8.962 :: General Relativity

Here is the rough plan of topics we shall cover in 8.962 during Spring 2008. Please don't take this plan as gospel; the actual pace may vary relative to that planned, and we may juggle topics around a bit as the term proceeds.

As term progresses, suggested readings will be posted here (and in some cases linked here).

Psets will be assigned on Thursdays, and are due the following Thursday.

Geometric viewpoint on physics in flat spacetime: vectors and dual vectors (1-forms), tensors, transformation laws, metric tensor. Special relativity.

Pset 1 assigned.

Suggested reading:

Carroll, Chapter 1. This chapter gets into topics we will cover in week 2.

Schutz, Chapters 2 and 3. This also covers topics we will save for week 2. Schutz Chapter 1 is good if you would like to review special relativity.

Blandford and Thorne, Chapter 1. This chapter (from a textbook in preparation by Roger Blandford and Kip Thorne; many thanks to Kip for permission to link to it here) gives a very concise and physical explanation of topics we will cover in the first two weeks of the class.

Geometric viewpoint on physics in flat spacetime continued: tensors more generally, derivatives of tensors. Energy and momentum, conserved currents and conservation laws. Integration in spacetime. Stress energy tensor (general considerations, important examples).

Pset 1 due.
Pset 2 assigned.

Suggested reading:

Carroll, Chapter 2. Carroll gets into quite a bit more technical detail than I would like to cover. In particular, the details of manifold mathematics are beyond the scope of what we intend to cover. The remainder of the chapter is quite useful for us (though again a bit more formal than my preferred approach, at least for a 1 semester class).

Schutz, Chapters 3 and 4.

Prelude to curved spacetime: Flat spacetime in curvilinear coordinates. Coordinate bases, differentiation, connection coefficients.

Pset 2 due.
Note: No class Feb 19 (Monday schedule).

Suggested reading:

Schutz, Chapter 5.

Orthonormal bases (particularly contrasted with coordinate bases). Introduction to the equivalence principle; inadequacy of special relativity when gravity is included. Need for curvature. Curved spacetime manifolds. Existence of local Lorentz frames. Differentiation of tensor fields --- covariant (parallel transport) and Lie derivatives.

Pset 3 assigned.

Suggested reading:

Carroll, Chapter 3, especially Sections 3.1 - 3.5.

Schutz, Chapter 6, especially Sections 6.1 - 6.4.

Symmetries and Killing vectors. Tensor densities, integration. Kinematics of freely falling bodies --- geodesics.

Pset 3 due.
Pset 4 assigned.

Last week's readings are still relevant for this week.

Curvature, curvature tensors. Variations on the curvature tensor. Breakdown of parallelism, geodesic deviation.

Pset 4 due.
Pset 5 assigned.

Suggested reading:

Carroll 3.6 - 3.10. Section 3.9 can be omitted; it covers a topic that is interesting, but not strictly necessary for our development. Also, I find Section 3.10 to be somewhat unsatisfying; please read it, but be aware that I will develop geodesic deviation somewhat differently.

Schutz 6.5 and 6.6 are good supplemental readings; Schutz develops curvature tensors in a somewhat more straightforward (and to my mind physical) way than Carroll does. There is a bit of hand-waving in places, though, which I hope to reduce when I develop these quantities in lecture.

Bianchi identity. Einstein's equation and gravitation; cosmological constant. Einstein-Hilbert action; Einstein's equation from least-action principle.

Pset 5 due.
Pset 6 assigned.

Suggested reading:

Carroll Chapter 4. Section 4.6 is not necessary for 8.962, but is interesting stuff and definitely worth reading. Section 4.8 does not have to be examined too closely, but is also worth reading (at least cursorily). Note, Section 4.8 makes it clear why Carroll's stuff typically includes "torsion terms"; I've been strictly ignoring them since torsion does not fall under the scope of general relativity.

Weak field/linearized general relativity. Gauge transformations in linear theory; Newtonian limit. Spacetime of an isolated, weakly gravitating body. Gauge invariant characterization of gravitational degrees of freedom in linear theory.

Pset 6 due.
Pset 7 assigned.

Suggested reading:

Carroll 7.1 - 7.4. Post-Spring Break material will focus on applications of general relativity, with a particular emphasis on astrophysical problems. As such, we are going to jump around in Carroll a bit.

Flanagan & Hughes, Sec. 2.2. This covers the gauge invariant reformulation of spacetime's degrees of freedom in linearized theory.

Gravitational waves. Quadrupole formula; radiation in an arbitrary background spacetime. Energy and momentum carried by waves.

Pset 7 due.
Pset 8 assigned.

Carroll 7.5 - 7.7.
The basics of gravitational-wave theory, by Flanagan and Hughes (optional).

Cosmology: Friedmann-Robertson-Walker solution, evolution of the universe.

Pset 8 due.


Suggested reading:

Carroll, Chapter 8. This is my favorite chapter in this textbook ... Sean really earns his royalty checks here!

Cosmology: Distance measures, redshift. Our universe today. Inflation.

Pset 9 assigned.

Note: No class Apr 22 (Patriot's Day).

Suggested reading:

Continue in Carroll, Chapter 8.

Schwarzschild line element. Birkhoff's theorem, metric of a spherical "star".

Pset 9 due.
Pset 10 assigned.

Suggested reading:

Carroll, Chapter 5.

Black holes. Collapse to black hole; orbits of a black hole. Kerr and Reissner-Nordstrom solutions.

Pset 10 due.
Pset 11 assigned.

Suggested reading:

Carroll, Chapters 5 and 6. Chapter 6 is somewhat more advanced than we have time to cover in detail, but is highly recommended for students who are interested in black hole physics.

Advanced topics and current research in general relativity.

Pset 11 due.