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 is an update to a set of handwritten notes that I have developed over the years and released to accompany the lectures that I recorded and released through OpenCourseWare in Spring 2020 (an "interesting" semester thanks to the interruption of the COVID-19 pandemic, which nearly derailed everything). A little more than half of these notes have now been typed up; some of the ones that were not fully typed up are accompanied by slides that include some technical details that go beyond what were in the original handwritten notes.

I was unable to complete the development of typewritten notes due to an overload of other obligations around week 8 of the Spring 2024 semester. I plan to complete the task of typing up my 8.962 notes in a future semester; for now, I leave things here. The previous handwritten notes which go with the Spring 2020 version of the course are still available at this link, should you be interested my original version for some reason.

Please note that these notes are provided *as is*; I don't
guarantee that they are free of typos or stupid mistakes. Indeed, by
comparing the new typewritten version with the older typed version,
you may very well find that various errors have been corrected.
However, it is not out of the question that new errors have been
introduced. 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!

1. Spacetime and geometry I: Introduction to the geometry of spacetime.

2. Spacetime and geometry II: Vectors and tensors; some important 4-vectors; derivatives of vectors; 1-forms.

3. Spacetime and geometry III: Basis objects; raising and lowering indices. Describing matter in geometric language; some conservation laws.

4. Spacetime and geometry IV: 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. Wrap up of stress-energy; prelude to curvature: Physical meaning of the stress-energy tensor. Special relativity and tensor analyses in curvilinear coordinates. Derivatives and the Christoffel symbol.

6. Wrap up of covariant derivative; introduction to curvature: Covariant derivative of other tensors; general formula for the Christoffel symbol given a metric. Curved manifolds: existence of locally flat regions of spacetime.

7. The principle of equivalence: The physical principle which local flatness allows us to introduce and use to put physics into a curved spacetime manifold. Derivatives on curved manifolds, and notions of transport. Parallel transport; Lie transport.

8. Spacetime symmetries and party tricks: Using Lie transport to characterize the symmetries of spacetime; Killing's equation. Fun and useful tricks based on the metric determinant.

9. Kinematics of point bodies: Geodesic trajectories via parallel transport and via an action principle. Things that are conserved along a geodesic trajectory; a hint of the Newtonian limit.

10. Spacetime curvature: The Riemann curvature tensor and its properties.

11. More curvature: The Ricci curvature; the Einstein tensor. The equation of geodesic deviation. The Bianchi identity.

12. An equation for gravity: Connecting spacetime curvature to a source.

13. An equation for gravity, path 2: The Einstein field equation via a variational principle.

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: A wave equation for gravity and its solution via the radiative Green's function. Characterization of the gauge-independent degrees of freedom in linearized theory.

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 gets into somewhat more advanced material; some of the key results are presented in a more schematic manner than many other lectures.

Note: Lectures from number 18 onward are handwritten, and are largely identical to the handwritten lectures presented previously (with a few small errors corrected). Many of them have accompanying slides that were projected during lecture, which include some technical details I did not want to write out on the chalkboard.

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 of cosmic inflation and the problems it solves
(leaving out the problems with this model -- take the cosmology
course to learn about that!).

19a. Slides that accompanied
Lecture 19, sketching data and recent results in cosmology.

20. Compact sources: A spherically 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.

20a. Slides that accompanied
Lecture 20, including some historical anecdotes and reminders of
someresults derived on a homework assignment which are an
important part of deriving the TOV equation.

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. Slides that accompanied
Lecture 21, including some results for how a body falls into the
Schwarzschild solution as a function of two different notions of
"time."

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. Slides that accompanied
Lecture 22, including some nice graphics showing the extended
manifold structure in Kruskal-Szekeres coordinates, as well as the
Penrose diagram of a Schwarzschild black hole.

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 that accompany Lecture 23,
including graphics illustrating many of the properties of black
hole orbits.

24. Wrap up of black hole orbits, including
considerations of Kerr black hole orbits. Beginnings of how to
solve the the Einstein field equations for realistic situations
for which there is no exact symmetry and spacetime is not nearly
flat. Introduction to the post-Newtonian expansion.

24a. Slides which accompany the
Lecture 24, including some of the details of Kerr black hole
orbits, and a key quantity in the discussion of post-Newtonian
theory.

25. Stepping beyond symmetry and linearized theory:
Iterating from the weak-field limit to develop the post-Newtonian
expansion; black hole perturbation theory.

25a. Slides which accompany the
discussion of the post-Newtonian expansion and black hole
perturbation theory.

26. No simplifying assumptions at all:
Reformulating the Einstein field equations for numerical
integration.

26a. Slides which accompany the
discussion of numerical relativity.

**NB:** Some of the material in Lecture 24, and all of the
material in Lectures 25 and 26, were not video recorded thanks to
the scheduling disruptions which occurred due to the COVID-19
emergency in Spring 2020.