A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity

A clear and enjoyable guide to the mathematics of relativity

To really understand relativity - one of the cornerstones of modern physics - you have to get to grips with the mathematics.  This user-friendly self-study guide is aimed at the general reader who is motivated to tackle that not insignificant challenge. The book is written using straightforward and accessible language, with clear derivations and explanations as well as numerous fully solved problems. For those with minimal mathematical background, the first chapter provides a crash course in foundation mathematics. The reader is then taken gently by the hand and guided through a wide range of fundamental topics, including Newtonian mechanics; the Lorentz transformations; tensor calculus; the Einstein field equations; the Schwarzschild solution (which gives a good approximation of the spacetime of our Solar System); simple black holes and relativistic cosmology. Following the historic 2015 LIGO (Laser Interferometer Gravitational-Wave Observatory) detection, there is now an additional chapter on gravitational waves, ripples in the fabric of spacetime that potentially provide a revolutionary new way to study the universe.



Special relativity helps explain a huge range of non-gravitational physical phenomena and has some strangely counter-intuitive consequences. These include time dilation, length contraction, the relativity of simultaneity, mass-energy equivalence and an absolute speed limit. General relativity, the leading theory of gravity, is at the heart of our understanding of cosmology and black holes.



Understand even the basics of Einstein's amazing theory and the world will never seem the same again.





Contents

Preface

Introduction

1 Foundation mathematics

2 Newtonian mechanics

3 Special relativity

4 Introducing the manifold

5 Scalars, vectors, one-forms and tensors

6 More on curvature

7 General relativity

8 The Newtonian limit

9 The Schwarzschild metric

10 Schwarzschild black holes

11 Cosmology

12 Gravitational waves

Appendix: The Riemann curvature tensor

Bibliography

Acknowledgements

January 2019. This third edition has been revised to make the material even more accessible to the enthusiastic general reader who seeks to understand the mathematics of relativity.