### GENERAL RELATIVITY

The topic of this course is the classical relativistic theory of gravity, General Relativity. This is a geometric theory: the key idea is that gravity is a manifestation of the curvature of space-time.

The course will consist of a brief review/introduction of special relativity, followed by a discussion of the differential geometry which provides the mathematical underpinning to the description of gravity as the curvature of spacetime. We will then discuss the field equations of general relativity, and explore the physical properties of interesting simple solutions, describing black holes, cosmology, and gravitational waves.

If you have questions about the course, e-mail me, or come and see me in my office, CM214/OC112. (Suggestions for the web page are also welcome.)  ### Course outline:

• Introduction to general relativity - gravity as geometry, spacetime
• Manifolds - spacetime as a manifold, coordinates, tensors and differential forms
• Derivatives and Connections - Lie and exterior differentiation, connection, parallel transport, geodesics
• Curvature - Riemann tensor, flatness, commutation of derivatives, geodesic deviation
• General relativity - Equivalence principles, physics in curved spacetime, Einstein's equations, Einstein-Hilbert action
• Black holes - Spherical symmetry, Schwarzschild solution, geodesics, event horizon and Kruskal coordinates
• Gravity in action - Perturbation theory, extra dimensions, string theory, ads/CFT (depends on time available and wishes of class!)

### Books

There are many good books, a selection of which are listed below. There are also many texts available on-line (see below); you may find it useful to look at relevant chapters of the lecture notes by Carroll.

• J. Hartle, GRAVITY: An introduction to Einstein's General Relativity, A great introduction to relativity via the physics. Fairly accessible and a good companion to the course.
• R. D'Inverno, Introducing Einstein's Relativity, Oxford (1992): Another introductory book, with somewhat more meat. A good book for independant study. May be useful for those who want to know a little more about topics I cover lightly.
• R.M. Wald, General Relativity, Chicago (1984). Somewhat brief treatment of the introductory topics; good discussions of advanced topics of current interest.

Be aware that different authors use different conventions; in this course, the conventions are as follows. Space-time coordinate indices are (lower-case) Greek, vierbein indices lower case roman (mostly); the metric signature is + -- -- --.  ### Useful web links

A good place to start is Relativity on the WWW. This contains plenty of movies etc, and a long list of on-line texts.

All the relativity-related links one could ever want are at the Syracuse relativity bookmarks. Carroll's lecture notes were already mentioned above. There's also an excellent guide to black holes with loads of movies and useful diagrams.

For further reading once you've learned the basics, I recommend the journal Living reviews in relativity, which publishes web-based reviews of current fields of interest in relativity.

Black hole accretion disc

Problem Sheet 1: (pdf)       Solutions: (pdf)

Problem Sheet 2: (pdf)       Solutions: (pdf)

Problem Sheet 3: (pdf)       Solutions: (pdf)

Problem Sheet 4: (pdf)       Solutions: (pdf)

Ruth Gregory (r.a.w.gregory@durham.ac.uk)