PHY 620 Relativity

Fall 2007 (MWF 9:35-10:30, ESS 177)

Warren Siegel

office consultation available on request

Textbook

As lecture notes I'll use parts of my book Fields (3rd edition).

Outline

Minimum prerequisites

graduate Classical Mechanics (PHY 501)
one semester of graduate Classical Electrodynamics (PHY 505)

The material to be covered this semester will be (more or less) determined on the first day of class, depending on the background of the students. The following is a rough outline, with references to the corresponding sections of Fields:

Special Relativity and related topics

coordinates: Poincaré & conformal groups (IA4-6, IB3, IC2)
spin: Weyl spinor notation (IIA)
actions (IIIA1&4)
particles (IIIB)
Yang-Mills (IIIC1-4, VIB1)

However, if everyone has already seen most of this, it may be skipped, or briefly reviewed.

General Relativity

cosmology without gravity (IVA7)
actions: symmetries, covariant derivatives, field equations (IXA)
gauges: coordinate systems, geodesics (IXB)
curved spaces: cosmology, Schwarzschild, classic experiments, black holes (IXC)

Introductions to more-advanced topics

As time permits, and depending on the interest and background of the audience, some of:

supergravity (XB4)
string theory (unless people are already taking it) (XIA3)
gauge theories for higher spin (IIB1-4)

Highlights

We will emphasize some important topics useful both conceptually and for calculations, but which are missing or left till the end in most other relativity texts and courses:

Special and general relativity apply to fields (waves), not just particles (and not just electromagnetism). This is particularly important for quantum theory and string theory.
Yang-Mills is a simpler analog of gravity. It describes spin 1 instead of spin 2, but also has self-interactions, local symmetry, covariant derivatives, etc.
Vierbeins (tetrads, local frames) are necessary for coupling to spinors and for supergravity. You actually have already seen them for flat space in freshman physics, as the basis of unit vectors for curvilinear coordinates.
Action principles, both for mechanics and field theory, are the simplest way to find field equations and examine symmetries, and are needed for quantum theory.
The lightcone gauge (coordinates) describes just propagating degrees of freedom, and so is the best gauge for describing gravitational waves (radiation). In most treatments it is derived as a second step after choosing a de Donder (Lorentz) gauge, whereas choosing the lightcone gauge directly is even simpler than that first step (at least for weak fields).
Local Weyl scale transformations are important for supergravity. They also provide the simplest way to obtain the most important solutions to Einstein's equations -- the cosmological and Schwarzschild solutions.

Additional references

More material can be found in various textbooks, the best of which are:

C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, 1970), 1279 pp.:
introductory, long-winded
S.W. Hawking and G.F.R. Ellis, The large-scale structure of spacetime (Cambridge University, 1973), 400 pp.:
mathematical; emphasis on singularity theorems and global properties (e.g., Penrose diagrams)
R.M. Wald, General relativity (University of Chicago, 1984), 492 pp.:
intermediate between the above two
S. Weinberg, Gravitation and cosmology (Wiley & Sons, 1972), 657 pp.:
old-fashioned, but with more detail on astrophysics and cosmology

Grading

Grading will be based entirely on homework. Problems will be taken from those in Fields. You may discuss problems with classmates, but the write-up must be your own. Homework is due one week after assignment, at the beginning of class. (Put it on my desk when you enter.) No late homework is accepted; it may be handed in early, but only to me in person.

University-required statements

These statements are required in all University syllabi. (They are the same in all course syllabi, so just read it once.)