The Theory of Relativity: Special and General

George Efstathiou

CONTENTS

Page

1  . . . . . . 1. Introduction to Special Relativity
9  . . . . . . 2. Tensors
19  . . . . . . 3. Relativistic kinematics
28  . . . . . . 4. Electromagnetism
35  . . . . . . 5. Constructing a theory of gravity
41  . . . . . . 6. Freely falling particles and affine connections
48  . . . . . . 7. Riemannian geometry
54  . . . . . . 8. Curvature
62  . . . . . . 9. Parallel transport
68  . . . . . . 10. Energy-momentum tensor
74  . . . . . . 11. The gravitational field equations
79  . . . . . . 12. Are the field equations unique?
74  . . . . . . 11. The gravitational field equations
84  . . . . . . 13. The Schwarzchild metric
89  . . . . . . 14. Motion in the Schwarzchild metric
96  . . . . . . 15. Precession of planetary orbits and radar echos
104 . . . . . . 16. Black holes
114 . . . . . . 17. Do black holes exist?
119 . . . . . . 18. Penrose diagrams and the Kerr metric

How to use these notes

There are lots of textbooks on special and general relativity. My personal favourite is Weinberg’s book Gravitation and Cosmology (John Wiley and Sons). This book is a bit old now and goes to a higher level than is required for this course. It is also expensive. For this course I recommend d’Inverno’s book Introducing Einstein’s Relativity (Oxford University Press) . It is cheap and well written. However, it approaches the subject from an applied mathematician’s point of view whereas I prefer to adopt a more intuitive ‘physics’ approach. I have therefore written these notes for three purposes. Firstly, I introduce the mathematics only as it is needed. The physics behind the theory drives the need for mathematics, not the other way round. Secondly, although I prefer a ‘physical’ approach to the subject, some of the advanced mathematical techniques developed by professional relativists (e.g. Penrose diagrams) are easy to understand and deserve exposure to an undergraduate audience. These advanced techniques can help you understand some of the more bizarre aspects of the theory that are often referred to in programmes like Star Trek. For example: What happens if you fall into a black hole? What is a wormhole? Can you travel to another Universe? Are these ideas pure science fiction, or do they have a basis in physical theory? Thirdly, General Relativity involves a lot of tedious algebra. Rather than plough through this algebra on the blackboard, which would bore you and take a long time, you can refer to these notes. Most of the algebra is worked through here. You will have seen cookery programmes where the cook prepares a dish, pops it in the oven, and then immediately pulls out another one saying ‘here is one that I prepared earlier’. That is how I want to approach the lectures. Instead of ploughing through complicated algebra, I will sometimes work partially through examples and then pull out results ‘ready made’ from these notes. There is no sleight of hand, the algebra is done in these notes. To get the most out of these notes, I would like you to work through all of the algebra. If you reproduce the calculations in these notes, you will gain experience in handling tensors and in solving problems in Special and General Relativity. In my experience, the only way of gaining an understanding of any branch of physics is by working through lots of examples.