*Of the general theory of relativity you will be convinced, once you have
studied it. Therefore I am not going to defend it with a single word.
A. Einstein in a letter to A. Sommerfield, 1916.*

*I can accept the theory of relativity as little as I can accept the
existence of atoms and other such dogmas. E. Mach, 1913.*

General relativity is a physical theory, in which gravitational effects are incorporated into the four dimensional space-time of special relativity by making it curved. The motion of particles in a gravitational field is simply described by saying that they take paths of extremal length (geodesics) in space-time. General relativity is needed to describe small effects in weak gravitational fields such as the gravitational time dilation (essential for precise timing as in the GPS navigation system: see Physics Today, May 2002) and the bending of light (leading to gravitational lensing effects) but the most spectacular predictions such as black holes are a consequence of the theory applied to strong gravitational fields. It is also needed to understand the large scale structure of the Universe, covered in the related level 4 unit in physics, Relativistic cosmology. The mathematics of curvature (Riemannian manifolds) also has a number of other applications, for example to spherical geometry and constrained systems.

This unit is structured in three parts: 1. Preliminaries (motivation, dimensions, Newtonian gravity, variational mechanics, special relativity, tensors), much of which will be somewhat familiar to you; 2. Mathematics of curvature (differentiable manifolds, metric, connection, geodesics, curvature); 3. General relativity (physics in curved spacetime, Einstein field equations, spherically symmetric solutions). Each of these three broad areas will take about a third of the lectures, but the exam will be more concentrated on the latter two.

If you are unsure about taking this unit, please contact me; I will advise you based on your marks, previous and current unit choices and general level of interest. If you are in your third year, you MUST contact me (as unit organiser and also DUS) as stated in the year 3/4 handbook; quite a few third year students have successfully taken this unit in the past.

Students on single or joint honours maths degrees: you already know how to register (if you haven't already). Students who are not on mathematics degrees (typically physics or incoming exchange students) note that all students MUST register with the maths department (Rebecca Staatz in room 1.10) by noon on Feb 9 or you may not be permitted to sit the exam. You will need to make sure your tutor agrees and your home department or university is aware of your unit choices.

- ``A first course in general relativity'', Bernard F Schutz, Cambridge University Press, 1985, 400 pages. First recommendation. Good beginner's book with very little assumed background. Covers the course but not much more.
- ``General Relativity'', Robert M Wald, U. Chicago press, 1984, 500 pages. Contains more recent developments. More mathematical approach. Recommended previously, but not well suited to beginners. Buy this book if you are a reasonably strong student and would like to understand this subject in greater depth or as a basis for postgraduate research. Only chapters 1-4,6 are directly relevant to this unit.
- ``A short course on general relativity'' J Foster and J D Nightingale, Springer-Verlag, 1995 250 pages. Good beginner's book, concise and logical, but maybe too short, ie not as much detailed explanation as Schutz, above.
- ``Relativity'' H Stephani, Cambridge, third ed. 2004, 400 pages. A little too advanced to use on its own, but good to extend a more elementary text. A more modern approach and more physical than Wald.
- ``A concise overview of the classical theory'' P A M Dirac, 1996 Princeton University Press, 80 pages. Concise and logical, but no modern developments, figures or problems.
- ``Gravitation'' C W Misner, K S Thorne, J A Wheeler, Freeman, 1973, 1300 pages. Exhaustive classic, not for beginners, but good for detailed pictorial arguments on many topics. Physical approach.
- ``Geometrical methods of mathematical physics'' Bernard F Schutz, Cambridge University Press, 1980, 250 pages. Not a text on general relativity per se, but delves deeper into the mathematical structure, and so useful to extend one of the above texts.

As stated in the unit description, you will get credit if you pass the exam (a mark of 50), or if you get a mark of 30 in the exam and hand in satisfactory attempts at a third of the total number of problems. Parts (a), (b) etc. of problems count as whole problems for this purpose. Please hand in work by the due date below, either in a lecture or to the envelope outside my office (room 3.15). Solutions will be accessible after the due date.

Number | Problems | Due date | Solutions |

1 | ps pdf | 9/2 | ps pdf |

2 | ps pdf | 16/2 | ps pdf |

3 | ps pdf | 23/2 | ps pdf |

4 | ps pdf | 2/3 | ps pdf |

5 | ps pdf | 13/3 | ps pdf |

6 | ps pdf | 19/3 | ps pdf |

7 | ps pdf | 27/4 | ps pdf |

8 | ps pdf | 4/5 | ps pdf |

9 | ps pdf | 11/5 | ps pdf |

Relativity on the web contains a wealth of information at all levels from the general public to current research. Students taking this unit should note that different sign conventions are in use for the metric, the Riemann curvature tensor and the Einstein tensor, and for placement of indices in the Riemann tensor: we use a timelike metric, otherwise conventions as in Schutz, above.

Living reviews in relativity. Online review articles that are updated by their authors. Very useful but may contain/require understanding of material beyond the scope of this unit.

GRTensor offers free software for doing relativistic tensor calculations (including many beyond the scope of the unit) within existing Maple or Mathematica systems.

- 16/10/02 Stronger evidence for a 3.7 million solar mass black hole at the centre of the galaxy Movie!
- 8/1/03 Measurement of the speed of gravity Discredited here!
- 18/2/03 LIGO gravity wave detector commissioned LIGO web site here.
- 3/3/03 Anisotropic Universe? Latest evidence from the Cosmic Microwave Background WMAP web site here.
- 10/9/03 Lowest frequency sound waves from black hole
- 25/9/03 Cassini mission tests photon effects to 0.00004
- 9/12/03 Black holes from galaxy merger
- 5/4/04 Gyroscope satellite to be launchedGravity Probe B website here, plans to report in April 2007.
- 16/4/04 Planet discovered through gravitational lensing
- 21/10/04 Frame dragging detected in Earth satellites (pre-empting Gravity Probe B).
- 12/1/05 Connections between Cosmic Microwave Background and large scale structure
- March 2005 First observation of system with two pulsars Tests on GR (Sept 2006). Jodrell Bank Observatory
- 15/9/05 Stephen Hawking recants on black hole information
- 19/4/06 Black hole merger modelled numerically (see LIGO website above).
- 26/6/06 Gravity wave detector GEO 600 online. website
- 9/1/07 Large black hole in a very small galaxy

Dr Carl Dettmann / carl.dettmann@bristol.ac.uk