ricci curvature; the equation of geodesic deviation; the Bianchi identity.
11a. Supplemental notes on geodesic deviation.

12. An equation for gravity: The Einstein curvature and the Einstein field equation.

13. An equation for gravity, path 2: The Einstein field equation via a variational principle. (Note, this lecture is particularly overdue for an overhaul.)

14. The linearized limit 1: How to sensibly define the linearized limit of the Einstein field equation. Gauge freedom; the Newtonian limit revisited.

15. The linearized limit 2: Characterization of the gauge-independent degrees of freedom in linearized theory. A wave equation for gravity.

16. Gravitational radiation 1: Characteristics of solutions to the wave equation for gravity. Action of these waves on matter; the quadrupole formula for the wave amplitude.

17. Gravitational radiation 2: Energy carried by gravitational waves; the quadrupole formula for wave power. Note that this lecture relies on somewhat more advanced material, and is presented in a more schematic manner than many other lectures.
17a. Some supplemental notes describing gauge transformations for perturbations on a curved background.

18. Cosmology 1: Solving the field equations by assuming an underlying symmetry. Maximally symmetric spaces and spacetimes; Robertson-Walker metrics; the Friedmann equations; matter and sources on cosmological scales.
18a. Supplemental notes written in response to student queries describing how to think about clocks in a Robertson-Walker spacetime.

19. Cosmology 2: How to connect models of the universe to things we can observe: redshift, distance measures. Sketch cosmic inflation and the problems it solves (leaving out the problems with this model -- take the cosmology course to learn about that!).

20. Compact sources: A spherical symmetric body that occupies a finite region of space. Birkhoff's theorem; interior and exterior metrics; the Schwarzschild solution. The Tolman-Oppenheimer-Volkov (TOV) equations of stellar structure. (Note that this PDF file is number 21 rather than 20; all handwritten files from now on are offset by 1. This is because Lecture 20 was an optional guest lecture the year I wrote these notes, covering a brief period of travel.)

21. Compact sources continued: Properties of solutions of the TOV equations; equations of state, the existence of a maximum mass for fluid bodies. Beginnings of black holes.
21a. A significantly improved presentation of how to use the first law of thermodynamics with polytropic equations of state.

22. Black holes: The nature of spacetimes that are Schwarzschild everywhere; event horizons, coordinate versus physical singularities. A brief discussion of other black hole solutions and the "no hair" theorem.
22a. Supplemental notes written in response to student queries describing in more detail the behavior of light propagating in black hole spacetimes.

23. Orbits of black holes: Equations governing the motion of a body orbiting a black hole. Conserved quantities; innermost stable orbits; highly non-Newtonian properties. Bending of light and the light ring.
23a. Slides which sketch how to apply the concepts discussed in this lecture to broader classes of black hole solutions.

NB: the next two lectures are somewhat more advanced material. They were not video recorded thanks to the scheduling disruptions which occurred due to the COVID-19 emergency in Spring 2020.

24. Beyond symmetry and linearized theory 1: How to solve the Einstein field equations for realistic situations for which there is no exact symmetry, and the spacetime is not nearly flat. Part 1: The post-Newtonian expansion and black hole perturbation theory.
24a. Slides which accompany the discussion of post-Newtonian theory.
24b. Slides which accompany the discussion of black hole perturbation theory.

25. Beyond symmetry and linearized theory 2: How to solve the Einstein field equations for realistic situations for which there is no exact symmetry, and the spacetime is not nearly flat. Part 2: Reformulating the Einstein field equations for numerical analysis; numerical relativity.
25a. Slides which accompany the discussion of numerical relativity.