General Relativity

This unit has not run since 2006-7.

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.

Unit registration

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.


This is the same list as in the course description, with slightly expanded advice. You can use any of the first three texts for this unit, perhaps with occasional reference to the greater detail of Misner for physics and the second Schutz book for mathematics. Blackwells availability

Problem sheets

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

Lecture notes

These are concise notes with no diagrams. They are not a substitute for lectures; I will be unsympathetic to any student who runs into difficulty as a result of not attending lectures and/or handing in problems. Minor corrections/additions are made from time to time; the first page gives the "last modified" date. Full lecture notes (39 pages, compact): ps pdf; (77 pages, room for annotations): ps pdf. You might also be interested to read the notes for my postgraduate lecture series on cosmology (13 pages) ps pdf and my relativity and cosmology research papers.

Past exams

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.

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